In this paper, similarity symplectic geometry for curves is proposed and studied. Explicit expressions of the symplectic invariants, Frenet frame and Prenet formulae for curves in similarity symplectic geometry are ob...In this paper, similarity symplectic geometry for curves is proposed and studied. Explicit expressions of the symplectic invariants, Frenet frame and Prenet formulae for curves in similarity symplectic geometry are obtained by using the equivariant moving frame method. The relationships between the Euclidean symplectic invariants, Frenet frame, Frenet formulae and the similarity symplectic invariants, Frenet frame, Frenet formulae for curves are established. Invariant curve flows in four-dimensional similarity symplectic geometry are also studied. It is shown that certain intrinsic invariant curve flows in four-dimensional similarity symplectic geometry are related to the integrable Burgers and matrix Burgers equations.展开更多
In this paper we present a new representation of curve, named parametric curve with an implicit domain(PCID), which is a curve that exists in parametric form defined on an implicit domain. PCID provides a bridge betwe...In this paper we present a new representation of curve, named parametric curve with an implicit domain(PCID), which is a curve that exists in parametric form defined on an implicit domain. PCID provides a bridge between parametric curve and implicit curve because the conversion of parametric form or implicit form to PCID is very convenient and efficient. We propose a framework model for mapping given points to the implicit curve in a homeomorphic manner. The resulting map is continuous and does not overlap. This framework can be used for many applications such as compatible triangulation, image deformation and fisheye views. We also present some examples and experimental results to demonstrate the effectiveness of the framework of our proposed model.展开更多
基金supported by National Natural Science Foundation of China(Grant Nos.11471174 and 11101332)Natural Science Foundation of Shaanxi Province(Grant No.2014JM-1002)the Natural Science Foundation of Xianyang Normal University of Shaanxi Province(Grant No.14XSYK004)
文摘In this paper, similarity symplectic geometry for curves is proposed and studied. Explicit expressions of the symplectic invariants, Frenet frame and Prenet formulae for curves in similarity symplectic geometry are obtained by using the equivariant moving frame method. The relationships between the Euclidean symplectic invariants, Frenet frame, Frenet formulae and the similarity symplectic invariants, Frenet frame, Frenet formulae for curves are established. Invariant curve flows in four-dimensional similarity symplectic geometry are also studied. It is shown that certain intrinsic invariant curve flows in four-dimensional similarity symplectic geometry are related to the integrable Burgers and matrix Burgers equations.
基金supported by National Natural Science Foundation of China(Grant Nos.11031007,11171322,61222206 and 11371341)One Hundred Talent Project of the Chinese Academy of Sciencesthe Program for New Century Excellent Talents in University(Grant No.NCET-11-0881)
文摘In this paper we present a new representation of curve, named parametric curve with an implicit domain(PCID), which is a curve that exists in parametric form defined on an implicit domain. PCID provides a bridge between parametric curve and implicit curve because the conversion of parametric form or implicit form to PCID is very convenient and efficient. We propose a framework model for mapping given points to the implicit curve in a homeomorphic manner. The resulting map is continuous and does not overlap. This framework can be used for many applications such as compatible triangulation, image deformation and fisheye views. We also present some examples and experimental results to demonstrate the effectiveness of the framework of our proposed model.
