By E. Hinton, et al.,

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3. Algorithm CIRCULAR and its worst-case analysis We will assume that \S\\ > 0 and IS2I > 0· We also assume that all the data points of S χ \^j S 2 do not lie on a straight line. If it is so [see Fig. 2(c)], we can determine S(S i,S 2 ), if exists, in 0(|Si| + |S 2 l)time. The step by step description of the algorithm CIRCULAR to compute S(5i^2) is described below. Input: Sets 51 and 5 2. Output: The boundary of S(51,^2) as a convex polygon. Begin Step 1: Determine a point χ0 ε S(S\yS2). If no such point exists then stop.

The icosahedron and the dodecahedron have the same group P, of size 60. If D is a bounded set in three dimensions, then GD may be a subgroup of Γ, W or P, or it consists of rotations about a line through the centroid of D and reflections in planes through the centroid. Again, the size of GD is bounded by a linear function of the size of D, and GD has a generating set containing at most one rotation, and at most one reflection. Even if D is unbounded, GD has a rather simple structure; see [7]. In Section 2 and 3 respectively, algorithms for two and three dimensional symmetry problems are described.

I-l. Now let a, be defined as in (1) above. , (rn ,αΠ )) is a one-to-one representation of D (up to congruence). A rotation on D induces a rotation on L (D ) and so rotational symmetries can be found by the Knuth-Morris-Pratt algorithm as described above. As for polygons, a reflection on D induces a reversal followed by a rotation. But here we need a little more care because the order of points with the same polar angle is not reversed. A run in L(D) is a maximal sublist with the same polar angle, that is, a sublist (&7 »<*/)» (^+ιΑ+ι)» ···» (rj>aj)) where a,·, a i + 1 ,...

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