On the Geometry of Rolling and Interpolation Curves on S n , SO n , and Grassmann Manifolds
Autor(a) principal: | |
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Data de Publicação: | 2007 |
Outros Autores: | |
Tipo de documento: | Artigo |
Idioma: | eng |
Título da fonte: | Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
Texto Completo: | http://hdl.handle.net/10316/7629 https://doi.org/10.1007/s10883-007-9027-3 |
Resumo: | Abstract We present a procedure to generate smooth interpolating curves on submanifolds, which are given in closed form in terms of the coordinates of the embedding space. In contrast to other existing methods, this approach makes the corresponding algorithm easy to implement. The idea is to project the prescribed data on the manifold onto the affine tangent space at a particular point, solve the interpolation problem on this affine subspace, and then project the resulting curve back on the manifold. One of the novelties of this approach is the use of rolling mappings. The manifold is required to roll on the affine subspace like a rigid body, so that the motion is described by the action of the Euclidean group on the embedding space. The interpolation problem requires a combination of a pullback/push forward with rolling and unrolling. The rolling procedure by itself highlights interesting properties and gives rise to a new, but simple, concept of geometric polynomial curves on manifolds. This paper is an extension of our previous work, where mainly the 2-sphere case was studied in detail. The present paper includes results for the n-sphere, orthogonal group SO n , and real Grassmann manifolds. In particular, we present the kinematic equations for rolling these manifolds along curves without slip or twist, and derive from them formulas for the parallel transport of vectors along curves on the manifold. |
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On the Geometry of Rolling and Interpolation Curves on S n , SO n , and Grassmann ManifoldsAbstract We present a procedure to generate smooth interpolating curves on submanifolds, which are given in closed form in terms of the coordinates of the embedding space. In contrast to other existing methods, this approach makes the corresponding algorithm easy to implement. The idea is to project the prescribed data on the manifold onto the affine tangent space at a particular point, solve the interpolation problem on this affine subspace, and then project the resulting curve back on the manifold. One of the novelties of this approach is the use of rolling mappings. The manifold is required to roll on the affine subspace like a rigid body, so that the motion is described by the action of the Euclidean group on the embedding space. The interpolation problem requires a combination of a pullback/push forward with rolling and unrolling. The rolling procedure by itself highlights interesting properties and gives rise to a new, but simple, concept of geometric polynomial curves on manifolds. This paper is an extension of our previous work, where mainly the 2-sphere case was studied in detail. The present paper includes results for the n-sphere, orthogonal group SO n , and real Grassmann manifolds. In particular, we present the kinematic equations for rolling these manifolds along curves without slip or twist, and derive from them formulas for the parallel transport of vectors along curves on the manifold.2007info:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/articlehttp://hdl.handle.net/10316/7629http://hdl.handle.net/10316/7629https://doi.org/10.1007/s10883-007-9027-3engJournal of Dynamical and Control Systems. 13:4 (2007) 467-502Hüper, K.Silva Leite, F.info:eu-repo/semantics/openAccessreponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãoinstacron:RCAAP2020-05-25T12:06:29Zoai:estudogeral.uc.pt:10316/7629Portal AgregadorONGhttps://www.rcaap.pt/oai/openaireopendoar:71602024-03-19T20:57:54.302589Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãofalse |
dc.title.none.fl_str_mv |
On the Geometry of Rolling and Interpolation Curves on S n , SO n , and Grassmann Manifolds |
title |
On the Geometry of Rolling and Interpolation Curves on S n , SO n , and Grassmann Manifolds |
spellingShingle |
On the Geometry of Rolling and Interpolation Curves on S n , SO n , and Grassmann Manifolds Hüper, K. |
title_short |
On the Geometry of Rolling and Interpolation Curves on S n , SO n , and Grassmann Manifolds |
title_full |
On the Geometry of Rolling and Interpolation Curves on S n , SO n , and Grassmann Manifolds |
title_fullStr |
On the Geometry of Rolling and Interpolation Curves on S n , SO n , and Grassmann Manifolds |
title_full_unstemmed |
On the Geometry of Rolling and Interpolation Curves on S n , SO n , and Grassmann Manifolds |
title_sort |
On the Geometry of Rolling and Interpolation Curves on S n , SO n , and Grassmann Manifolds |
author |
Hüper, K. |
author_facet |
Hüper, K. Silva Leite, F. |
author_role |
author |
author2 |
Silva Leite, F. |
author2_role |
author |
dc.contributor.author.fl_str_mv |
Hüper, K. Silva Leite, F. |
description |
Abstract We present a procedure to generate smooth interpolating curves on submanifolds, which are given in closed form in terms of the coordinates of the embedding space. In contrast to other existing methods, this approach makes the corresponding algorithm easy to implement. The idea is to project the prescribed data on the manifold onto the affine tangent space at a particular point, solve the interpolation problem on this affine subspace, and then project the resulting curve back on the manifold. One of the novelties of this approach is the use of rolling mappings. The manifold is required to roll on the affine subspace like a rigid body, so that the motion is described by the action of the Euclidean group on the embedding space. The interpolation problem requires a combination of a pullback/push forward with rolling and unrolling. The rolling procedure by itself highlights interesting properties and gives rise to a new, but simple, concept of geometric polynomial curves on manifolds. This paper is an extension of our previous work, where mainly the 2-sphere case was studied in detail. The present paper includes results for the n-sphere, orthogonal group SO n , and real Grassmann manifolds. In particular, we present the kinematic equations for rolling these manifolds along curves without slip or twist, and derive from them formulas for the parallel transport of vectors along curves on the manifold. |
publishDate |
2007 |
dc.date.none.fl_str_mv |
2007 |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/article |
format |
article |
status_str |
publishedVersion |
dc.identifier.uri.fl_str_mv |
http://hdl.handle.net/10316/7629 http://hdl.handle.net/10316/7629 https://doi.org/10.1007/s10883-007-9027-3 |
url |
http://hdl.handle.net/10316/7629 https://doi.org/10.1007/s10883-007-9027-3 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
Journal of Dynamical and Control Systems. 13:4 (2007) 467-502 |
dc.rights.driver.fl_str_mv |
info:eu-repo/semantics/openAccess |
eu_rights_str_mv |
openAccess |
dc.source.none.fl_str_mv |
reponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação instacron:RCAAP |
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Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação |
instacron_str |
RCAAP |
institution |
RCAAP |
reponame_str |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
collection |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
repository.name.fl_str_mv |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação |
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1799133869294747648 |