PROJECTION AND CROSSRATIOS
§ 12. If we join a point A to a point S, then the point where the line SA cuts a fixed plane ,r is called the projection of A on the plane a from S as centre of projection. If we have two planes it and Zr and a point S, we may project every point A in 7 to the other plane, If A' is the projection of A, then A is also the projection of A', so that the relations are reciprocal. To every figure in ,r we get as its projection a corresponding figure in .
We shall determine such properties of figures as remain true for the projection, and which are called projective properties. For this purpose it will be sufficient to consider at first only constructions in one plane.
Let us suppose we have given in a plane two lines p and p' and a centre S (fig. 4); we may then project the points in p from S to p',
Let A', B' .. be the projections of A, B . . ., the point at infinity in p which we shall denote by I will be projected into a finite point
single point in the line AB.
[Relations between segments of lines are interesting as showing an application of algebra to geometry. The genesis of such relations
•
C
I' in p', viz. into the point where the parallel to p through S cuts
Similarly one point J in p will be projected into the point JJ' at infinity in p'. This point J is of course the point where the parallel to p' through S cuts p. We thus see that every point in p is projected into a single point in p'.
Fig. 5 shows that a segment AB will be projected into a segment A'B' which is not equal to it, at least not as a rule; and also that the ratio AC: CB is not equal to the ratio A'C': C'B' formed by the projections. These ratios will become equal only if p and p' are parallel, for in this case the triangle SAB is similar to the triangle SA'B'. Between three points in a line and their pro
tctions there exists therefore in general no relation. But between four points a relation does exist. § i. Let A, B, C, D be four points in p, A', B', C', D' their projections in p', then the ratio of the two ratios AC:CB and AD:DB into which C and D divide the segment AB is equal to the corresponding expression between A', B', C', D'. In symbols we have
AC AD A'C' A'D'
CB DB = C'B' D'B' '
This is easily proved by aid of similar triangles.
Through the points A and B on p draw parallels to p', which cut the projecting rays in
s C2, D2, B2 and Al, Cl,
DI, as indicated in fig. 6. The two triangles
©s ACC2 and BCC' will be similar, as will also be the triangles ADD2 and BDDI.
The proof is left to A the reader.
This result of fundamental importance.
The expression A' c• d _s' AC/CB:AD/DB has been
called by Chasles the
" anharmonic ratio of the
four points A, B, C, D."
Professor Clifford pro
posed the shorter name of " crossratio." We shall adopt the latter. We have then the
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