The Similarity Between Shapes Under Affine Transformation

J Hong

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Then we need — steps.
In fact, we can make use of the property |^(ri)-(r2) |^Z, |ri — r2 1 still further. As illustrated in Fig. 2, we draw line S of slope —L through point (a, ^(a)) and a line T through point (fe, (fe)). These two lines intersect at point (u, v). The Lipsiz condition guarantees that if a^x^b then (x)^v. It is easy to show that v=(^(a) + ^(b))/2-Lib-a)/2, therefore ^(x)^i^{a) + ^(b))/2-Lib-a)/2 '^(X) Based on this fact, we will try to skip as many computations as we can.
N...ow we divide our computation into phases. At phase i, we compute (r) for all values of 2A:, ir. It, = 1/2', 3/2', 5/2'' (2'-l)/2'.
In other words.
Phase 0: Jto=l; Phase 1; iti=l/2; Phase 2: it2 = l/4, 3/4; Phase 3: A3 = 1 /8, 3/8, 5/8. 7/«.
Assume that we have finished phase i and m is the minimum value of $ that we have ever found. By the reason mentioned above, we can easily see that if an inter- val [;72', (; + l)/2'] has the property that (cI)(;72') + :/n Then we can guarantee that no points in this internal can have a lower value than m, therefore we can remove all these kinds of intervals.

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