794 PROJECTION $ 2. We at once get the following properties : The projection of a point is a point, and one point only. The projection of a line, (straight line] is a line ;. for all points in a line arc projectc I by rays which lie in the plane determined by S and the line, and this plane cuts the plane IT in a line which is the projection of tin* 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 inns the projections A , !> of the points A, B. For the projecting piano of the line AB contains the rays SA, SB which project the points A, B. T/ie prycction of the point of intersection of two lines a, I is the point of intersection of the projections a , V of those lines. Similarly we get The projection of a curve icill be a curve. Thf. 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 ob tained 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 homographic. 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 tvo planes ir and IT are perspective, then their line of inter section 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 Ecery 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 ir a triangle ABC, and let the lines BO, 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 AA , BB , CC meet in a point, viz., at S, and the points of intersection of a and a , b and b , c and c 1 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. DESARGTTE S THEOREM. If two triangles, whether in the same plane or not, are co-linear tJicy 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 arc co-axal they arc co-linear. Or If the intersection of the sides of two triangles ABC and A B C , viz., of KG and B C , o/CA and C A , <ml of AB and A B , lie in a line, tlicn the lines A A , BB , and CC meet in a point. Proof. Let 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 SAP,. In tlie first lie the points B, C ami also B , C". Hence the lines BC ami B C will intersect at some point P, because any two lines in the same plane in ersect. Similarly CA and C A will intersect at some point <), and AB anil A B at some point R. These points P, Q, R lie in the pi me of the triangle ABC because they are points on the sides of this triangle, and similarly in ill- 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 tiia igles 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 Sj, and in EA, EB, KC take points Aj, B^ Cj respectively, but so that Sj, Aj, BJ, Cj are not in a plane. In the plane ESA which projects the line SjAj lie then the line SjA a and also EA ; these will therefore meet in a point A, of which A will be the projec tion. Similarly points B j, C are found. Hence we have now in space two triangles A 1 6 1 C 1 and A jB jC j which are co-linear. They are therefore co-axal, that is, the points Pj, Qj, R 1( where AjB 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 Dcsargue s theorem we can now prove a funda mental 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 ir 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 arc perspective, then if the one plane be turned about the axis through any angle, especially if the one plane be turned till it coincides ivith the other, the two planes will remain perspective ; corresponding lines will 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 ir, which is kept fixed, and whose plant is perpendicular to the axis. The last part will be proved presently. As the plane ir 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) IT, TT be the planes intersecting in the axis s whilst S is the centre of projection. To project a point A in ir we join A to S and see where this line cuts ir . This gives the point A . But if we draw through S any line parallel to TT, then this line will cut ir in some point I , and if all lines through S lie drawn which are parallel to ir these will form a plane parallel to ir which will cut the plane ir 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 w 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 ir, whilst all other points are projected to finite points. We say therefore that all points in the plane ?r 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 ir whieh is projected to infinity in ir ; this projection shall be denoted by/ so that i and/ are Hues 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 SI parallel to ir and SJ parallel to ir , then the lines through I and J parallel to the axis will be the lines i and,/. At the same time a parallelogram SJTI S has been formed. If now the plane ir be turned about the axis, then the points I 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 ir 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. 8. The plane ir may be turned either in the sense, indicated by the arrow at Z or in the opposite sense till ir falls into ir. In the first case we get a figure like fig. 3 ; i and ./ will be on the same side of the axis, and on this side will also lie the centre S ;
J.