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PROJECTION AND
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See also:PROJECTION AND See also:CROSS-RATIOS § 12. If we join a point A to a point S, then the point where the See also:line SA cuts a fixed See also: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 See also:algebra to See also:geometry. The See also: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 See also: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 See also:case the triangle SAB is similar to the triangle SA'B'. Between three points in a line and their See also:pro- tctions there exists therefore in See also: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 See also: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 See also: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 See also:proof is See also:left to A the reader. This result of fundamental importance. The expression A' c• d _s' AC/CB:AD/DB has been called by See also:Chasles the " anharmonic ratio of the four points A, B, C, D." See also:Professor See also:Clifford pro- posed the shorter name of " cross-ratio." We shall adopt the latter. Additional information and CommentsThere are no comments yet for this article.
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