The classification of homotopy classes of bounded curvature paths

Ayala, José L.; Rubinstein, J. Hyam

doi:10.48550/arxiv.1403.5314

Cited by 4 publications

(25 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In general, it is not an easy task to construct κ-constrained homotopies between two given curves (see [6]). In Proposition 3.8 we will see that the existence of parallel tangents leads to a method for constructing κ-constrained homotopies.…”

Section: A Fundamental Lemmamentioning

confidence: 99%

“…In particular, we conclude that these trapped curves correspond to a homotopy class of embedded κ-constrained curves. (On piecewise constant curvature κ-constrained curves) As seen in [6], constructing explicit κ-constrained homotopies is not a simple matter. In subsection 4.1 we will discuss a process applied to κ-constrained curves called normalisation (see [4,2,6]).…”

Section: Homotopy Classes In Spaces Of κ-Constrained Plane Curvesmentioning

confidence: 99%

“…(On piecewise constant curvature κ-constrained curves) As seen in [6], constructing explicit κ-constrained homotopies is not a simple matter. In subsection 4.1 we will discuss a process applied to κ-constrained curves called normalisation (see [4,2,6]). The normalisation of a κ-constrained curve σ is a piecewise constant curvature κ-constrained curve corresponding to a finite number of concatenated pieces called components.…”

Section: Homotopy Classes In Spaces Of κ-Constrained Plane Curvesmentioning

confidence: 99%

“…It is important to note that both σ and its normalisation are curves in the same connected component in Σ(x, y). Our efforts in [4,2,6] have been made in order to overcome the difficulty of constructing explicit κ-constrained homotopies. After normalising σ we apply a reduction process, also described in subsection 4.1.…”

Section: Homotopy Classes In Spaces Of κ-Constrained Plane Curvesmentioning

confidence: 99%

“…This process consists on manipulating piecewise constant curvature κ-constrained curves while reducing length and complexity and without violating the curvature bound. With the intention of simplifying our arguments, in [6], we defined the so called operations of type I and II. The Operations of type III will be defined to be the ones performed under the existence of parallel tangents (see Proposition 3.8).…”

Section: Homotopy Classes In Spaces Of κ-Constrained Plane Curvesmentioning

confidence: 99%

See 4 more Smart Citations

On the topology of the spaces of curvature constrained plane curves

Ayala

2017

Advances in Geometry

Self Cite

View full text Add to dashboard Cite

Abstract. It is well known that plane curves with the same endpoints are homotopic. An analogous claim for plane curves with the same endpoints and bounded curvature still remains open. In this work we find necessary and sufficient conditions for two plane curves with bounded curvature to be deformed, one to another, by a continuous one-parameter family of curves also having bounded curvature. We conclude that the space of these curves has either one or two connected components, depending on the distance between the endpoints. The classification theorem here presented answers a question raised in 1961 by L. E. Dubins.

show abstract

Section: A Fundamental Lemmamentioning

confidence: 99%

Section: Homotopy Classes In Spaces Of κ-Constrained Plane Curvesmentioning

confidence: 99%

Section: Homotopy Classes In Spaces Of κ-Constrained Plane Curvesmentioning

confidence: 99%

Section: Homotopy Classes In Spaces Of κ-Constrained Plane Curvesmentioning

confidence: 99%

Section: Homotopy Classes In Spaces Of κ-Constrained Plane Curvesmentioning

confidence: 99%

See 3 more Smart Citations

On the topology of the spaces of curvature constrained plane curves

Ayala

2017

Advances in Geometry

Self Cite

View full text Add to dashboard Cite

show abstract

Length minimising bounded curvature paths in homotopy classes

Ayala

2015

Topology and its Applications

Self Cite

View full text Add to dashboard Cite

Available online xxxx MSC: primary 49Q10, 55Q05 secondary 90C47, 51E99, 68R99Choose two points in the tangent bundle of the Euclidean plane (x, X), (y, Y ) ∈ T R 2 . In this work we characterise the immersed length minimising paths with a prescribed bound on the curvature starting at x, tangent to X; finishing at y, tangent to Y , in each connected component of the space of paths with a prescribed bound on the curvature from (x, X) to (y, Y ).

show abstract

História e cultura: negros do Rosário de Pombal

Ayala¹

View full text Add to dashboard Cite

Os agradecimentos começam pelos "negros do Rosário" de Pombal, com quem aprendi muito no decorrer da pesquisa e que se dispuseram a conversar, gravar entrevistas, fornecer as informações pedidas. Sou grato a todos, embora só mencione alguns deles: o juiz da Irmandade do Rosário, Antonio Felismino de Sousa; Aurélio Gomes de Farias, ligado aos Pontões e à Irmandade e grande conhecedor de suas atividades, embora não pertença a nenhum dos grupos culturais; Clóvis Rufino de Jesus, capitão dos Pontões; Lourenço Rufino de Jesus, ex-integrante do mesmo grupo; Maria de Barros, ex-participante dos Congos; João Pedro de Souza, rei dos Congos; João Luiz de Andrade, secretário dos Congos; e Chico de Bembém, chefe do Reisado.Outras pessoas de Pombal colaboraram para a realização deste trabalho: dona Rita Maria de Sousa Lucena, na casa de quem me hospedava em Pombal, que forneceu informações sobre os grupos; seu filho, Gilberto de Sousa Lucena; e Luís Barbosa Neto, ex-aluno da Universidade Federal da Paraíba, que se interessa pela festa e tem fornecido apoio material aos brinquedos.Agradeço, também à profS drâ Maria de Lourdes Monaco Janotti, que orientou este trabalho durante parte do tempo e ao prof. dr. Elias Thomé Saliba, que a substituiu.Marineis, minha mulher, não só esteve comigo em Pombal, como em outros lugares onde se realizam danças e festas populares, que pesquisamos juntos há bastante tempodesde que comecei a estudar estas práticas culturais. Também leu, discutiu e revisou os textos, além de incentivar-me e dar todo o apoio indispensável à realização do trabalho.A CAPES, através do PICD, forneceu a bolsa para a realização do Doutorado.

show abstract

The classification of homotopy classes of bounded curvature paths

Cited by 4 publications

References 0 publications

On the topology of the spaces of curvature constrained plane curves

On the topology of the spaces of curvature constrained plane curves

Length minimising bounded curvature paths in homotopy classes

História e cultura: negros do Rosário de Pombal

Contact Info

Product

Resources

About