Entropy and Integral Geometry on Motion Spaces

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Entropy and Integral Geometry on Motion Spaces


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Entropy and Integral Geometry on Motion Spaces Gregory S. Chirikjian, Bernard Shiffman∗ Department of Mechanical Engineering ∗ Department of Mathematics Johns Hopkins University gregc@jhu.edu, shiffman@math.jhu.edu May 16, 2014 Abstract Let X denote n-dimensional Euclidean space, let G denote the group of orientation- preserving rigid-body motions of X, and let Γ be a crystallographic (discrete co-compact) subgroup of G. In classical integral geometry, the Principal Kinematic Formula com- putes the integral [1, 3, 4] ∫ G χ(A ∩ gB)dg = n∑ i=0 cinµi(A)µn−i(B) (1) where χ(·) denotes the Euler-Poincar´e characteristic, dg is the Haar measure for G, cin are known real constants, µi(·) is the ith invariant curvature measure of a body bounded by a smooth, closed, orientable surface, and A and B are arbitrary smooth bodies in X. Moreover, when the bodies are convex, χ(·) can be replaced with the set indicator function, i(·). Under mild constraints on the bodies, an analogous formula holds for computing the volume in G corresponding to one convex body, B, moving inside another, C, without their bounding surfaces colliding [6]. Both formulas can be used together to compute the change in entropy of a freely moving convex body, B, moving inside a body, C, when an obstacle, A, is introduced in such a way that body B can circumvent it at any orientation without becoming jammed. This is true because if V denotes the volume of free motion, then the probability of finding the moving body in a feasible region of configuration space is 1/V , and the entropy is then S = log V , and the change in entropy resulting from the presence of an obstacle is ∆S = log V1(B, C) − log V2(A, B, C). We seek a formula analogous to (1) for the coset space Γ\G as a way to compute the entropy associated with all configurations of bodies arranged with crystallographic symmetry. This coset space is a smooth manifold of dimension n(n + 1)/2, but because Γ is not normal in G, Γ\G is not a group. Γ\G is called a motion space [2]. Though it is not a group, a mapping π : (Γ\G) × X → X can be defined for each Γg ∈ Γ\G and x ∈ X as π[Γg, x] . = (Γg)x = ∪ γ∈Γ γ · (g · x). We derive a formula for the volume in Γ\G corresponding to arrangements such that π[Γg, B] is a collision-free configuration of bodies. That is, we derive a closed-form 1 expression analogous to (1) for the integral V1 = ∫ FΓ\G i  (g · B) ∩ ∪ γ∈Γ−e (γ ◦ g) · B   dg where FΓ\G is a fundamental domain for the action of Γ on G, V1 is the volume in Γ\G corresponding to bodies in collision, and V2 = V (FΓ\G) − V1 is the volume of the free space. S(Γ, B) = log V2 is the entropy of all possible collision-free crystallographic arrangements of copies of body B with symmetry group Γ. Moreover, we use the derived formula to examine changes in entropy as a function of the shape and size of the body B, and we point toward applications in the emerging field of “geometric frustration” [5]. References [1] Chern, S.-S., “On the Kinematic Formula in the Euclidean Space of N Dimensions,” American Journal of Mathematics, Vol. 74, No. 1 (Jan., 1952), pp. 227-236. [2] Chirikjian, G.S., Yan, Y., “Mathematical Aspects of Molecular Replacement: II. Geometry of Motion Spaces,” Acta. Cryst. A (2012). [3] Klain, D.A., Rota, G.-C., Introduction to Geometric Probability, Cambridge Uni- versity Press, 1997. [4] Schneider, R., Weil, W., Stochastic and Integral Geometry, Springer-Verlag, Berlin, 2008. [5] Sadoc, J.-F., Mosseri, R., Geometrical Frustration, Cambridge University Press, 2006. [6] Yan, Y., Chirikjian, G.S., “Closed-form characterization of the Minkowski sum and difference of two ellipsoids,” Geometriae Dedicata (to appear) 2