On global equilibria of finely discretized curves and surfaces
aa r X i v : . [ m a t h . DG ] O c t ON GLOBAL EQUILIBRIA OF FINELY DISCRETIZED CURVES ANDSURFACES
G ´ABOR DOMOKOS AND ZSOLT L ´ANGI
Abstract.
In our earlier work [7] we identified the types and numbers of static equilib-rium points of solids arising from fine, equidistant n -discretrizations of smooth, convexsurfaces. We showed that such discretizations carry equilibrium points on two scales:the local scale corresponds to the discretization, the global scale to the original, smoothsurface. In [7] we showed that as n approaches infinity, the number of local equilibriafluctuate around specific values which we call the imaginary equilibrium indices associ-ated with the approximated smooth surface. Here we show how the number of globalequilibria can be interpreted, defined and computed on such discretizations. Our resultsare relevant from the point of view of natural pebble surfaces, they admit a comparisonbetween field data based on hand measurements and laboratory data based on 3D scans. Introduction
Static equilibria of convex bodies correspond to the singularities of the gradient vectorfield characterizing their surface. The study of equilibria of rigid bodies is a classic chapterof mathematics and mechanics; initiated by Archimedes [1], the theory was revived inmodern times by the works of Cayley [3] and Maxwell [12] yielding results on the globalnumber of stationary points. Further generalization by Poincar´e and Hopf led to thePoincar´e-Hopf Theorem [2] on topological invariants. If applied to generic, convex bodies,represented by gradient fields defined on the sphere, this theorem states that the number S of ‘sinks’ (stable equilibria), the number U of ‘sources’ (unstable equilibria) and thenumber N of saddles always satisfy the equation(1) S + U − N = 2 . This formula, the so-called Poincar´e-Hopf formula can be regarded as a generalization ofthe well-known Euler’s formula [10] for convex polyhedra.We also mention results on polyhedra; monostatic polyhedra (i.e. polyhedra with just S = 1 stable equilibrium point) have been studied in [11], [4],[5] and [6] and more recentlyin [14].The total number T of equlibria ( T = S + U + N ) has also been in the focus ofresearch. In planar, homogeneous, convex bodies (rolling along their boundary on ahorizontal support), we have T ≥ T = 2 exist in the three-dimensional space (cf. [15]). Zamfirescu [16] showed that for typical convex bodies, T is infinite, suggesting that equilibria in abundant numbers mayoccur in physically relevant scenarios.Natural pebbles exhibit similar behavior: their convex hull is a multi-faceted polyhe-dron P carrying many static equilibria [9] appearing in strongly localized flocks . In [7] we Mathematics Subject Classification.
Key words and phrases. equilibrium, convex surface, Poincar-Hopf formula, polyhedral approximation.The authors gratefully acknowledge the support of the J´anos Bolyai Research Scholarship of theHungarian Academy of Sciences and support from OTKA grant 104601. studied this phenomenon and showed that if P is defined as a sufficiently dense, equidis-tant discretization of a smooth surface M then flocks are indeed strongly localized in thevicinity of (isolated) equilibria of M and the number, type and geometrical arrangementof equilibrium points inside any single flock can be expressed by the principal curvaturesand the distance to the center of gravity of M .Here we make one further step: we give definitions and prove statements based onwhich, for polyhedra P defined by sufficiently dense discretizations, the number andtype of maxima and minima of M can be determined and the number of saddles maybe obtained via equation (1). These results, defining the number of flocks on a densediscretization P corresponding to a smooth surface M , may help to bridge the gap betweenhand experiments on pebbles, identifying the equilibria of M and 3D computer scans,identifying the equilibria of P [9]. We formulate our results for general functions in twovariables, however, all results are valid for convex surfaces interpreted as the distancefunction measured form the center of gravity. We also formulate the results for functionsin one variable (2D convex shapes) where the statements are rather simple.2. Main results
We start with some preliminary assumptions and definitions.Let f : [0 , a ] × [0 , b ] → R be a C -class function. Consider a division D n of the rectangle D = [0 , a ] × [0 , b ] into n × n congruent rectangles. We call the vertices of these rectangles grid vertices , and denote the grid vertex (cid:0) in a, jn b (cid:1) by p i,j . The neighbors of the grid vertex p i,j are the four grid vertices p i ± ,j and p i,j ± . The two pairs p i ± ,j and p i,j ± are called opposite neighbors of p i,j .Recall that a point p ∈ R is called a stationary point of f , if f ′ x ( p ) = f ′ y ( p ) = 0. Definition 1.
A grid vertex p is stationary , if for any opposite pair { q, q ′ } of its neighbors, f ( p ) ≥ max { f ( q ) , f ( q ′ ) } or f ( p ) ≤ min { f ( q ) , f ( q ′ ) } is satisfied. If p i,j is a grid vertex, then the grid circle of centre p i,j and radius r is the set C r ( p i,j ) = { p l,m : max {| l − i | , | m − j |} ≤ r } . During the consideration, we assume that f has finitely many stationary points, eachin the interior of the domain D , and the determinant of the Hessian of f at each of themis not zero. We assume that the grids we use are nondegenerate; more specifically, that if p = p ′ are two grid vertices, then f ( p ) = f ( p ′ ). Theorem 1.
Let p = ( x , y ) ∈ int D . (1) If p is not a stationary point of f , then p has a neighborhood U ⊂ D such that forany n ≥ , if the grid vertex p i,j of D n , and each of its neighbors, is contained in U , then p i,j is not a stationary grid vertex. (2) If p is a local minimum of f , then p has a neighborhood U and some suitable valueof r such that for every sufficiently large n , there is exactly one grid vertex p i,j of D n in U , which is minimal within its grid circle C r ( p i,j ) . (3) If p is a local maximum of f , then p has a neighborhood U and some suitable valueof r such that for every sufficiently large n , there is exactly one grid vertex p i,j of D n in U , which is maximal within its grid circle C r ( p i,j ) . (4) If p is a saddle point of f , then p has a neighborhood of and some suitable value of r such that for every sufficiently large n , any grid vertex p i,j of D n in U is neithera local maximum, nor a local minimum within its grid circle C r ( p i,j ) . N GLOBAL EQUILIBRIA 3
Proof.
First, we prove (1). Let L be the line through the origin, perpendicular to grad f ( p ).Note that the derivative of f is zero in this direction. Let ε > A be the union of the lines, through p , the angles of which with L is not greater than ε .Note that by the continuity of grad f , p has a neighborhood U such that for any q ∈ U ,grad f ( q ) is perpendicular to some line in A . This implies that if q ∈ U , and A containsno line parallel to the vector u , then f ′ u ( q ) = 0. Without loss of generality, we may assumethat U is a Euclidean disk in R .Now, consider any division D n , and assume that the grid vertex p i,j and all its neighborsare contained in U . Since ε > x -axis, or the y -axis is not parallelto any line in A . Without loss of generality, let the x -axis have this property. We showthat the sequence f ( p i − ,j ) , f ( p i,j ) and f ( p i +1 ,j ) is strictly monotonous. Indeed, if, forexample, f ( p i,j ) ≥ max { f ( p i − ,j ) , f ( p i +1 ,j ) } , then by the Lagrange Theorem, for some q , q ∈ U , we have f ′ x ( q ) ≤ ≤ f ′ x ( q ), which, by the continuity of f x , yields that forsome q ∈ U , we have f ′ x ( q ) = 0. Nevertheless, it contradicts the definition of A . If f ( p i,j ) ≤ min { f ( p i − ,j ) , f ( p i +1 ,j ) } , we can reach a contradiction in a similar way. Thus, p i,j is not a stationary grid vertex.In the next part, we prove (2). Without loss of generality, assume that f ( p ) = 0. Notethat since p is a local minimum, both eigenvalues λ ≤ λ of the Hessian of f at p arepositive. Let P denote the second order Taylor polynomial of f around p . Then P is aquadratic form with eigenvalues λ > λ >
0, and the curve P = 1 is an ellipse.Now, since f is C -class, there is some ¯ L ∈ R such that for every ( x, y ) ∈ D , we have | f ( x, y ) − P ( x, y ) | < ¯ L √ (cid:0) | x | + x | y | + | x | y + | y | (cid:1) = ¯ L √ | x | + | y | ) (cid:0) x + y (cid:1) ≤ L (cid:0) x + y (cid:1) / , which yields that for some suitable L ∈ R , we have | f ( q ) − P ( q ) | ≤ ( P ( q )) / for every q ∈ D .Let ε > U of p such that • for every q ∈ U , we have f ( q ) >
0, and | f ( q ) − P ( q ) | < εP ( q ), • f is convex in U .Observe that the second condition holds for any convex neighborhood of p , where theHessian of f has only positive eigenvalues, and, the existence of such a neighborhoodfollows from the fact that f is C -class. Now, since P ( q ) is homogeneous, every point q ∈ D , with f ( q ) = α , is contained between the ellipses P ( q ) = (1 − ε ) α and P ( q ) = (1 + ε ) α .Note that if ε is sufficiently small, for any value of α and any point q of the level curve f ( x, y ) = α , the angle between the two tangent lines of the ellipse P ( x, y ) = (1 − ε ) α ,passing through q , is at least π .Fix any division D n , and consider the level curves f ( x, y ) = α , as α ≥ p be the first grid vertex that reaches the boundary of such a curve. Clearly, f (¯ p isminimal among all the grid vertices in U . Let(2) r ≥ max ( √ a + b { a, b } , λ λ √ a + b min { a, b } r ε − ε ) , where τ = λ λ ≥ f at p . In theremaining part of the proof of (2), we show that there is no other grid vertex in U whichis minimal within its grid circle of radius r .Assume, for contradiction, that the grid vertex q is minimal within C r ( q ), and let f ( q ) = β . Then the level curve f ( x, y ) = β already contains some grid vertex q ′ in its G. DOMOKOS AND Z. L ´ANGI interior. Note that the semi-axes of the ellipse P ( x, y ) = t are of length q tλ i , where i = 1 ,
2. Recall that the curve f ( x, y ) = β is contained in the ellipse P ( x, y ) = (1 + ε ) β ,and the diameter of the latter curve is 2 q ε ) βλ . Since, according to our assumption, q ′ is contained in the interior of P ( x, y ) = (1 + ε ) β , and f ( q ′ ) < f ( q ), we obtain that(3) rδ < s ε ) βλ , where δ = min (cid:8) an , bn (cid:9) denotes the minimal distance between any two grid vertices.Let w be the point of P ( x, y ) = (1 − ε ) β closest to q . Let ∆ = √ a + b n = √ a + b min { a,b } δ , andobserve that any circle of diameter ∆ contains a grid vertex. We show that the circle C of diameter ∆, touching the ellipse P ( x, y ) = (1 − ε ) β at w from inside, is contained inthe ellipse. By Blaschke’s Rolling Ball Theorem, to do this it suffices to show that ∆2 isnot greater than any radius of curvature of the ellipse. It is a well-known fact that theradius of curvature at any point of an ellipse with semi-axes M ≥ m is at least m M and atmost M m . Thus, a simple computation yields that what we need to show is(4) ∆ ≤ p − ε ) βλ λ . To show (4), we can combine (3) with the definition of r in (2).Let ¯ C be the circle of radius ∆ that touches the tangent lines of the ellipse P ( x, y ) =(1 − ε ) β through q . Since f is convex in U , the level curve f ( x, y ) = β is also convex,and thus, this circle is also contained inside the level curve f ( x, y ) = β . On the otherhand, ¯ C as any other circle of diameter ∆, contains a grid vertex q ′′ . Then, our previousobservation yields that f ( q ′′ ) < β = f ( q ). To finish the proof, we show that ¯ C is containedin the circle of radius rδ , centered at q , which implies that q ′′ is contained in the gridcircle of radius r , centered at q .Assume, for contradiction, that it is not so. Let φ be the angle between the twotangent lines of the ellipse P ( x, y ) = (1 − ε ) β , through q . Since the angle betweenthese two tangent lines is at least π , a simple computation yields that the distance ofthe centre of ¯ C and q is at most ∆, and hence no point of ¯ C is farther from q than ∆ = √ a + b { a,b } δ ≤ rδ , which finishes the proof of (2).To prove (3), we can apply (2) for the function − f .Finally, we prove (4). Let f ( p ) = 0. Then, in a neighborhood U of q , the set { f ( q ) = 0 } , q ∈ U can be decomposed into the union of two C -class curves, crossing each other at q ,and for any α = 0, the set { f ( q ) = α } , q ∈ U is the union of two disjoint, C -class curves.Furthermore, if U is sufficiently small, there is some sufficiently small φ > ε > q ∈ U • there is a closed angular domain A with apex q and angle φ such that for anypoint q ′ ∈ A with 0 < | q ′ − q | < ε , we have f ( q ) < f ( q ′ ); • there is a closed angular domain B with apex q and angle φ such that for anypoint q ′ ∈ B with 0 < | q ′ − q | < ε , we have f ( q ) > f ( q ′ ).Clearly, for a sufficiently large r (chosen independently of q ), any such closed angulardomain in U contains a vertex of C r ( q ), which yields the assertion. ✷ N GLOBAL EQUILIBRIA 5
Theorem 2. Le f have s local minima and u local maxima. Then there is some r suchthat for any sufficiently large n , exactly s grid vertices of D n are minimal, and exactly u grid vertices of D n are maximal within their grid circles of radius r .Proof. Fix some r such that any stationary point q of f has some neighborhood thatsatisfies the corresponding conditions in (2), (3) or (4) of Theorem 1. Observe that wecan choose ε , ε > • if q is a stationary point, the assertion in (2), (3) or (4) Theorem 1 holds in the ε -neighborhood U q of q ; • if q is not a stationary point, and its distance from any stationary point is at least ε , then (1) holds in the ε -neighborhood U q of q .Now, let n be large enough such that for any point q ∈ D , C r ( q ) ⊂ U ( q ), and for anystationary point, (2), (3) and (4) can be applied, and then, the theorem follows. ✷ Remark 1.
Note that we may apply the following theorem for a parametrized convexsurface r = r ( u, v ) , with x, y as ( u, v ) , and z = f ( x, y ) as the distance function k r ( u, v ) k . Note that the one-dimensional case of the problem is straightforward. More specifically,the following holds.
Remark 2.
Let f : [ a, b ] → R be a C -class function with finitely many stationary points,each in the interval ( a, b ) , such that the second derivative of f at each such point isnot zero. Let x , x , . . . , x k denote the local minima, and x ′ , x ′ , . . . , x ′ l denote the localmaxima of f . Let P = { p , p , . . . , p n − } denote the set of the vertices of the equidistant n -element partition of [ a, b ] , contained in ( a, b ) , and assume that for any i = 0 , , . . . , n − , f ( p i ) = f ( p i +1 ) . Then, if n is sufficiently large, there are exactly k local minima, and l local maxima in P . Applications
Our results may help to relate hand experiments on pebbles to the results of 3D com-puter scans. The latter identify the exact convex hull as a multi-faceted polyhedron (oftenwith several thousand faces) and locate the equilibrium points of this polyhedron. As pre-dicted by [7], these appear in flocks and the number and type of equilibria observed inside each flock appears to be well approximated by the numbers predicted in [7]. Hand ex-periments, on the other hand, tend to identify each flock as one single equilibrium point,associated with an (imaginary) smooth surface. In [9] we introduced a ‘fudge’ parame-ter µ describing the uncertainty of hand experiments, µ = 0 corresponding to the exactmeasurement which is identical to the computer output.When plotting the number T of equilibria versus µ we observed that after a steep initialdrop, the plot has a long plateau extending often over several orders of magnitude of µ and the function value T ⋆ of this plateau we associated with the number of equilibriaof the (imaginary) smooth surface. Our current note gives an independent definition forthis number. Comparing those values may not only shed light on the applicability of our‘fudge’ parameter but also may significantly contribute to the evaluation of geologicalfield experiments. References [1] T. I. Heath (ed.)
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Int. J. Comput. Geom. Appl., (2014) 39[15] Varkonyi, P.L., Domokos G., Static equilibria of rigid bodies: dice, pebbles and the Poincar´e-HopfTheorem , J. Nonlinear Science (2006), 255-281.[16] Zamfirescu, T., How do convex bodies sit? , Mathematica (1995), 179-181. G´abor Domokos, Dept. of Mechanics, Materials and Structures, Budapest Universityof Technology, M˝uegyetem rakpart 1-3., Budapest, Hungary, 1111
E-mail address : [email protected] Zsolt L´angi, Dept. of Geometry, Budapest University of Technology and Economics,Budapest, Egry J´ozsef u. 1., Hungary, 1111
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