By By (author) Z. A. Melzak
A two-volume remedy in one binding, this supplementary textual content stresses intuitive attraction and ingenuity. It employs actual analogies, encourages challenge formula, and offers problem-solving equipment. 1973 and 1976 versions.
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Additional info for Companion to Concrete Mathematics: Vol. I: Mathematical Techniques and Various Applications
C) 32 GEOMETRY corner of the square, continuity argument illustrated in Figure 5c suggests that the ray be required to turn back on its path. Consider an arbitrary point P(x, y) in S; then there are four infinite sequences of the multiple images of P under reflections (x + 2n, y + 2m), (2 - x + 2n, y + 2m), (x + 2n, 2 - y + 2m), (2-x+2n,2-y+2m); (1) here m and n range independently over all integers. The main point of using the reflections is that the images of the segments constituting the path form a straight line L.
It will be noticed that the first problem is a special case of the third one, and that all three problems are similar except that in the second one we have a non-Euclidean space, the sphere, and its group of motions, instead of Euclidean cases. Since the measure such as m 3 (A, B) is defined· in terms of motion of one set A relative to another set B, this measure is also called a kinematic measure, and a formula which expresses it in terms of A and B is also called a kinematic formula. To motivate the insistence on the invariance of our measures we consider the so-called Bertrand paradox.
INTEGRAL GEOMETRY 39 gruent to C 1 under a rigid motion. Or, we ask about the measure m 2 (C) of all great circles on a sphere which cut a specified set C, asking additionally that m2(C) == m 2(C1 ) if C and C1 are congruent under a rotation of the sphere. Again, we ask for the n1easure m 3 (A, B) of all positions of an n-dimensional set A in En such that it intersects another such set B which is fixed; here we demand the invariance of m 3 (A, B) under all rigid motions applied to B. It will be noticed that the first problem is a special case of the third one, and that all three problems are similar except that in the second one we have a non-Euclidean space, the sphere, and its group of motions, instead of Euclidean cases.
Companion to Concrete Mathematics: Vol. I: Mathematical Techniques and Various Applications by By (author) Z. A. Melzak