Abstract: The purpose of this paper is to give a combinatorial characterization and construct representations of the chaotic fundamental groups of the chaotic submanifolds of chaotic flat space by using some geometrical transformations. The chaotic homotopy groups of the limit folding for chaotic flat space are presented. The chaotic fundamental groups of some types of chaotic geodesics in chaotic flat space are deduced.
Abstract: In this article, we would like to show that there is no cut point of any point in a plane, but there exists the cut locus of a point in a flat torus. By the results, we would like to determine the structure of cut locus of a flat torus.
Abstract: In this paper, we studied some properties of geodesic on some open surfaces: Hyperboloid, Paraboloid and Funnel Surface. Geodesic equation in the v-Clairaut parameterization was calculated and reduced to definite integral. Some geodesics on some open surfaces as mention above were classified by Clairaut's relation.
Abstract: It is an important problem to compute the geodesics on
a surface in many fields. To find the geodesics in practice, however,
the traditional discrete algorithms or numerical approaches can only
find a list of discrete points. The first author proposed in 2010 a new,
elegant and accurate method, the geodesic-like method, for
approximating geodesics on a regular surface. This paper will present
by use of this method a computation of the Bezier geodesic-like curves
on spheres.