Asymptotic approximation of convex curves

Ludwig, Monika

doi:10.1007/bf01189576

Cited by 15 publications

(14 citation statements)

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“…Given a convex curve γ , one wants to approximate it by circumscribed polygons using area as the distance between the curve and an approximating polygon; this problem makes sense in the Euclidean, hyperbolic and spherical geometries. Formula (2) provides an asymptotic expansion for the distance between γ and its best approximating n-gon; in the Euclidean case, the term a 1 was found by Fejes Toth [35,36] (see [20] for a complete proof, [18] for the value of the term a 2 and [7,8] for surveys on approximating convex bodies by polytops). The approach via interpolating Hamiltonians provides a novel view point in the approximation theory of smooth convex curves by polygons.…”

Section: Remarkmentioning

confidence: 99%

Dual billiards in the hyperbolic plane*

Tabachnikov

2002

Nonlinearity

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We study an area preserving map of the exterior of a smooth convex curve in the hyperbolic plane, defined by a natural geometrical construction and called the dual billiard map. We consider two problems: stability and the area spectrum. The dual billiard map is called stable if all its orbits are bounded. We show that both stable and unstable behaviours may occur. If the map at infinity has a hyperbolic periodic orbit, then the dual billiard map has orbits escaping to infinity. On the other extreme, if the map at infinity is smoothly conjugated to a Diophantine irrational rotation of the circle, then the dual billiard map is stable.The area spectrum is the set of extremal areas of n-gons, circumscribed about the dual billiard curve; this is to the dual billiard what the length spectrum is to the usual, inner, one. We show that the area spectrum has an asymptotic expansion in even negative powers of n as n → ∞. The first coefficient of this expansion is the area of the dual billiard curve, and the next is, up to a constant, the cubed integral of the cube root of its curvature. We describe the curves that are relative extrema of these two functionals and show that they are the trajectories of the pseudospherical pendulum with various gravity directions.

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Section: Remarkmentioning

confidence: 99%

Dual billiards in the hyperbolic plane*

Tabachnikov

2002

Nonlinearity

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“…The next proposition is a consequence of (8) (consider the cases x − p ≤ α and x − p ≥ α separately): (22) For any ξ > 0 there is > 0 such that for all sufficiently large n we have the following: for each p ∈ bd C and x ∈ H p with x − p ≥ /n…”

Section: Facets and Dirichlet-voronoi Cellsmentioning

confidence: 99%

“…For d = 2 Ludwig [22] specified the second term of an asymptotic series development of δ V (C, P c (n) ). The complete series was given by Tabachnikov [27] in the form of a result on periodic trajectories of the "dual" or "exterior" billiard determined by C.…”

Section: 2mentioning

confidence: 99%

“…/| bd C| and choose corresponding to ξ as in (22) and σ corresponding to as in (24). Propositions (26), (24) and (22) then show that for an infinite set of (sufficiently large) n the following holds: there is a measurable set M n in bd P n (a union of pieces of the facets of P n ) such that |M n | ≥ 3| bd P n |/4 and x − x π ≥ ξ/n 2/(d−1) for each x ∈ M n .…”

Section: Facets and Dirichlet-voronoi Cellsmentioning

confidence: 99%

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Error of Asymptotic Formulae for Volume Approximation of Convex Bodies in

Gruber

2002

Mh Math

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Abstract. We estimate the error of asymptotic formulae for volume approximation of sufficiently differentiable convex bodies by circumscribed convex polytopes as the number of facets tends to infinity. Similar estimates hold for approximation with inscribed and general polytopes and for vertices instead of facets. Our result is then applied to estimate the minimum isoperimetric quotient of convex polytopes as the number of facets tends to infinity.

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“…There it was also shown that for a smooth convex disc K, the vertices of any sequence of best approximating inscribed polygons are uniformly distributed in bd K with respect to affine length. For sequences of best approximating circumscribed polygons it is shown there that the points where the sides of the polygons touch K are uniformly distributed in bd K with respect to affine length (see also [12]). That this also holds for general convex discs with positive affine length, follows from the next theorem, since the vertices of best approximating inscribed polygons lie on bd K and the sides of best approximating circumscribed polygons touch bd K.…”

Section: Theorem 4 For Everymentioning

confidence: 99%