# The Advanced Geometry of Plane Curves and Their Applications by C. Zwikker

By C. Zwikker

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Additional info for The Advanced Geometry of Plane Curves and Their Applications (Dover Phoenix Editions)

Example text

CD =1-15. BD~ in words: The product of the diagonals of a quadrilateral inscribed in a circle equals the sum of the products of the opposite sides. 4. Conformal transforms, inversion. Bya transformation W= f(z), where wand Z are complex values, one or more points of the complex w-plane correspond with one or more points of the z-plane. Let, furthermore, the function f be of such a nature, that the derivative dwldz is a single-valued function of z, by which we mean that it is independent of the direction of dz.

We first construct the line Za+Z4U which is , , " , Fig. 39 showR in the same way the construction of (1): III. 44 The straight line Now move one of the here mentioned uniform scales until both are parallel and at the same time perspective and the chain of operations leading from (1) to (2) is closed. e... Perspective projection leaves certain properties intact; these are called projective properties, other properties may be lost in the projection; as opposed to the projective properties they are called metric properties.

E is. ~nvaria'[lt. To prove this, take four points U 1U 2U aU 4 on the line (2) . and calculate the cross ratio. We find: D= Z' (u1)-z' (ua) --=-- z' (u 2)-z' (ua) = u1-ua -,- U2-Ua z' (u 1) - z' (u 4) . Z' (u 2)- z' (u4) U1- u 4 . U2- U4 and this is entirely independent of za' Z4' m and n. ). o Fig 40 The cross ratio is, so to say, picked up by a projecting pencil of rays and transferred to a second line. We expect that it 6. Projective geometry 45 can be expressed in the angles of the pencil without the help of a secant.