A Kind Of Definition Of Curvature On Discrete Surface

A new subdivision rule is proposed in this paper. Applying this rule, we give a kind of definition of discrete Gauss curvature and mean curvature on the discrete parametric curves net. At last, we illustrate the validity and superiority of this definition by numerical experiments.Chapter 1 is devoted to investigating the idea, history and development of the classic differential geometry. And then, we review the history, development and actuality of the discrete differential geometry and subdivision method. Correspondingly, some dissertations and achievements on these subjects were listed, which include domestic and abroad.Chapter 2 introduces some propaedeutics of classic differential geometry, which include the first basic form, the second basic form, normal curvature, main curvature, mean curvature, Gauss curvature, unit normal vector, main vector, etc. And then, the paper introduces some definitions of discrete curvature on triangular net surfaces and quadrilateral net surfaces. At last, we give the definitions of parametric curves net, isothermal net, conjugate net, line of curvature net and an important theory in [1].Considering the [2] and barycentric subdivision method, chapter 3 imports some discrete objects-discrete parametric curves net. And then, using Matlab, we give some examples on regular surfaces. For example, plane, helicoidal, 3-dimensional Gauss profile, etc.Chapter 4 uses barycentric subdivision method on arbitrary discrete parametric curves net. And then, we gain some definitions that in relation to regular surfaces which include discrete Gauss curvature and discrete mean curvature. In the end, using Matiab, we compare the discrete curvatures and real curvatures.Chapter 5 includes conclusion and conceive.