Components of spaces of curves with constrained curvature on flat surfaces

Saldanha, Nicolau C.; Zühlke, Pedro

doi:10.2140/pjm.2016.281.185

Cited by 5 publications

(5 citation statements)

References 38 publications

(70 reference statements)

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“…The study of the spaces of locally convex curves started in the seventies with the works of Litte on the 2-sphere. But the research on the topological aspects on these spaces of curves on the spheres of higher dimension as in related spaces is very productive area, here we mention some other relevant works: [9], [19], [26], [27], [28], [31], [32], [33] and [34]. A very hard and interesting question in this topic is to determine the homotopy type of the spaces of locally convex curves on the n-sphere, for n ≥ 3.…”

Section: Final Considerationsmentioning

confidence: 99%

Characterization of some convex curves on the 3-sphere

Alves

2020

Preprint

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Section: Final Considerationsmentioning

confidence: 99%

Characterization of some convex curves on the 3-sphere

Alves

2020

Preprint

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“…We mention only [10,13,17,21,23,28,30]. Literature on the topology and geometry of spaces of bounded curvature paths can be found in [2,3,4,5,6,15,16,24,26,29].…”

Section: Preludementioning

confidence: 99%

Census of bounded curvature paths

Díaz

Ayala

2019

Geom Dedicata

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A bounded curvature path is a continuously differentiable piece-wise C2 path with bounded absolute curvature connecting two points in the tangent bundle of a surface. These paths have been widely considered in computer science and engineering since the bound on curvature models the trajectory of the motion of robots under turning circle constraints. Analyzing global properties of spaces of bounded curvature paths is not a simple matter since the length variation between length minimizers of arbitrary close endpoints or directions is in many cases discontinuous. In this note, we develop a simple technology allowing us to partition the space of spaces of bounded curvature paths into one-parameter families. These families of spaces are classified in terms of the type of connected components their elements have (homotopy classes, isotopy classes, or isolated points) as we vary a parameter defined in the reals. Consequently, we answer a question raised by Dubins (Pac J Math 11(2):471481, 1961).

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“…More explicitly, we have many times concatenated locally convex arcs Γ 1 and Γ 2 on Spin n+1 regardless of the differentiability of the resulting path Γ 1 * Γ 2 at the welding point, and considered it nonetheless as a locally convex curve, omitting thereby a tedious smoothening out argument. This appendix therefore plays the the same role as Section 2 in [53], Section 1 in either [56] or [57].…”

Section: B Hilbert Manifolds Of Curvesmentioning

confidence: 99%

Combinatorialization of spaces of nondegenerate spherical curves

Goulart¹,

Saldanha²

2018

Preprint

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A parametric curve γ of class C n on the n-sphere is said to be nondegenerate (or locally convex) when det γ(t), γ (t), • • • , γ (n) (t) > 0 for all values of the parameter t. We orthogonalize this ordered basis to obtain the Frenet frame Fγ of γ assuming values in the orthogonal group SO n+1 (or its universal double cover, Spin n+1 ), which we decompose into Schubert or Bruhat cells. To each nondegenerate curve γ we assign its itinerary: a word w in the alphabet S n+1 {e} that encodes the succession of non open Schubert cells pierced by the complete flag of R n+1 spanned by the columns of Fγ . Without loss of generality, we can focus on nondegenerate curves with initial and final flags both fixed at the (non oriented) standard complete flag. For such curves, given a word w, the subspace of curves following the itinerary w is a contractible globally collared topological submanifold of finite codimension. By a construction reminiscent of Poincaré duality, we define abstract cell complexes mapped into the original space of curves by weak homotopy equivalences. The gluing instructions come from a partial order in the set of words. The main aim of this construction is to attempt to determine the homotopy type of spaces of nondegenerate curves for n > 2. The reader may want to contrast the present paper's combinatorial approach with the geometry-flavoured methods of previous works.

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Components of spaces of curves with constrained curvature on flat surfaces

Cited by 5 publications

References 38 publications

Characterization of some convex curves on the 3-sphere

Characterization of some convex curves on the 3-sphere

Census of bounded curvature paths

Combinatorialization of spaces of nondegenerate spherical curves

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