This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1910 Excerpt: ...x1 = 0; the point Jl by the fact that x2 = 0; and the point Jf by the fact that x1 = x3. Theorem 15. In homogeneous coordinates a projectivity on a line is represented by a linear homogeneous transformation in two variables, (1) K--+ fcq1, (ad-bc+0) px2 = cx1 + dx3, where p is an arbitrary factor of proportionality. (A ...
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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1910 Excerpt: ...x1 = 0; the point Jl by the fact that x2 = 0; and the point Jf by the fact that x1 = x3. Theorem 15. In homogeneous coordinates a projectivity on a line is represented by a linear homogeneous transformation in two variables, (1) K--+ fcq1, (ad-bc+0) px2 = cx1 + dx3, where p is an arbitrary factor of proportionality. (A, E, P) Proof. By division, this clearly leads to the transformation (2) rf- - w cx + d provided x2 and x2 are both different from 0. If x3 = 0, the transformation (1) gives the point (x/, x2') = (a, c); i.e. the point H. = (1, 0) is transformed by (1) into the point whose nonhomogeneous coordinate is a/c. And if x2'=0, we have in (1) (xv x2) = (d, --c); i.e. (1) transforms the point whose nonhomogeneous coordinate is--d/c into the point R. By reference to Theorem 11 the validity of the theorem is therefore established. As before, the matrix (, J of the coefficients may conveniently be used to represent the projectivity. The double points of the projectivity, if existent, are obtained in homogeneous coordinates as follows: The coordinates of a double point (xv x3) must satisfy the equations px--ax1 + fo px3 = cx1 + dxr These equations are compatible only if the determinant of the system 3) (a-p)x1+bx2=0, K' cx1 + (d-p)x2=0, vanishes. This leads to the equation a--p b c d--p for the determination of the factor of proportionality p. This equation is called the characteristic equation of the matrix representing the projectivity. Every value of p satisfying this equation then leads to a double point when substituted in one of the equations (3); viz., if p1 be a solution of the characteristic equation, the point (xv x3) = (-b, a-P1) = (d-pv-c) is a double point. In homogeneous coordinates the cross ratio lb(AB, CD) of four points A = («.
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