Repository: UIHistories Project: Mathematical Models by Arnold Emch - Series 4 (1928) [PAGE 30]

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D. MATHEMATICAL M O D E L S

46. Desargues Theorem. If two triangles A B C and A'B' C correspond to each other such that the joins of corresponding points A, A'; B, B''; C, Cf pass through a point S, then the points of intersection Ai, Bi, Ci of corresponding sides B C , B ' C ; CA, C'A'; A B , A'B' are collinear on a line s. Conversely when two trilaterals abc and a'b'c' correspond to each other such that the intersections of corresponding sides a, a ' '; b, b ' c, c' l e on a line s, then the joins a y b y c of corresponding '; i \ i \ vertices be, bV'; ca, c'a'; ab, a'bf are concurrent at a point S. The model illustrating this is mounted on an aluminum base on which sets the pyramid S A B C made of wood with white enamel paint. The intersecting plane, made of glass cuts the pyramid at A'B'C, and rests on the base along the line s. Projecting this from a generic point upon a generic plane, the theorem stated above results. 47. Perspective. This model illustrates relation between two planes 2 (x, y) and 2' (x'} y') in perspective, and its interpretation in one plane (2 and 2' superposed). T o the line at infinity q in 2 corresponds q' (vanishing line = horizon) in 2'. The line whose perspective r' is at infinity is denoted by r. The planes 2 and 2' intersect in the pointwise invariant line s. The lines s, r, q' are parallel. If the center of perspective is denoted by 0, then the distance from 0 to q' is equal to the distance of r from s, or oq' = rs. This is still true when 2 and 2' are superposed, and any such assumption is a necessary and sufficient condition for the data of a perspective. Fig. 20 shows the construction of the perspective K' of a circle K. Choose any line / (conveniently parallel to ^-axis) cutting K in A and B. Let / cut s at S. Draw line through 0 parallel to / , cutting q' at Q'. Then the join of S and Q' is the perspective I of ' / O A and O B cut /' at A' and B', two points of K'. As K inter. sects r in two points T\ and T2, with t and t as tangents, the per\ 2 spectives of h and t will be tangents t and t to K' at the infinite 2 x 2 points Ti and T2, i e , t and t are the asymptotes of K'. The pole .. x 2 M of r with respect to K is transformed into the center M ' of K'. Denoting the distances Oq' and q's by a and b, the analytic form of the perspective is easily found as

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