§ I. Projection of Plane Figures.—Let us suppose we have in
space two planes rr and 1.-'. In the plane 1r a figure is given having
known properties; then we have the problem to find its projection
from some centre S to the plane n~', 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 rr we have to join it to the
centre S and find the point A' where this ray SA cuts the plane
1r'; it is the projection of A. On the other hand if A' is given in
the plane 1r', then A will be its projection in 1r. Hence if one
figure in 1r' is the projection of another in -rr, then conversely the latter
is also the projection of the former.
A point and its rejection are therefore also called corresponding points, and similarij/ we speak of corresponding lines and curves, &c. § 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 are projected by rays which lie in the plane determined by S and the line, and this plane cuts the plane 1r 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 projection 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 is 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. § 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 the are obtained. They are then still said to be one the projection ofy the other, or to be projective or homo graphic. 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 §§ I, 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 rr and rr' 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-Evcry line meets its projection on the axis. § 4. The property that the lines joining corres onding points all pass through a common point, that any pair oi) corresponding points and the centre are in a line, is also expressed by sayin that the figures are co-linear or co-polar; and the fact that both figures have a line, the axis, in common on which corres onding lines meet is expressed by saying that the figures are co-axaii The connexion between these properties has to be investigated. For this purpose we consider in the plane -rr 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 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' 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. If two triangles, whether in the same plane or not, are co-linear they are co-axal. Or-
f the lines AA', 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. of BC and B'C', of 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.-Let us first suppose the triangles to be in different places. By supposition the lines AA', BB', CC' (fig. I) meet in a point S. But three intersecting lines determine three planes, SCB, 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 ie in the plane of the triangle ABC ecause 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. This line (PQR in fig. I) is called the axis of perspective or homolo§ Y» and the intersection of AA', BB', CC, i.e. S in the figure, the centre of perspective. 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 triangles 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 Sl, and in EA, EB,
EC take points Al, Bl, C1 respectively, but so that Sl, Al, Bl, C1
are not in a plane. In the plane
ESA which projects the line SIA;
lie then the line SIA; and also
EA'; these will therefore meet in
a point A1', of which A' will be the projection. Similarly points Bt, C1' are found. Hence we have now in space two triangles A1B1C1 and A1'B1'C1' which are co-linear. They are therefore coaxal, that is, the points P1, Qi, Rl, where A1B1, &c., meet will lie in a line. Their projections therefore lie in a line. But these pre the points P, Q, R, which were to be proved to lie in a ine.
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 this 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 1r 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.
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 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 -rr, which is kept fixed, and whose plane is perpendicular to the axis.
The last part will be proved presently. As the plane -rr' may be turned about the axis in one or the opposite sense, there will be two perspective positions ossible when the planes coincide. § 6. Let (fig. 2) rr, rr' be the planes intersecting in the axiss 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 1r'. This gives Z
the point A'. But if we draw
through S any line parallel
to rr, then this line will cut
-rr' in some point I', and if
all lines through S be drawn A
which are parallel to -rr these E s
§
S
R A
v r”
{ '*
FIG. 1. B
will form a plane parallel to
rr which will cut the plane
rr' in a line i' parallel to the
axis s.. If we say that a line
parallel to a plane cuts the
after at an infinite distance,
we may say that all points
at an infinite distance in rr
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 rr, whilst all other points are projected to finite points. We say therefore that all points in the plane -rr 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 (cf. G. §§ 2-4). Similarly there is a line j in 1r which is projected to infinity in 1r'; this projection will be denoted by j' so that i and j' 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 SI parallel to -lr and S] parallel to 1r', then the lines through I' and ] lY
A
Fig. 2.
