§ 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...