PROJECTION. If from a fixed point S in space lines or rays be drawn to different points A,B,C, . . . in space, and if these rays are cut by a plane in points A',B',C', . . . the latter are called the projections of the given points on the plane. Instead of the plane another surface may be taken, and then the points are projected to that surface instead of to a plane. In this manner any figure, plane or in space of three dimensions, may be projected to any surface from any point which is called the centre of projection. If the figure projected is in three dimensions then this projection is the same as that used in what is generally known as perspective.
In modern mathematics the word projection is often taken with a slightly different meaning, supposing that plane figures are projected into plane figures, but three-dimensional ones into three-dimensional figures. Projection in this sense, when treated by coordinate geometry, leads in its algebraical aspect to the theory of linear substitution and hence to the theory of invariants and co-variants.
In this article projection will be treated from a purely geometrical point of view.
We shall first and principally treat of the projection of plane figures into plane figures, and consider a number of special cases due to special positions of the two planes or of the centre of projection. We shall next consider the representation of figures of three dimensions by plane figures (orthographic projections, drawing in plan and elevation, &c.), then treat of perspective in its ordinary sense, and speak shortly of projections to curved surfaces.
References like (G. § 87) relate to section II. of the article Geometry, vol. x. pp. 388 sq.
§ 1. Projection of Plane Figures.— Let us suppose we have in space two planes π and π'. In the plane π a figure is given having known properties; then we have the problem to find its projection from some centre S to the plane ιr', and to deduce from the known properties of the given figure the properties of the new one.
If a point A is given in the plane π we have to join it to the centre S and find the point A' where this ray SA cuts the plane √; it is the projection of A. On the other hand if A' is given in the plane √, then A will be its projection in τr. Hence if one figure in √ is the projection of another in ιr, then conversely the latter is also the projection of the former.
A point and its projection are therefore also called corresponding points, and similarly we speak of corresponding lines and curves, &c.
§ 2. We at once get the following properties :—
The projection of a point is a point, mid one point only. The projection of a line {straight line) is a line ; for all points in a line are projecte I by rays which lie in the plane determined by S and the line, and this plane cuts the plane π in a line which is the projection of the given line.
If a point lies in a line its projection lies in the projection of the line. The projection of the line joining two points A, B is the line which joins the projections A', B' of the points A, B. For the projecting plane of the line AB contains the rays SA, SB which project the points A, B.
The prelection of the point of intersection of two lines a, b is the point of intersection of the projections a', b' of those lines.
Similarly we get—
The projection of a curve will be a curve. The projections of the points of intersection of two curves are the points of intersection of the projections of the given curves.
If a line cuts a curve in n points, then the projection of the line cuts the projection of the curve in n points. Or The order of a curve remains unaltered by projection. The projection of a tangent to a curve is a tangent to the projection of the curve. For the tangent is a line which has two coincident points in common with a curve.
The number of tangents that can be drawn from a point to a curve remains unaltered by projection. Or The class of a curve remains unaltered by projection. Example. — The projection of a circle is a curve of the second order and second class.
§ 3. Two figures of which one is a projection of the other obtained in the manner described may be moved out of the position in which ^they are obtained. They are then still said to be one the projection of the other, or to be projective or homographie. But when they are in the position originally considered they are said to be in perspective position, or (shorter) to be perspective.
All the properties stated in §§ 1, 2 hold for figures which are projective, whether they are perspective or not. There are others which hold only for projective figures when they are in perspective position, which we shall now consider.
If two planes π and √ are perspective, then their line of intersection is called the axis of projection. Any point in this line coincides with its projection. Hence All points in the axis are their own projections. Hence also Every line meets its projection on the axis.
The property that the lines joining corresponding points all pass through a common point, that any pair of corresponding points and the centre are in a line, is also expressed by saying that the figures are co-linear; and the fact that both figures have a line, the axis, in common on which corresponding lines meet is expressed by saying that the figures are co-axal.
The connexion between these properties has to be investigated.
For this purpose we consider in the plane π a triangle ABC, and let the lines BC, CA, AB be denoted by a, b, c. The projection will consist of three points A', B', C' and three lines a', b', c'. These have such a position that the lines A A', BB', CC' meet in a point, viz., at S, and the points of intersection of a and a', b and b', c and c' lie on the axis (by § 2). The two triangles therefore are said to be both co-linear and co-axal. Of these properties either is a consequence of the other, as will now be proved.
§ 4. Desargue’s theorem.— If two triangles, whether in the same plane or not, are co-linear they are co-axal. Or If the lines A A', BB', CC' joining the vertices of two triangles meet in a point,, then the intersections of the sides BC and B'C', CA and C'A', AB and A'B' are three points in a line. Conversely, If two triangles are co-axal they are co linear. Or If the intersection of the sides of two triangles ABC and A'B'C', viz., ofRC and B'C', o∕CA and C'A', and of AB and A'B', lie in a line, then the lines AA', BB', and CC' meet in a point. Proof.—het us first suppose the triangles to be in different planes. By supposition the lines AA', BB', CC' (fig. 1) meet in a point S. But three intersecting lines determine three planes, SBC, SCA, and SAB. In the first lie the points B, C and also B', C'. Hence the lines BC and B'C' will intersect at some point P, because any two lines in the same plane intersect. Similarly CA and C'A' will intersect at some point Q, and AB and A'B' at some point R. These points P, Q, R lie in the •pluie of the triangle ABC because they are points on the sides of this triangle, and similarly in the plane of the triangle A'B'C ’. Hence they lie in the intersection of two planes,—that is, in a line.
Secondly, If the triangles ABC and A'B'C' lie both in the same plane the above proof does not hold. In this case we may consider the plane figure as the projection of the figure in space of which we have just proved the theorem. Let ABC, A'B'C' be the co-linear tiiaigles with S as centre, so that AA', BB', CC' meet at S. Take now any point in space, say your eye E, and from it draw the rays projecting the figure. In the line ES take any point S 1 , and in EA, EB, EC take points A 1 , B 1 , C 1 respectively, but so that S 1 , A 1 , B 1 , C 1 are not in a plane. In the plane ESA which projects the line S j A j lie then the line S 1 A α and also EA'; these will therefore meet in a point A' 1 , of which A' will be the projection. Similarly points B' 1 , C' 1 are found. Hence we have now in space two triangles AjB 1 C 1 and A' 1 B' 1 C'ι which are co-linear. They are therefore co-axal, that is, the points P 1 , Q 1 , R 1 , where A 1 B 1 , &c., meet will lie in a line. Their projections therefore lie in a line. But these are the points P, Q, R, which were to be proved to lie in a line.
This proves the first part of the theorem. The second part or converse theorem is proved in exactly the same way. For another proof see (G. § 37).
§ 5. By aid of Desargue’s theorem we can now prove a fundamental property of two projective planes.
Let s be the axis, S the centre, and let A, A' and B, B' be two pairs of corresponding points which we suppose fixed, and C, C' any other pair of corresponding points. Then the triangles ABC and A'B'C' are co-axal, and they will remain co-axal if the one plane √ be turned relative to the other about the axis. They will therefore, by Desargue’s theorem, remain co-linear, and the centre will be the point S', where AA' meets BB'. Hence the line joining any pair of corresponding points C, C' will pass through the centre S'. The figures are therefore perspective. This will remain true if the planes are turned till they coincide, because Desargue’s theorem remains true.
Theorem.— If two planes are perspective, then if the one plane be turned about the axis through any angle, especially if the one plane be turned till it coincides with the other, the two planes will remain perspective; corresponding lines toill still meet on a line called the axis, and the lines joining corresponding points will still pass through a common centre S situated in the. plane. Whilst the one plane is turned this point S will move in a circle whose centre lies in the plane π, which is kept fixed, and whose plane is perpendicular to the axis.
The last part will be proved presently. As the plane ιr' may be turned about the axis in one or the opposite sense, there will be two perspective positions possible when the planes coincide.
§ 6. Let (fig. 2) ιr, π' be the planes intersecting in the axis s whilst S is the centre of projection. To project a point A in π we join A to S and see where this line cuts π'. This gives the point A'. But if we draw through S any line parallel to ττ, then this line will cut ιr' in some point Γ, and if all lines through S be draλvn which are parallel to π these will form a plane parallel to π which will cut the plane τr' in a line i' parallel to the axis s. If we say that a line parallel to a plane cuts the latter at an infinite distance, we may say that all points at an infinite distance in n are projected into points which lie in a straight line i', and conversely all points in the line are projected to an infinite distance in π, whilst all other points are projected to finite points. We say therefore that all points in the plane τ at an infinite distance may be considered as lying in a straight line, because their projections lie in a line. Thus we are again led to consider points at infinity in a plane as lying in a line (comp. G. §§ 2 .~ 4 >∙
Similarly there is a line/ in π which is projected to infinity in π'; this projection shall be denoted by j' so that i anàj' are lines at infinity.
§ 7. If we suppose through S a plane drawn perpendicular to the axis s cutting it at T, and in this plane the two lines SΓ parallel to π and SJ parallel to √, then the lines through I' and J parallel to the axis will be the lines i' and/. At the same time a parallelogram SJTΓS has been formed. If now the plane √ be turned about the axis, then the points Γ and J will not move in their planes; hence the lengths TJ and TI', and therefore also SI' and SJ, will not change. If the plane w is kept fixed in space the point J will remain fixed, and S describes a circle about J as centre and with SJ as radius. This proves the last part of the theorem in § 5. ι I
§ 8. The plane √ may be turned either in the sense indicated by the arrow at Z or in the opposite sense till π' falls into π. In the first case we get a figure like fig. 3; i' and j will be on the same side of the axis, and on this side will also lie the centre S;
and then ST=SJ + SI' or SΓ = JT, SJ = I'T. In the second case (fig. 4) i' and J will be on opposite sides of the axis, and the centre
S will lie between them in such a position that I'S = TJ and I'T = SJ. If ΓS = SJ, the point S will lie on the axis.
It follows that any one of the four points S, T, J, Γ is completely determined by the other three: if the axis, the centre, and one of the lines i' or j are given the other is determined; the three lines s, i', j determine the centre; the centre and the lines i', j determine the axis.
§ 9. We shall now suppose that the two projective planes π, √ are perspective and have been made to coincide.
Theorem.— If the centre, the axis, and either one pair of corresponding points on a line through the centre or one pair of corresponding lines meeting on the axis are given, then the whole projection is determined. Proof— If A and A' (fig. 1) are given corresponding points, it has to be shown that we can find to every other point B the corresponding point B'. Join AB to cut the axis in R. Join RA'; then B' must lie on this line. But it must also lie on the line SB. Where both meet is B'. That the figures thus obtained are really projective can be seen by aid of Desargue’s theorem. For, if for any point C the corresponding point C' be found, then the triangles ABC and A'B'C' are, by construction, co-linear, hence co-axal; and s will be the axis, because AB and AC meet their corresponding lines A'B' and A'C' on it. BO and B'C' therefore also meet on s.
If on the other hand a, a' are given corresponding lines, then any line through S will cut them in corresponding points A, A' which may be used as above.
§ 10. Rows and pencils which are projective or perspective have been considered in the article Geometry (G. §§ 12-40). All that has been said there holds, of course, here for any pair of corresponding rows or pencils. The centre of perspective for any pair of corresponding rows is at the centre of projection S, whilst the axis contains coincident corresponding elements. Corresponding pencils on the other hand have their axis of perspective on the axis of projection whilst the coincident rays pass through the centre.
We mention here a few of those properties which are independent of the perspective position :—
The correspondence between two projective rows, or pencils, is completely determined if to three elements in one the corresponding ones in the other are given. If for instance in two projective rows three pairs of corresponding points are given, then we can find to every other point in either the corresponding point (G. §§ 29-36).
If A, B, C, D are four points in a row and A', B ’, C', D' the corresponding points, then their cross-ratios are equal (ABCD) = (A'B'C'D'), —where (ABCD) = AC/CB: AD∕DB.
If in particular the point D lies at infinity we have (ABCD) = - AC/CB = AC∕BC. If therefore the points D and D' are both at infinity we have AC∕BC = AD∕BD, and the rows are similar (G. § 39). This can only happen in special cases. For the line joining corresponding points passes through the centre; the latter must therefore lie at infinity if D, D' are different points at infinity. But if p and D' coincide they must lie on the axis, that is, at the point at infinity of the axis unless the axis is altogether at infinity.
Hence— In two perspective planes every row which is parallel to the axis is similar to its corresponding roιo, and in general no other row has this property. But if the centre or the axis is at infinity then every row is similar to its corresponding row.
In either of these two cases the metrical properties are particularly simple. If the axis is at infinity the ratio of similitude is the same for all rows and the figures are similar. If the centre is at infinity we get parallel projection; and the ratio of similitude changes from row to row (see §§ 16, 17).
In both cases the mid-points of corresponding segments will be corresponding points.
_ § 11. Involution. —If the planes of two projective figures coincide, then every point in their common plane has to be counted twice, once as a point A in the figure τr, once as a point B' in the figure π'. The points A' and B corresponding to them will in general be different points; but it may happen that they coincide. Here a theorem holds similar to that about rows (G. §§ 76 sq. ). Theorem.— If two projective planes coincide, and if it happens that to one point in their common plane the same point corresponds, whether we consider the point as belonging to the first or to the second plane, then the same will happen for every other point—that is to say, to every point will correspond the same point in the first as in the second plane.
In this case the figures are said to be in involution. Proof.— Let (fig. 5) S be the centre, s the axis of projection, and let a point which has the name A in the first plane and B' in the second have the property that the points A' and B corresponding to them again coincide. Let C and D' be the names which some other point has in the two planes. If the line AC cuts the axis in X, then the point where the line XA' cuts SC will be the point C' corresponding to C (§ 9). The line B'D' also cuts the axis in X, and therefore the point D corresponding to D' is the point where XB cuts SD'. But this is the same point as C'. Q. E. I).
This point C' might also be got by drawing CB and joining its intersection Y with the axis to B'. Then C' must be the point where B'Y meets SC. This figure, which now forms a complete quadrilateral, shows that in order to get involution the corresponding points A and A' have to be harmonic conjugates with regard to S and the point T where AA' cuts the axis.
Theorem.— If two perspective figures arc in involution, two corresponding points are harmonic conjugates with regard to the centre and the paint where the line joining them cuts the axis. Similarly—
Any two corresponding lines are harmonic conjugates with regard to the axis and the line from their point of intersection to the centre.
Conversely— If in two perspective planes one pair of corresponding points are harmonic conjugates with regard to the centre and the point where the line joining them cuts the axis, then every pair of corresponding points has this property and the planes are in involution.
§ 12. Projective Planes which are not in perspective position. —We return to the case that two planes π and √ are projective but not in perspective position, and state in some of the more important cases the conditions which determine the correspondence between them. Here it is of great advantage to start with another definition λvhich, though at first it may seem to be of far greater generality, is in reality equivalent to the one given before.
Definition.— We call two planes projective if to every pointin one corresponds a paint in the other, to every line a line, and to a point in a line a point in the corresponding line, in such a manner that the cross-ratio of four points in a line, or of four rays in a pencil, is equal to the cross-ratio of the corresponding points or rays.
The last part about the equality of cross-ratios can be proved to be a consequence of the first. But as space does not allow us to give an exact proof for this we include it in the definition.
If one plane is actually projected to another we get a correspondence which has the properties required in the new definition. This shows that a correspondence between two planes conform to this definition is possible. That it is also definite we have to show. It follows at once that—
Corresponding rows, and likewise corresponding pencils, are projective in the old sense (G. §§ 25, 30). Further, If two planes are projective to a third they are projective to each other. Theorem.— The correspondence between two projective planes π and π' is determined if we have given either two rows u, v in π and the corresponding rows u', v' in π', the point where u and v meet corresponding to the points where u' and v' meet, or two pencils U, V in τr and the corresponding pencils U', V' in π', the ray UV joining the centres of the pencils in ιr corresponding to the ray U'V'.
It is sufficient to prove the first part. Let any line a cut u, v in the points A and B. To these will correspond points A' and B' in u' and v' which are known. To the line a corresponds then the line A'B'. Thus to every line in the one plane the corresponding line in the other can be found, hence also to every point the corresponding point.
§ 13. Theorem.— If the planes of two projective figures coincide, and if either four points, of which no three lie in a line, or else four lines, of which no three pass through a point, in the one coincide with their corresponding points, or lines, in the other, then every point and every line coincides with its corresponding point or line so that the figures are identical.
If the four points A, B, C, D coincide with their corresponding points, then every line joining two of these points will coincide with its corresponding line. Thus the lines AB and CD, and therefore also their point of intersection E, will coincide with their corresponding elements. The row AB has thus three points A, B, E coincident with their corresponding points, and is therefore identical with it (§ 10). As there are six lines which join two and two of the four points A, B, C, D, there are six lines such that each point in either coincides with its corresponding point. Every other line will thus have the six points in which it cuts these, and therefore all points, coincident with their corresponding points. The proof of the second part is exactly the same. It follows—
§ 14. If two projective figures which are not identical lie in the same plane, then not more than three points which are not in a line, or three lines which do not pass through a point, can be coincident with their corresponding points or lines.
If the figures are in perspective position, then they have in common one line, the axis, with all points in it, and one point, the centre, with all lines through it. No other point or line can therefore coincide with its corresponding point or line without the figures becoming identical.
It follows also that—
The correspondence between two projective planes is completely determined if there are given —cither to four points in the one the corresponding four points in the other provided that no three of them lie in a line, or to any four lines the corresponding lines provided that no three of them pass through a point.
To show this we observe first that two planes π, π' may be made projective in such a manner that four given points A, B, C, D in the one correspond to four given points A,' B', C', D' in the other; for to the lines AB, CD will correspond the lines A'B' and C'D', and to the intersection E of the former the point E' where the latter meet. The correspondence between these rows is therefore determined, as we know three pairs of corresp mding points. But this determines a correspondence (by § 12). To prove that in this case and also in the case of § 12 there is but one correspondence possible, let us suppose there were two, or that we could have in the plane √ two figures which are each projective to the figure in π and which have each the points A'B'C'D' corresponding to the points ABCD in τr. Then these two figures will themselves be projective and have four corresponding points coincident. They are therefore identical by § 13.
Theorem. — Two projective planes will be in perspective position if one row coincides with its corresponding row. The line containing these rows will be the axis of projection. Proof.— As in this case every point on s coincides with its corresponding point, it follows that every row a meets its corresponding row a' on s where corresponding points are united. The two rows a, a' are therefore perspective (G. § 30), and the lines joining corresponding points will meet in a point S. If r be any one of these lines cutting a, a' in the points A and A' and the line 4 at K, then to the line AK corresponds A'K, or the ray r corresponds to itself. The points B, B' in which r cuts another pair b, b' of corresponding rows must therefore be corresponding points. Hence the lines joining corresponding points in b and b' also pass through S. Similarly all lines joining corresponding points in the two planes π and √ meet in S; hence the planes are perspective.
The following proportion is proved in a similar way :—
Theorem.— Two projective planes will be in perspective position if one pencil coincides with its corresponding one. The centre of these pencils will be the centre of perspective.
In this case the two planes must of course coincide, whilst in the first case this is not necessary.
§ 15. We shall now show that two planes which are projective according to definition § 12 can be brought into perspective position, hence that the new definition is really equivalent to the old. We use the following property:—If two coincident planes τr and π' are perspective with S as centre, then any two corresponding rows are also perspective with S as centre. This therefore is true for the rows j and j' and for i and i', of which i and j' are the lines at infinity in the two planes. If now the plane ιr' be made to slide on π so that each line moves parallel to itself, then the point at infinity in each line, and hence the whole line at infinity in √, remains fixed. So does the point at infinity on j, which thus remains coincident with its corresponding point on j', and therefore the rows j and j' remain perspective, that is to say the rays joining corresponding points in them meet at some point T. Similarly the lines joining corresponding points in i and i' will meet in some point T'. These two points T and T' originally coincided with each other and with S.
Conversely, if two projective planes are placed one on the other, then as soon as the lines/ and rare parallel the two points T and Τ' can be found by joining corresponding points in j and/, and also in i and i'. If now a point at infinity is called A as a point in n and B' ¾s a point in √, then the point A' will lie on i' and B on j, so that the line AA' passes through T' and BB' through T. These two lines are parallel. If then the plane √ be moved parallel to itself till Τ' comes to T, then these two lines will coincide with each other, and with them will coincide the lines AB and A'B'. This line and similarly every line through T will thus now coincide with its corresponding line. The two planes are therefore according to the last theorem in § 14 in perspective position.
It will be noticed that the plane π' may be placed on π in two different ways, viz., - if we have placed π' on π we may take it off and turn it over in space before we bring it back to π, so that what was its upper becomes now its lower face. For each of these positions we get one pair of centres T, T', and only one pair, because the above process must give every perspective position. It follows—
In two projective plains there are in general two and only two pencils in either such that angles in one are equal to their corresponding angles in the other. If one of these pencils is made coincident with its corresponding one, then the planes will be perspective.
This agrees with the fact that two perspective planes in space can be made coincident by turning one about their axis in two different ways (§ 8).
In the reasoning employed it is essential that the lines / and i' are finite. If one lies at infinity, say j, then i and ∕ coincide, hence their corresponding lines i' and j' will coincide; that is, i' also lies at infinity, so that the lines at infinity in the two planes are corresponding lines. If the planes are now made coincident and perspective, then it may happen that the lines at infinity correspond point for point, or can be made to do so by turning the one plane in itself. In this case the line at infinity is the axis, whilst the centre may be a finite point. This gives similar figures (see § 16). In the other case the line at infinity corresponds to itself without being the axis; the lines joining corresponding points therefore all coincide with it, and the centre S lies on it at infinity. The axis will be some finite line. This gives parallel projection (see § 17). For want of space we do not show how to find in these cases the perspective position, but only remark that in the first case any pair of corresponding points in π and √ may be taken as the points T and T', whilst in the other case there is a pencil of parallels in π such that any one line of these can be made to coincide point for point with its corresponding line in √, and thus serve as the axis oι projection. It will therefore be possible to get the planes in perspective position by first placing any point A' on its corresponding point A and then turning π' about this point till lines joining corresponding points are parallel.
§ 16. Similar Figures. —If the axis is at infinity every line is parallel to its corresponding line. Corresponding angles are therefore equal. The figures are similar, and (§10) the ratio of similitude of any two corresponding rows is constant.
If similar figures are in perspective position they are said to be similarly situated, and the centre of projection is called the centre of similitude. To place two similar figures in this position, we observe that their lines at infinity will coincide as soon as both figures are put in the same plane, but the rows on them are not necessarily identical. They are projective, and hence in general not more than two points on one will coincide with their corresponding points in the other (G. § 34). To make them identical it is either sufficient to turn one figure in its plane till three lines in one are parallel to their corresponding lines in the other, or it is necessary before this can be done to turn the one plane over in space. It can be shown that in the former case all lines are, or no line is, parallel to its corresponding line, whilst in the second case there are two directions, at right angles to each other, which have the property that each line in either direction is parallel to its corresponding line. We also see that—
If in two similar figures three lines, of which no two are parallel, are parallel respectively to their corresponding lines, then every line has th is q>roperty and the two figures are similarly situated ; or Two similar figures are similarly situated as soon as two corresponding triangles are so situated.
If two similar figures are perspective without being in the same plane, their planes must be parallel as the axis is at infinity. Hence Any plane figure is projected from any centre to a parallel plane into a similar figure.
If two similar figures are similarly situated, then corresponding points may either be on the same or on different sides of the centre. If, besides, the ratio of similitude is unity, then corresponding points will be equidistant from the centre. In the first case therefore the two figures will be identical. In the second case they will be identically equal but not coincident. They can be made to coincide by turning one in its plane through two right angles about the centre of similitude S. The figures are in involution, as is seen at once, and they are said to be symmetrical with regard to the point S as centre. If the two figures be considered as part of one, then this is said to have a centre. Thus regular polygons of an even number of sides and parallelograms have each a centre, which is a centre of symmetry.
§ 17. Parallel Projection.— If, instead of the axis, the centre be moved to infinity,' all the projecting rays will be parallel, and we get what is called Parallel Projection. In this case the line at infinity passes through the centre and therefore corresponds to itself, — but not point for point as in the case of similar figures. To any point I at infinity corresponds therefore a point Γ also at infinity but different from the first. Hence to parallel lines meeting at I cor-
respond parallel lines of another direction meeting at Γ. Further, in any two corresponding rows the two points at infinity are corresponding points; hence the rows are similar. This gives the principal properties of parallel projection :—
To parallel lines correspond parallel lines ; or To a parallelogram corresponds a parallelogram. The correspondence of parallel projection is completely determined as soon as for any parallelogram in the one figure the corresponding parallelogram in the other has been selected, as follows from the general case in § 14.
Corresponding rows are similar (§ 10).
The ratio of similitude for these rows changes with the direction: If a row is parallel to the axis, its corresponding row, which is also parallel to the axis, will be equal to it, because any two pairs AA' and BB' of corresponding points will form a parallelogram.
Another important property is the following: —
The areas of corresponding figures have a constant ratio.
We prove this first for parallelograms. Let ABCD and EFGH be any two parallelograms in it, A'B'C'D' and E'F'G'H' the corresponding parallelograms in τr'. Then to the parallelogram KLMN which lies (fig. 6) between the lines AB, CD and EF, GH will correspond a parallelogram K'L'M'N' formed in exactly the same manner. As ABCD and KLMN are between the same parallels their areas are as the bases. Hence
ABCD AB 1 . . 1 1 A'B'C'D A'B ’ KLMN = KL’ and sιm,larl y K'L'M'N' K'L' '
But AB∕KL = A'B'∕K'L', as the rows AB and A'B'are similar. Hence
ABCD KLMN , . . 1 1 EFGH KLMN
A'B'C'D ’ K'L'M'N' , and sπnιlarl y E'F'G'H' K'L'M'N'*
Hence also
ABCD _ EFGHA'B'C'D' E'F'G'H'*
This proves the theorem for parallelograms and also for their ιalv^s, that is, for any triangles. As polygons can be divided into τiangles the truth of the theorem follows at once for them, and is hen by the well-known method of exhaustion extended to areas >ounded by curves by inscribing polygons in, and circumscribing )olygons about, the curves.
Just as (G. § 8) a segment of a line is given a sense, so a sense nay be given to an area. This is done as follows. If we go round he boundary of an area, the latter is either to the right or to the left. f we turn round and go in the opposite sense, then the area will be;o the left if it was first to the right, and vice versa. If we give he boundary a definite sense, and go round in this sense, then the urea is said to be either of the one or of the other sense according as he area is to the right or to the left. The area is generally said to >e positive if it is to the left. The sense of the boundary is indi-ated either by an arrowhead or by the order of the letters which lenote points in the boundary. Thus, if A, B, C be the vertices of L triangle, then ABC shall denote the area in magnitude and sense, he sense being fixed by going round the triangle in the order rom A to B to C. It will then be seen that ABC and ACB denote he same area but with opposite sense, and generally ABC = BCA = √AB= -ACB= -BAC= -CBA; that is, an interchange of two etters changes the sense. Also, if A and A' are two points on >pposite sides of, and equidistant from, the line BC, then IBC= -A'BC.
Taking account of the sense, we may make the following state-nent :—
If A, A' are two corresponding points, if the line A A' cuts the Lxis in B, and if C is any other point in the axis, then the triangles VBC and A'BC are corresponding, and
ABC AB _ AB A'BC A'B^ BA' ’
>r The constant ratio of corresponding areas is equal and opposite o the ratio in which the axis divides the segment joining two corre-ponding points.
§ 18. Several special cases of parallel projection are of interest.
Orthographic Projection.— If the two planes π and √ have a lefinite position in space, and if a figure in π is projected to π' by ays perpendicular to this plane, then the projection is said to be )rthographic. If in this case the plane π be turned till it coincides vith τr' so that the figures remain perspective, then the projecting 'ays will be perpendicular to the axis of projection, because any oħe of these rays is, and remains during the turning, perpendicular to the axis.
The constant ratio of the area of the projection to that of the original figure is, in this case, the cosine of the angle between the two planes π and π', as will be seen by projecting a rectangle which has its base in the axis.
Orthographic projection is of constant use in geometrical drawing and will be treated of fully later on in this article (§ 28 sq.). Shear.— If the centre of projection be taken at infinity on the axis, then the projecting rays aie parallel to the axis; hence corresponding points will be equidistant from the axis. In this case therefore areas of corresponding figures will be equal.
If A, A' and B, B' (fig. 7) are two pairs of corresponding points on the same line, parallel to the axis, then, as corresponding segments parallel to the axis are equal, it follows that AB = A'B', hence also AA' = BB'. If these points be joined to any point 0 on the axis, then AO and ΑΌ will be corresponding lines; they will therefore be cut by any line parallel to the axis in corresponding points. In the figure therefore C, C' and also D, D' will be pairs of corresponding points and CC' = DD'. As the ratio CC'/AA' equals the ratio of the distances of C and A from the axis, therefore—
Two corresponding figures may be got one out of the other by moving all points in the one parallel to a fixed line, the axis, through distances which are proportional to their own distances from the axis. Points in a line remain hereby in a line.
Such a transformation of a plane figure is produced by a shearing stress in any section of a homogeneous elastic solid. For this reason Sir William Thomson has given it the name of shear.
A shear of a plane figure is determined if we are given the axis and the distance through which one point has been moved; for in this case the axis, the centre, and a pair of corresponding points are given.
§ 19. Symmetry and Skew-Symmetry. —If the centre is not on the axis, and if corresponding points are at equal distances from it, they must be on opposite sides of it. The figures will be in involution (§ 11). In this case the direction of the projecting rays is said to be conjugate to the axis.
The conjugate direction may be perpendicular to the axis. If the line joining two corresponding points A, A' cuts the axis in B, then AB = BA'. Therefore, if the plane be folded over along the axis, A will fall on A'. Hence by this folding over every point will coincide with its corresponding point. The figures therefore are identically equal or congruent, and in their original position they are symmetrical with regard to the axis, which itself is called an axis of symmetry. If the two figures are considered as one this one is said to be symmetrical with regard to an axis, and is said to have an axis of symmetry or simply an axis. Every diameter of a circle is thus an axis; also the median line of an isosceles triangle and the diagonals of a rhombus are axes of the figures to which they belong.
In the more general case where the projecting rays are not perpendicular to the axis we have a kind of twisted symmetry which maybe called skew-symmetry. It can be got from symmetry by giving the whole figure a shear. It will also be easily seen that we get skew-symmetry if we first form a shear to a given figure and then separate it from its shear by folding it over along the axis of the shear, which thereby becomes an axis of skew-symmetry.
Skew-symmetrical and therefore also symmetrical figures have the following properties :—
Corresponding areas are equal, but of opposite sense. Any two corresponding lines are harmonic conjugates with regard to the axis and a line in the conjugate direction.
If the two figures be again considered as one whole, this is said to be skew-symmetrical and to have an axis of skew-symmetry. Thus the median line of any triangle is an axis of skew-symmetry, the side on which it stands having the conjugate direction, tħe other sides being conjugate lines. From this it follows, for instance, that the three median lines of a triangle meet in a point. For two median lines will be corresponding lines with regard to the third as axis, and must therefore meet on the axis.
An axis of skew-symmetry is generally called a diameter. Thus every diameter of a conic is an axis of skew-symmetry, the conjugate direction being the direction of the chords which it bisects.
§ 20. We state a few properties of these figures useful in mechanics, but we omit the easy proofs :—
If a plane area has an axis of skew-symmetry, then the mass-centre (centre of mean distances or centre of inertia) lies on it. If a figure undergoes a shear, the mass-centre of its area remains the mass-centre ; and generally—
In parallel projection the mass-centres of corresponding areas {or of groups of points, but not of curves') are corresponding points.
The moment of inertia of a plane figure does not change if the figure undergoes a shear in the direction of the axis with regard to which the moment has been taken. If a figure has an axis of skew-symmetry, then this axis and the conjugate direction are conjugate diameters of the momental ellipse for every point in the axis. If a figure has an axis of symmetry, then this is an axis of the momental ellipse for every point in it.
The truth of the last propositions follows at once from the fact that the product of inertia for the lines in question vanishes.
It is of interest to notice how a great many propositions of Euclid are only special cases of projection. The theorems Euel. I. 35-41 about parallelograms or triangles on equal bases and between the same parallels are examples of shear, whilst I. 43 gives a case of skew-symmetry, hence of involution. Figures which are identically equal are of course projective, and they are perspective when placed so that they have an axis or a centre of symmetry (comp. Henriei, Elementary Geometry, Congruent Figures). In this case again the relation is that of involution. The importance of treating similar figures when in perspective position has long been recognized; we need only mention the well-known proposition about the centres of similitude of circles.
Applications to Conics.
§ 21. Theorem.— Any conic can be projected into any other conic. This may be done in such a manner that three points on one conic and the tangents at two of them are projected to three arbitrarily selected points and the tangents at two of them on the other. Proof.— If u and u' are any two conics, then we have to prove that we can project u in such a manner that five points on it will be projected to points on u'. As the projection is determined as soon as the projections of any four points or four lines are selected, we cannot project any five points of u to any five arbitrarily selected points on √. But if A, B, C be any three points on u, and if the tangents at B and C meet at D, if further A', B', C' are any three points on u', and if the tangents at B 'and C' meet at D', then the plane of u may be projected to the plane of u' in such a manner that the points A, B, C, D are projected to to A', B', C', D'. This determines the correspondence (§ 14). The conic u will be projected into a conic, the points A, B, C and the tangents BD and CD to the points A', B', C' and the lines B'D' and C'D', which are tangents to u' at B' and C'. The projection of u must therefore (G. § 52) coincide with u', because it is a conic which has three points and the tangents at two of them in common with u'.
Similarly we might have taken three tangents and the points of contact of two of them as corresponding to similar elements on the other.
If the one conic be a circle which cuts the line j, the projection will cut the line at infinity in two points; hence it will be an hyperbola. Similarly, if the circle touches j, the projection will be a parabola; and, if the circle has no point in common with j, the projection will be an ellipse. These curves appear thus as sections of a circular cone, for in case that the two planes of projection are separated the rays projecting the circle form such a cone.
Any conic may be projected into itself.
If we take any point S in the plane of a conic as centre, the polar of this point as axis of projection, and any two points in which a line through S cuts the conic as corresponding points, then these will be harmonic conjugates with regard to the centre and the axis. We therefore have involution (§ 11), and every point is projected into its harmonic conjugate with regard to the centre and the axis,— hence every point A on the conic into that point A' on the conic in which the line SA' cuts the conic again, as follows from the harmonic properties of pole and polar (G. § 62 sq.). Two conics which cut the line at infinity in the same two points are similar figures and similarly situated,—the centre of similitude being in general some finite point.
To prove this, we take the line at infinity and the asymptotes of one as corresponding to the line at infinity and the asymptotes of the other, and besides a tangent to the first as corresponding to a parallel tangent to the other. The line at infinity will then correspond to itself point for point; hence the figures will be similar and similarly situated.
§22. Areas of Parabolic Segments.— One parabola may always be considered as a parallel projection of another in such a manner that any two points A, B on the one correspond to any two points A', B' on the other; that is, the points A, B and the point at infinity on the one may be made to correspond respectively to the points A', B' and the point at infinity on the other, whilst the tangents at A and at infinity of the one correspond to the tangent at B' and at infinity of the other. This completely determines the correspondence, and it is parallel projection because the line at infinity corresponds to the line at infinity. Let the tangents at A and B meet at C, and those at Λ', B' at C'; then C, C 7 will correspond, and so will the triangles ABC and A'B'C' as well as the parabolic segments cut off by the chords AB and A'B'. If (AB) denotes the area of the segment cut off by the chord AB we have therefore
(AB)∕ABC-(A'B')∕A'B'C'; or The area of a segment of a parabola stands in a constant ratio to the area of the triangle formed by the chord of the segment and the tangents at the end points of the chord.
If then (fig. 8) we join the point C to the mid-point Μ of AB, then this line I will be bisected at D by the parabola (G. § 74), and the tangent at D will be parallel to AB. Let this tangent cut AC in E and CB in F, then by the last theorem
(AB) (AD) (BD) m
ABC ADE BFD
where m is some number to be determined. The figure gives
(AB) = ABD + (AD) + (BD).
Combining both equations, we have
ABD = τn (ABC - ADE - BFD).
But we have also ABD = ⅜ ABC, and ADE = BFD — ⅛ ABC; hence ⅜ ABC = 7iι (1 — ⅛ — ⅜ ) ABC, or wι=f.
The area of a parabolic segment equals two-thirds of the area of the triangle formed by the chord and the tangents at the end points of the chord.
§ 23. Elliptic Areas. —To consider one ellipse a parallel projection of another we may establish the correspondence as follows. If AC, BD are any pair of conjugate diameters of the one and A'C', B'D' any pair of conjugate diameters of the other, then these may be made to correspond to each other, and the correspondence will be completely determined if the parallelogram formed by the tangents at A, B, C, D is made to correspond to that formed by the tangents at A', B', C', D' (§§ 17 and 21). As the projection of the first conic has the four points A', B', C', D' and the tangents at these points in common with the second, the two ellipses are projected one into the other. Their areas will correspond, and so do those of the parallelograms ABCD and A'B'C'D'. Hence The area of an ellipse has a constant ratio to the area of any inscribed parallelogram ιohose diagonals are conjugate diameters, and also to every circumscribed parallelogram whose sides are parallel to conjugate diameters.
It follows at once that All parallelograms inscribed in an ellipse whose diagonals' are conjugate diameters are equal in area ; and All parallelograms circumscribed about an ellipse whose sides are parallel to conjugate diameters are eqzwt,l in area.
If a, b are the length of the semi-axes of the elli∣ se, then the area of the circumscribed parallelogram will be 4αδ and of the inscribed one 2ab.
For the circle of radius r the inscribed parallelogram becomes the square of area 2r 2 and the circle has the area r⅛; the constant ratio of an ellipse to the inscribed parallelogram has therefore also the value ⅜τr. Hence The area of an ellipse equals ab∙κ.
§ 24. Projective Properties. —The properties of the projection of a figure depend partly on the relative position of the planes of the figure and the centre of projection, but principally on the properties of the given figure. Points in a line are projected into points in a line, harmonic points into harmonic points, a conic into a conic; but parallel lines are not projected into parallel lines nor right angles into right angles, neither are the projections of equal segments or angles again equal. There are then some properties which remain unaltered by projection, whilst others change. The former are called projective or descriptive, the latter metrical properties of figures, became the latter all depend on measurement.
To a triangle and its median lines correspond a triangle and three lines which meet in a point, but which as a rule are not median lines.
In this case, if we take the triangle together with the line at infinity, we get as the projection a triangle ABO, and some other line j which cuts the sides a, b, c of the triangle in the points A 1 , B 1 , C 1 . If we now take on BC the harmonic conjugate A 2 to A 1 and similarly on CA and AB the harmonic conjugates to B 1 and C 1 respectively, then the lines AA 2 , BB 2 , CC 2 will be the projections of the median lines in the given figure. Hence these lines must meet in a point.
As the triangle and the fourth line we may take any four given lines, because any four lines may be projected into any four given lines (§ 14). This gives a theorem :—
If each vertex of a triangle be joined to that point in the apposite side which is, with regard to the vertices, the harmonic conjugate of the point in which the side is cut by a given line, then the three lines thus obtained meet in a point.
We get thus out of the special theorem about the median lines
of a triangle a more general one. But before this could be done we had to add the line at infinity to the lines in the given figure.
In a similar manner a great many theorems relating to metrical properties can be generalized by taking the line at infinity or points at infinity as forming part of the original figure. Conversely special cases relating to measurement are obtained by projecting some line in a figure of known properties to infinity. This is true for all properties relating to parallel lines or to bisection of segments, but not immediately for angles. It is, however, possible to establish for every metrical relation the corresponding projective property. To do this it is necessary to consider imaginary elements. These have originally been introduced into geometry by aid of coordinate geometry, where imaginary quantities constantly occur as roots of equations.
Their introduction into pure geometry is due principally to Poncelet, who by the publication of his great work Traité des Propriétés Projectives des Figures became the founder of projective geometry in its widest sense. Monge had considered parallel projection and had already distinguished between permanent and accidental properties of figures, the latter being those which depended merely on the accidental position of one part to another. Thus in projecting two circles which lie in different planes it depends on the accidental position of the centre of projection whether the projections be two conics which do or do not meet. Poncelet introduced the principle of continuity in order to make theorems general and independent of those accidental positions which depend analytically on the fact that the equations used have real or imaginary roots. But the correctness of this principle remained without a proof. Von Staudt has, however, shown how it is possible to introduce imaginary elements by purely geometrical reasoning, and we shall now try to give the reader some idea of his theory.
§ 25. Imaginary Elements. —If a line cuts a curve and if the line be moved, turned for instance about a point in it, it may happen that two of the points of intersection approach each other till they coincide. The line then becomes a tangent. If the line is still further moved in the same manner it separates from the curve and two points of intersection are lost. Thus in considering the relation of a line to a conic we have to distinguish three cases—the line cuts the conic in two points, touches it, or has no point in common with it. This is quite analogous to the fact that a quadratic equation with one unknown quantity has either two, one, or no roots. But in algebra it has long been found convenient to express this differently by saying a quadratic equation has always two roots, but these may be either both real and different, or equal, or they may be imaginary. In geometry a similar mode of expressing the fact above stated is not less convenient.
We say therefore a line has always two points in common with a conic, but these are either distinct, or coincident, or invisible. The word imaginary is generally used instead of invisible; but, as the points have nothing to do with imagination, we prefer the word “invisible” recommended originally by Clifford.
Invisible points occur in pairs of conjugate points, for a line loses always two visible points of intersection with a curve simultaneously. This is analogous to the fact that an algebraical equation with real coefficients has imaginary roots in pairs. Only one real line can be drawn through an invisible point, for two real lines meet in a real or visible point. The real line through an invisible point contains also its conjugate.
Similarly there are invisible lines—tangents, for instance, from a point within a conic—which occur in pairs of conjugates, two conjugates having a real point in common.
The introduction of invisible points would be nothing but a play upon words unless there is a real geometrical property indicated which can be used in geometrical constructions—that it has a definite meaning, for instance, to say that two conics cut a line in the same two invisible points, or that we can draw one conic through three real points and the two invisible ones which another conic has in common with a line that does not actually cut it. We have in fact to give a geometrical definition of invisible points. This is done by aid of the theory of involution (G. § 76 sq. y ).
An involution of points on a line has (according to G. § 77 (2)) either two or one or no foci. Instead of this we now say it has always two foci which may be distinct, coincident, or invisible. These foci are determined by the involution, but they also determine the involution. If the foci are real this follows from the fact that conjugate points are harmonic conjugates with regard to the foci. That it is also the case for invisible foci will presently appear. If we take this at present for granted we may replace a pair of real, coincident, or invisible points by the involution of which they are the foci.
^Now any two pairs of conjugate points determine an involution Hence any point-pair, whether real or invisible, is completely determined by any two pairs of conjugate points of the involution which has the given point-pair as foci and may therefore be replaced by them.
Two pairs of invisible points are thus said to be identical if, and only if, they are the foci of the same involution.
We know (G. § 82) that a conic detιrmines on every line an involution in which conjugate points are conjugate poles with regard to the conic—that is, that either lies on the polar of the other. This holds whether the line cuts the conic or not. Furthermore, in the former case the points common to the line and the conic are the foci of the involution. Hence we now say that this is always the case, and that the invisible points common to a line and a conic are the invisible f<⅛ei of the involution in question. If then we state the problem of drawing a conic which passes through two points given as the intersection of a conic and a line as that of drawing a conic which determines a given involution on the line, we have it in a form in which it is independent of the accidental circumstance of the intersections being real or invisible. So is the solution of the problem, as we shall now show.
§ 26. We have seen (§ 21) that a conic may always be projected into itself by taking any point S as centre and its polar s as axis of projection, corresponding points being those in which a line through S cuts the conic. If then (fig. 9) A, A' and B, B' are pairs of corresponding points so that the lines AA' and BB' pass through S, then the lines AB and A'B', as corresponding lines, will meet at a point Ron the axis, and the lines AB' and A'B will meet at another point R' on the axis. These points R, R' are conjugate points in the involution which the conic determines on the line s, because the triangle RSR' is a polar triangle (G. § 62), so that R' lies on the polar of R.
This gives a simple means of determining for any point Q on the line s its conjugate point Q'. We take any two points A, A' on the conic which lie on a line through S, join Q to A by a line cutting the conic again in C, and join C to A'. This line will cut s in the point Q' required.
Problem. — To draw some conic which shall determine on a line s a given involution. Solution.— We have here to reconstruct the fig. 9, having given on the line s an involution. Let Q, Q' and R, R' (fig. 9) be two pairs of conjugate points in this involution. We take any point B and join it to R and R', and another point C to Q and Q'. Let BR and CQ meet at A, and BR' and CQ' at A'. If now a point P be moved along s its conjugate point P' will also move and the two points will describe projective rows. The two rays AP and A'P' will therefore describe projective pencils, and the intersection of corresponding rays will lie on a conic which passes through A, A', B, and C. This conic determines on s the given involution.
Of these four points not only B and C but also the point A may be taken arbitrarily, for if A, B, C are given, the line AB will cut s in some point R. As the involution is supposed known, we can find the point R' conjugate to R, which we join to B. In the same way the line CA will cut s in some point Q. Its conjugate point Q' we join to C. The line CQ' will cut BR' in a point A', and then AA' will pass through the pole S (comp. fig. 9). We may now interchange A and B and find the point B'. Then BB' will also pass through S, which is thus found. At the same time five points A, B, C, A', B' on the conic have been found, so that the conic is completely known which determines on the line s the given involution. Hence—
Theorem.— Through three points we can always draw one conic, and only one, which determines on a given line a given involution, all the same whether the involution has real, coincident, or invisible foci.
In the last case the theorem may now also be stated thus :—
Theorem. — It is always possible to draw a conic which passes through three given real points and through two invisible points which any other conic has in common with a line.
§ 27. The above theory of invisible points gives rise to a great number of interesting consequences of which we state a iew.
The theorem at the end of § 21 may now be stated :—
Theorem.— Any two conics are similar and similarly situated if they cut the line at infinity in the same two points—real, coincident, or invisible.
It follows that Any two parabolas are similar; and they are similarly situated as soon as their axes are parallel.
The involution which a circle determines at its centre is circular (G. § 79); that is, every line is perpendicular to its conjugate line. This will be cut by the line at infinity in an involution which has the following property:—The lines which join any finite point to two conjugate points in the involution are at right angles to each other. Hence all circular involutions in a plane determine the same involution on the line at infinity. The latter is therefore called the circular involution on the line at infinity, and the involution which a circle determines at its centre is called the circular involution at that point. All circles determine thus on the line at infinity the same involution; in other w∩rds, they have the same two invisible points in common with the line at infinity.
Theorem.— All circles may be considered as passing through the same two points at infinity.
These points a>∙e called the circular points at infinity, and by Prof. Cayley the absolute in the plane. They* are the foci of the circular involution in the line at infinity.
Conversely— Every conic which passes through the circular points is a circle ; because the involution at its centre is circular, hence conjugate diameters are at right angles, and this property only circles possess.
We now see why we can draw always one and only one circle through any three points: these three points together with the circular points at infinity are five points through which one conic only can be drawn.
Any two circles are similar and similarly situated because they have the same points at infinity (§ 21).
Any two concentric circles may be considered as having double contact at infinity, because the lines joining the common centre to the circular points at infinity are tangents to both circles at the circular points, as the line at infinity is the polar of the centre.
Any two lines at right angles to one another are harmonic conjugates with regard to the rays joining their intersection to the circular points, because these rays are the focal rays of the circular involution at the intersection of the given lines.
To bisect an angle with the vertex A means (G. § 23) to find two rays through A which are harmonic conjugates with regard to the limits of the angle and perpendicular to each other. These rays are therefore harmonic with regard to the limits of the given angle and with regard to the rays through the circular points. Thus perpendicularity and bisection of an angle have been stated in a projective form.
It must not be forgotten that the circular points do not exist at all; but to introduce them gives us a short way of making a statement which would otherwise be long and cumbrous.
We can now generalize any theorem relating to metrical properties. For instance, the simple fact that the chord of a circle is touched by a concentric circle at its mid point proves the theorem:
If two conics have double contact, then the points where any tangent to one of them cuts the other are harmonic with regard to the point of contact and the point where the tangent cuts the chord of contact.
DESCRIPTIVE GEOMETRY.
For many, especially technical, purposes it is of the utmost importance to represent solids and other figures in three dimensions by a drawing in one plane.
A variety of methods have been introduced for this purpose. The most important is that which towards the end of the last century was invented by Monge under the name of “descriptive geometry.” We give the elements of his method. It is based on parallel projections to a plane by rays perpendicular to the plane. Such a projection is called orthographic (§ 18). If the plane is horizontal the projection is called the plan of the figure, and if the plane is vertical the elevation. In Monge’s method a figure is represented by its plan and elevation. It is therefore often called drawing in plan and elevation, and sometimes simply orthographic projection.
§ 28. We suppose then that we have two planes, one horizontal, the other vertical, and these we call the planes of plan and of elevation respectively, or the horizontal and the vertical plane, and denote them by the letters π 1 and π 2 . Their line of intersection is called the axis, and shall be denoted by the letter x.
If the surface of the drawing naper is taken as the plane of the plan, then the vertical plane will be the plane perpendicular to it through the axis x. To bring this also into the plane of the drawing paper we turn it about the axis till it coincides with the horizontal plane. This process of turning one plane down till it coincides with another is called rabatting one to the other. Of course there is no necessity to have one of the two planes horizontal, but even when this is not the case it is convenient to retain the above names.
The whole arrangement will be better understood by a glance at fig. 10. A point A in space is there projected by the perpendicular AAj and AA 2 to the planes w 1 and π 2 , so that A 1 and A 2 are the horizontal and vertical projections of A.
If we remember that a line is perpendicular to a plane that is perpendicular to every line in the plane if only it is perpendicular to any two intersecting lines in the plane, we see that the axis which is perpendicular both to AA 1 and to AA 2 is also perpendicular to A 1 Λ 0 and to A 2 A 0 because these four lines are all in the same plane. Hence, if the plane w 2 be turned about the axis till it coincides with the plane ιr 1 , then A 2 A 0 will be the continuation of A 1 A 0 ∙ This position of the planes is represented in fig. 11, in which the line A 1 A 2 is perpendicular to the axis x.
Conversely any two points A 1 , A 2 in a line perpendicular to the axis will be the projections of some point in space when the plane ιr 2 is turned about the axis till it is perpendicular to the plane π 1 , because in this position the two perpendiculars to the planes τr 1 and π 2 through the points A 1 and A 2 will be in a plane and therefore meet at some point A.
Representation of Points. —We havethusthefollowingmethod of representing in a single plane the position of points in space :— we take in the plane a line x as the axis, and then any pair of points A 1 , A 2 in the plane on a line perpendicular to the axis represent a point A in space. If the line A 1 A 2 cuts the axis at A o , and if at A 1 a perpendicular be erected to the plane, then the point A will be in it at a height A 1 A = A 0 A 2 above the plane. This gives the position of the point A relative to the plane ιr 1 . In the same way, if in a perpendicular to π 2 through A 2 a point A be taken such that A 2 A = A 0 A 1 , then this will give the point A relative to the plane τr 2 .
§ 29. The two planes w 1 , π 2 in their original position divide space into four parts. These are called the four quadrants. We suppose that the plane τr 2 is turned as indicated in fig. 10, so that the point P comes to Q and R to S, then the quadrant in which the point A lies is called the first, and we say that in the first quadrant a point lies above the horizontal and in front of the vertical plane. Now we go round the axis in the sense in which the plane ιr 2 is turned and come in succession to the second, third, and fourth quadrant. In the second a point lies above the plane of the plan and behind the plane of elevation, and so on. In fig. 12, which represents a side view of the planes in fig. 10 the quadrants are marked, and in each a point with its projection is taken. Fig. 11 shows how these are represented when the plane π 2 is turned down. We see that A point lies in the first quadrant if the plan lies below, the elevation above the axis; in the second if plan and elevation both lie above; in the third if the plan lies above, the elevation below; in the fourth if plan and elevation both lie below the axis. If a point lies in the horizontal plane, its elevation lies in the axis and the plan coincides with the point itself. If a point lies in the vertical plane, its plan lies in the axis and the elevation coincides with the point itself. If a point lies in the axis, both its plan and elevation lie in the axis and coincide with it.
Of each of these propositions, which will easily be seen to be time, the converse holds also.
§ 30. Representation of a Plane. —As we are thus enabled to represent points in a plane, we can represent any finite figure by representing its separate points. It is, however, not possible to represent a plane in this way, for the projections of its points completely cover the planes π 1 and π 2 , and no plane would appear different from any other. But any plane α cuts each of the planes π,, π 2 in a line. These are called the traces of the plane. They cut each other in the axis at the point where the latter cuts the plane α.
A plane is determined by its two traces, which are two lines that meet on the axis, and, conversely, any two lines which meet on the axis determine a plane.
If the plane is parallel to the axis its traces are parallel to the axis. Of these one may be at infinity; then the plane will cut one of the planes of projection at infinity and will be parallel to it. Thus a plane parallel to the horizontal plane of the plan has only one finite trace, viz., that with the plane of elevation.
If the plane passes through the axis both its traces coincide with the axis. This is the only case in which the representation of the plane by its two traces fails. A third plane of projection is therefore introduced, which is best taken perpendicular to the other two. We call it simply the third nlane, and denote it by τr 3 . As it is perpendicular to π 1 , it may υe taken as the plane of elevation, its line of intersection y with ιr 1 being the axis, and be turned down to coincide with π 1 . This is represented in fig. 13, OC is the axis x whilst OA and OB are the traces of the third plane. They lie in one line y. The plane is rabatted about y to the horizontal plane. A plane a through the axis x will then show in it a trace α 3 . In fig. 13 the lines OC and OP will thus be the traces of a plane through the axis x which makes an angle POQ with the horizontal plane.
We can also find the trace which any other plane makes with ιr 3 . In rabatting the plane τr 3 its trace OB with the plane w 3 will come to the position OD. Hence a plane β having the traces CA and CB will have with the third plane the trace β 3 , or AD if OD = OB.
It also follows immediately that—
If a plane a is perpendicular to the horizontal plane, then every point in it has its horizontal projection in the horizontal trace of the plane, as all the rays projecting these points lié in the plane itself.
Any plane which is perpendicular to the horizontal plane has its vertical trace perpendicular to the axis. Any plane which is perpendicular to the vertical plane has its horizontal trace perpendicular to the axis and the vertical projections of all points in the plane lie in this trace.
§ 31. Representation of a Line.— A line is determined either by two points in it or by two planes through it. We get accordingly two representations of it either by projections or by traces.
First.— A line a is represented by its projections a 1 and a 2 on the two planes τr 1 and π 2 . These may be any two lines, for, bringing the planes π 1 , π 2 into their original position, the planes through these lines perpendicular to π 1 and τr 2 respectively will intersect in some line a which has a 1 , a 2 as its projections.
Secondly. — A line a is represented by its traces—that is, by the points in which it cuts the two planes π 1 , π 2 . Any two points may be taken as the traces of a line in space, for it is determined when the planes are in their original position as the line joining the two traces. This representation becomes undetermined if the two traces coincide in the axis. In this case we again use a third plane, or else the projections of the line.
§ 32. The fact that there are different methods of representing points and planes, and hence two methods of representing lines, suggests the principle of duality (G. § 41). It is worth while to keep this in mind. It is also worth remembering that traces of planes or lines always lie in the planes or lines which they represent. Projections do not as a rule do this excepting when the point or line projected lies in one of the planes of projection.
§ 33. Having now shown how to represent points, planes, and lines, we have to state the conditions which must hold in order that these elements may lie one in the other, or else that the figure formed by them may possess certain metrical properties. It will be found that the former are very much simpler than the latter.
Before we do this, however, we shall explain the notation used; for it is of great importance to have a systematic notation. We shall denote points in space by capitals A, B, C; planes in space by Greek letters a, β, y ; lines in space by small letters a, b, c; horizontal projections by suffixes 1, like A 1 , α 1 ; vertical projections by suffixes 2, like A 2 , a 2 ; traces by single and double dashes a a, a' a". Hence P 1 will be the horizontal projection of a point P in space; a line a will have the projections α 1 , a 2 and the traces a' and a" ; a plane α has the traces a and a.
§ 34. If a point lies in a line, the projections of the point lie in the projections of the line. If a line lies in a plane, the traces of the line lie in the traces of the plane.
These propositions follow at once from the definitions of the projections and of the traces.
If a point lies in two lines its projections must lie in the projections of both. Hence If two lines, given by their projections, intersect, the intersection of their plaits and the intersection of their elevations must lie in a line perpendicular to the axis, because they must be the projections of the point common to the two lines.
Similarly— If two lines given by their traces lie in the same plane or intersect, then the lines joining their horizontal and vertical traces respectively must meet on the axis, because they must be the traces of the plane through them.
§ 35. To find the projections of a line which joins two points A, B given by their projections A 1 , A 2 and B υ B 2 , we join A 1 , Bj and A 2 , Ba; these will be the projections required. For example, the traces of a line are two points in the line whose projections are known or at all events easily found. They are the traces themselves and the feet of the perpendiculars from them to the axis.
Hence if a', a" (fig. 14) are the traces of a line a, and if the perpendiculars from them cut the axis in P and Q respectively^ then the line α'Q will be the horizontal and a"P the vertical projection of the line.
Conversely, if the projections a l , a 2 of a line are given, and if these cut the axis in Q and P respectively, then the perpendiculars Pa' and Qa' , to the axis drawn through these points cut the projections a x and a 2 in the traces a' and a". To find the line of intersection of two planes, we observe that this line lies in both planes; its traces must therefore lie in the traces of both. Hence the points where the horizontal traces of the given planes meet will be the horizontal, and the point where the vertical traces meet the vertical trace of the line required.
§ 36. To decide whether a point A, given by its projections, lies in a plane a, given by its traces, we draw a line p by joining A to some point in the plane α and determine its traces. If these lie in the traces of the plane, then the line, and therefore the point A, lies in the plane; otherwise not. This is conveniently done by joining A 1 to some point p' in the trace a ; this gives p 1 ; and the point where the perpendicular from p' to the axis cuts the latter we join to A 2 ; this gives p 2 . If the vertical trace of this line lies in the vertical trace of the plane, then, and then only, does the line p, and with it the point A, lie in the plane a.
§ 37. Parallel planes have parallel traces, because parallel planes are cut by any plane, hence also by π 1 and by π 2 , in parallel lines.
Parallel lines have parallel projections, because points at infinity are projected to infinity.
If a line is parallel to a plane, then lines through the traces of the line and parallel to the traces of the plane must meet on the axis, because these lines are the traces of a plane parallel to the given plane.
§ 38. To draw a plane through two intersecting lines or through two parallel lines, we determine the traces of the lines; the lines joining their horizontal and vertical traces respectively will be the horizontal and vertical traces of the plane. They will meet, at a finite point or at infinity, on the axis if the lines do intersect.
To draw a plane through a line and a point without the line, we join the given point to any point in the line and determine the plane through this and the given line.
To draw a plane through three points which are not in aline, we draw two of the lines which each join two of the given points and draw the plane through them. If the traces of all three lines AB, BC, CA be found, these must lie in two lines which meet on the axis.
§ 39. We have in the last example got more points, or can easily get more points, than are necessary for the determination of the figure required—in this case the traces of the plane. This will happen in a great many constructions and is of considerable importance. It may happen that some of the points or lines obtained are not convenient in the actual construction. The horizontal traces of the lines AB and AC may, for instance, fall very near together, in which case the line joining them is not well defined. Or, one or both of them may fall beyond the drawing paper, so that they are practically non-existent for the construction. In this case the traces of the line BC may be used. Or, if the vertical traces of AB and AC are both in convenient position, so that the vertical trace of the required plane is found and one of the horizontal traces is got, then we may join the latter to the point where the vertical trace cuts the axis.
Furthermore the draughtsman will never forget that the lines which he draws are not mathematical lines without thickness. For this reason alone every drawing is affected by some errors. And inaccuracies also come in in drawing the lines required in the construction. It is therefore very desirable to be able constantly to check the latter. Such checks always present themselves when the same result can be obtained by different constructions, or when, as in the above case, some lines must meet on the axis, or if three points must lie in a line. A careful draughtsman will always avail himself of these checks.
§ 40. To draw a plane through a given point parallel to a given plane a, we draw through the point two lines which are parallel to the plane <x, and determine the plane through them; or, as we know that the traces of the required plane are parallel to those of the given one (§ 37), we need only draw one line I through the point parallel to the plane and find one of its traces, say the vertical trace Γ ; a line through this parallel to the vertical trace of a will be the vertical trace 0" of the required plane β, and a line parallel to the horizontal trace of a meeting 0" on the axis will be the horizontal trace β'.
Let A 1 A 2 (fig. 15) be the given point, a a" the given plane, a line ξ through A 1 parallel to a and a horizontal line Z 2 through A s will be the projections of a line I through A parallel to the plane, because the horizontal plane through this line will cut the plane a in a line c which has its horizontal projection c 1 parallel to α'.
§ 41. We now come to the metrical properties of figures.
A line is perpendicular to a plane if the projections of the line are perpendicular to the traces of the plane. We prove it for the horizontal projection. If a line p is perpendicular· to a plane α, every plane through p is perpendicular to α; hence also the vertical plane which projects the line p to p i . As this plane is perpendicular both to the horizontal plane and to the plane a, it is also perpendicular to their intersection —that is, to the horizontal trace of a. It follows that every line in this projecting plane, therefore also p 1 , the plan of p, is perpendicular to the horizontal trace of a. Q.E.D.
To draw a plane through a given point A perpendicular to a given line p, we first draw through some point 0 in the axis lines y', y" perpendicular respectively to the projections p 1 and p 2 of the given line. These will be the traces of a plane y which is perpendicular to the given line. We next draw through the given point A a plane parallel to the plane y ; this will be the plane required.
Other metrical properties depend on the determination of the real size or shape of a figure.
In general the projection of a figure differs both in size and shape from the figure itself. But figures in a plane parallel to a plane of projection will be identical with their projections, and will thus be given in their true dimensions. In other cases there is the problem, constantly recurring, either to find the true shape and size of a plane figure when plan and elevation are given, or, conversely, to find the latter from the known true shape of tire figure itself. To do this, the plane is turned about one of its traces till it is laid down into that plane of projection to which the trace belongs. This is technically called rabatting the plane respectively into the plane of the plan or the elevation. As there is no difference in the treatment of the two cases, we shall consider only the case of rabatting a plane α into the plane of the plan. The plan of the figure is a parallel (orthographic) projection of the figure itself. The results of parallel projection (§§ 17 and 18) may therefore now be used. The trace a! will hereby take the place of what formerly was called the axis of projection. Hence we see that corresponding points in the plan and in the rabatted plane are joined by lines which are perpendicular to the trace α' and that corresponding lines meet on this trace. We also see that the correspondence is completely determined if we know for one point or one line in the plan the corresponding point or line in the rabatted plane.
Before, however, we treat of this we consider some special cases.
§ 42. To determine the distance between two points A, B given by their projections A 1 , B 1 and A 2 , B 2 .
Solution.— The two points A, B in space lie vertically above their plans A 1 , B 1 (fig. 16) and A 1 A = A 0 A 2 , B 1 B=B o B 2 , The four points A, B, A 1 , B 1 therefore form a plane quadrilateral on the base A 1 B 1 and having right angles at the base. This plane we rabatt about A 1 B 1 by drawing A 1 A and B 1 B j>erpendicular to A 1 B 1 and making A 1 A = A 0 A 2 , B 1 B = B 0 B 2 . Then AB will give the length required.
The construction might have been performed in the elevation by making A 2 A = A 0 A 1 and B 2 B = B 0 B 1 on lines perpendicular to A 2 B 2 , Of course AB must have the same length in both cases.
This figure may be turned into a model. We cut the paper along A 1 A, AB, and BB 1 and fold the piece A 1 ABB 1 over along A 1 B 1 till it stands upright at right angles to the horizontal plane. The points A, B will then be in their true position in space relative to τr 1 . Similarly if B 2 BAA 2 be cut out and turned along A 2 B i through a right angle we shall get AB in its true position relative to the plane a·,. Lastly we fold the whole plane of the paper along the axis x till the plane π 2 is at right angles to π 1 . In this position the two sets of points ÀB will coincide n tne drawing has been accurate.
Models of this kind can be made in many cases and their construction cannot be too highly recommended in order to realize orthographic projection.
§ 43. To find the angle between two given lines a, b of which the projections a 1 , b 1 and α 2 , b 2 are given. Solution. — Let a l , b 1 (fig. 17) meet in P 1 , α 2 , δ 2 in T, then if the line P 1 T is not perpendicular to the axis the two lines will not meet. In this case we draw a line parallel to b to meet the line a. This is easiest done by drawing first the line P 1 P 2 perpendicular to the axis to meet α 2 in P 2 , and then drawing through P 2 a line c 2 parallel to ⅛ 2 ; then b 1 , c s will be the projections of a line c which is parallel to b and meets a in P. The plane α which these two lines determine we rabatt to the plan. We determine the traces a! and c' of the lines a and c ; then a'c , is the trace α' of their plane. Ou rabatting the point P comes to a point S on the line P 1 Q perpendicular to a'c', so that QS = QP. But QP is the hypothenuse of a triangle PP 1 Q with a right angle P 1 . This we construct by making QR=P 0 P 2 ; then P 1 R=PQ. The lines α'S and c'S will therefore include angles equal to those made by the given lines. It is to be remembered that two lines include two angles which are supplementary. Which of these is to be taken in any special case depends upon the circumstances.
To determine the angle between a line and a plane, we draw through any point in the line a perpendicular to the plane (§ 41) and determine the angle between it and the given line. The complement of this angle is the required one.
To determine the angle between two planes, we draw through any point two lines perpendicular to the two planes and determine the angle between the latter as above.
In special cases it is simpler to determine at once the angle between the two planes by taking a plane section perpendicular to the intersection of the two planes and rabatt this. This is especially the case if one of the planes is the horizontal or vertical plane of projection.
Thus in fig. 18 the angle P 1 QR is the angle which the plane a makes with the horizontal plane.
§ 44. We return to the general case of rabatting a plane α of which the traces a a" are given.
Here it will be convenient to determine first the position which the trace α"—which is a line in a— assumes when rabatted. Points in this line coincide with their elevations. Hence it is given in its true dimension, and we can measure off along it the true distance between two pointsinit. If therefore (fig. 18) P is any point in a" originally coincident with its elevation P 2 , and if 0 is the point where a" cuts the axis x, so that 0 is also in a , , then the point P will after rabatting the plane assume such a position that OP=OP 2 ∙ At the same time the plan is an orthographic projection of the plane a. Hence the line joining P to the plan P j will after rabatting be perpendicular to a. But P 1 is known; it is the foot of the perpendicular from P 2 to the axis x. We draw therefore, to find P, from P 1 a perpendicular P 1 Q to α' and find on it a point P such that OP = OP 2 ∙
Then the line OP will be the position of a" when rabatted. This line corresponds therefore to the plan of a"— that is, to the axis x, corresponding points on these lines being those which lie on a perpendicular to α'.
We have thus one pair of corresponding lines and can now find for any point B 1 in the plan the corresponding point B in the rabatted plane. We draw a line through B 1 , say B 1 P 1 , cutting a in C. To it corresponds the line CP, and the point where this is cut by the projecting ray through B 1 , perpendicular to α', is the required point B.
Similarly any figure in the rabatted plane can be found when the plan is known; but this is usually found in a different manner without any reference to the general theory of parallel projection. As this method and the reasoning employed for it have their peculiar advantages, we give it also.
Supposing the planes π 1 and ιr 2 to be in their positions in space ρeιpendicular to each other, we take a section of the whole figure by a plane perpendicular to the trace a about which we are going to rabatt the plane a. Let this section pass through the point Q in α'. Its traces will then be the lines QP 1 and P 1 P 2 (fig. 18). These will be at right angles, and will therefore, together with the section QP 2 of the plane α, form a right-angled triangle QP 1 P 2 with the right angle at Pj, and having the sides P 1 Q and P 1 P 2 which both are given, in their true lengths. This triangle we rabatt about its base P 1 Q, making P 1 R=P 1 P 2 ∙ The line QR will then give the true length of the line QP in space. If now the plane α be turned about α' the point P will describe a circle about Q as centre with radius QP—QR, in a plane perpendicular to the trace a. Hence when the plane a has been rabatted into the horizontal plane the point P will lie in the perpendicular P 1 Q to α', so that QP = QR.
If A 1 is the plan of a point A in the plane a, and if A 1 lies in QP 1 , then the point A will lie vertically above A 1 in the line QP. On turning down the triangle QP 1 P 2 , the point A will come to A o , the line A 1 A 0 being perpendicular to QP 1 . Hence A will be a point in QP such that QA=QA 0 ∙
If B 1 is the plan of another point, but such that A 1 B 1 is parallel to a, then the corresponding line AB will also be parallel to a. Hence, if through A a line AB be drawn parallel to α', and B 1 B perpendicular to a, then their intersection gives the point B. Thus of any point given in plan the real position in the plane a, when rabatted, can be found by this second method. This is the one most generally given in books on geometrical drawing. The first method explained is, however, in most cases preferable as it gives the draughtsman a greater variety of constructions. It requires a somewhat greater amount of theoretical knowledge.
If instead of our knowing the plan of a figure the latter is itself given; then the process of finding the plan is the reverse of the above and needs little explanation. We give an example.
§ 45. Problem.— It is required to draw the plan and elevation of a polygon of which the real shape and position in a given plane a are known. Solution.— We first rabatt the plane α (fig. 19) as before so that P 1 comes to P, hence
OP 1 to OP. Let the given polygon in a be the figure ABODE. We project, not the vertices, but the sides. To project the line AB, we produce it to cut α' in F and OP in G, and draw GG 1 perpendicular to a ; then G 1 corresponds to G, therefore FG 1 to FG. In the same manner we might project all the other sides, at least those which cut OF and OP in convenient points. It will be best, however, first to produce all the sides to cut OP and α' and then to draw all the projecting rays through A, B,C . . . perpendicular to a', and in the same direction the lines G,G 1 , &c. By drawing FG wβ get the points A 1 , B 1 on the projecting ray through A and B. We then join B to the point Μ where ÉC produced meets the trace α'. This gives C 1 . So we go on till we have found E 1 . The line A 1 E 1 must then meet AE in α', and this gives a check. If one of the sides cuts a' or OP beyond the drawing paper this method fails, but then we may easily find the projection of some other line, say of a diagonal, or directly the projection of a point, by the former methods. The diagonals may also serve to check the drawing, for two corresponding diagonals must meet in the trace α'.
Having got the plan we easily find the elevation. The elevation of G is above G 1 in a", and that of F is at F 2 in the axis. This gives the elevation F 2 G 2 of FG and in it we get A 2 B 2 in the verticals through A 1 and B 1 . As a check we have OG=OG 2 , Similarly the elevation of the other sides and vertices are found.
§ 46. We have now obtained the ABC of descriptive geometry, and proceed to give some applications to the representation of solids and of the solution of problems connected with them.
Problem.— Of a pyramid are given its base, the length of the perpendicular from the vertex to the base, and the point where this perpendicular cuts the base; it is required first to develop the whole surface of the pyramid into one plane, and second to determine its section by a plane which cuts the plane of the base in a given line and makes a given angle with it. Solution.— (1) As the planes of projection are not given we can take them as we like, and we select them in such a manner that the solution becomes as simple as possible. We take the plane of the base as the horizontal plane and the vertical plane perpendicular to the plane of the section. Let then (fig. 20) ABCD be the base of the pyramid, V 1 the plan of the vertex, then the elevations of A, B, C, D will be in the axis at A 2 , B 2 , C 2 , D 2 , and the vertex at some point V 2 above V 1 at a known distance from the axis. The lines V 1 A, V 1 B, &c., will be the plans and the lines V 2 A 2 , V 2 B 2 , &c., the elevations of the edges of the pyramid, of which thus plan and elevation are known.
We develop the surface into the plane of the base by turning each lateral face about its lower edge into the horizontal plane by the method used in § 43. If one face has been turned down, say ABV to ABP, then the point Q to which the vertex of the next face BCV comes can be got more simply by finding on the line V 1 Q perpendicular to BC the point Q such that BQ = BP, for these lines represent the same edge BY of the pyramid. Next R is found by making CR=CQ, and so on till we have got the last vertex—in this case S. The fact that AS must equal AP gives a convenient check.
(2) The plane α whose section we have to determine has its horizontal trace given perpendicular to the axis, and its vertical trace makes the given angle with the axis. This determines it. To find the section of the pyramid by this plane there are two methods applicable: we find the sections of the plane either with the faces or with the edges of the pyramid. We use the latter.
As the plane a is perpendicular to the vertical plane, the trace a" contains the projection of every figure in it; the points E 2 , F 2 , G 2 , H 2 where this trace cuts the elevations of the edges will therefore be the elevations of the points where the edges cut a. From these we find the plans E 1 , F 1 , G 1 , H v and by joining them the plan of the section. If from E 1 , F 1 lines be drawn perpendicular to AB, these will determine the points E, F on the developed face in which the plane α cuts it; hence also the line EF. Similarly on the other faces. Of course BF must be the same length on BP and on BQ. If the plane α be rabatted to the plan, we get the real shape of the section as shown in the figure in EFGH. This is done easily by making F 0 F-OF 1 , Ac. If the figure representing the development of the pyramid, or better a copy of it, is cut out, and if the lateral faces be bent along the lines AB, BC, &c., we get a model of the pyramid with the section marked on its faces. This may be placed on its plan A BCD and the plane of elevation bent about the axis x. The pyramid stands then in front of its elevations. If next the plane a with a hole cut out representing the true section be bent along the trace α' till its edge coincides with a", the edges of the hole ought to coincide with the lines EF, FG, &c., on the faces.
§ 47. Polyhedra like the pyramid in § 46 are represented by the projections of their edges and vertices. But solids bounded by curved surfaces, or surfaces themselves, cannot be thus represented.
For a surface we may use, as in case of the plane, its traces—that is, the curves in which it cuts the planes of projection. We may also project points and curves on the surface. A ray cuts the surface generally in more than one point; hence it will happen that some of the rays touch the surface, if two of these points coincide. The points of contact of these rays will form some curve on the surface and this will appear from the centre of projection as the boundary of the surface or of part of the surface. The outlines of all surfaces of solids which we see about us are formed by the points at which rays through our eye touch the surface. The projections of these contours are therefore best adapted to give an idea of the shape of a surface.
Thus the tangents drawn from any finite centre to a sphere form a right circular cone, and this will be cut by any plane in a conic. It is often called the projection of a sphere, but it is better called the contour-line of the sphere, as it is the boundary of the projections of all points on the sphere.
If the centre is at infinity the tangent cone becomes a right circular cylinder touching the sphere along a great circle, and if the projection is, as in our case, orthographic, then the section of this cone by a plane of projection will be a circle equal to the great circle of the sphere. We get such a circle in the plan and another in the elevation, their centres being plan and elevation of the centre of the sphere.
Similarly the rays touching a cone of the second order will lie in two planes which pass through the vertex of the cone, the contour-line of the projection of the cone consists therefore of two lines meeting in the projection of the vertex. These may, however, be invisible if no real tangent rays can be drawn from the centre of projection; and this happens when the ray projecting the centre of the vertex lies within the cone. In this case the traces of the cone are of importance. Thus in representing a cone of revolution with a vertical axis we get in the plan a circular trace of the surface whose centre is the plan of the vertex of the cone, and in the elevation the contour, consisting of a pair of lines intersecting in the elevation of the vertex of the cone. The circle in the plan and the pair of lines in the elevation do not determine the surface, for an infinite number of surfaces might be conceived which pass through the circular trace and touch two planes through the contour lines in the vertical plane. The surface becomes only completely defined if we write down to the figure that it shall represent a cone. The same holds for all surfaces. Even a plane is fully represented by its traces only under the silent understanding that the traces are those of a plane.
§ 48. Some of the simpler problems connected with the representation of surfaces are the determination of plane sections and of the curves of intersection of two such surfaces. The former is constantly used in nearly all problems concerning surfaces. Its solution depends of course on the nature of the surface.
To determine the curve of intersection of two surfaces, we take a plane and determine its section with each of the two surfaces, rabatting this plane if necessary. This gives two curves which lie in the same plane and whose intersections will give us points on both surfaces. It must here be remembered that two curves in space do not necessarily intersect, hence that the points in which their projections intersect are not necessarily the projections of points common to the two curves. This will, however, be the case if the two curves lie in a common plane. By taking then a number of plane sections of the surfaces we can get as many points on their curve of intersection as we like. These planes have, of course, to be selected in such a way that the sections are curves as simple as the case permits of, and such that they can be easily and accurately drawn. Thus when possible the sections should be straight lines or circles. This not only saves time in drawing but determines all points on the sections, and therefore also the points where the two curves meet, with equal accuracy.
§ 49. We give afew examples how these sections have to be selected.
A cone is cut by every plane through the vertex in lines, and if it is a cone of revolution by planes perpendicular to the axis in circles.
A cylinder is cut by every plane parallel to the axis in lines, and if it is a cylinder of revolution by planes perpendicular to the axis in circles.
A sphere is cut by every plane in a circle.
Hence in case of two cones situated anywhere in space we take sections through both vertices. These will cut both cones in lines. Similarly in case of two cylinders we may take sections parallel to the axis of both. In case of a sphere and a cone of revolution with vertical axis, horizontal sections will cut both surfaces in circles whose plans are circles and whose elevations are lines, whilst vertical sections through the vertex of the cone cut the latter in lines and the sphere in circles. To avoid drawing the projections of these circles, which would in general be ellipses, we rabatt the plane and then draw the circles in their real shape. And so on in other cases.
Special attention should in all cases be paid to those points in which the tangents to the projection of the curve of intersection are parallel or perpendicular to the axis x, or where these projections touch the contour of one of the surfaces.
