Algorithm For Generating Frame Fields On Surfaces

With the advancement and development of computer technology,the application of three-dimensional data is more and more extensive.Mesh is an important represen-tation of 3D data.On the discrete two-dimensional manifold represented by the mesh,the frame field is an important tool.The finite element splitting guided by the frame field,the remesh of the alignment frame field,the alignment of the frame field and the texture mapping and other mesh operations have a good effect.However,the existing methods of generating the frame field often need to solve more complicated optimiza-tion problems,and the calculation process is quite time consuming,so the algorithm for efficiently generating the frame field is of great significance.This paper is based on a triangular mesh calculation where the trend of the frame field is close to the curvature.First,the second fundamental tensor defined at each point on the two-dimensional manifold is a matrix whose eigenvalue is the main curvature and the eigenvector is the direction of the main curvature.In order to make the frame field have the property of curvature,the second fundamental tensor can be used to define the Riemann metric.Then,in the sense of this Riemannian metric,the anisotropic frame field is regarded as an isotropic orthogonal field,and the isotropic field is optimally aligned.Finally,the generated isotropic orthogonal field is mapped to an anisotropic frame field according to the Riemann metric of its location.Each step of the process solves a Laplacian equation.In order to speed up the solution,a multi-resolution hi-erarchy is established here,and the Gauss-Seidel iteration method is used to solve the optimization problem.Here,some triangle meshes were randomly selected and a comparative experiment was performed.When the mesh is sparse and the feature precision is not high enough,the speed of the algorithm is obviously improved,but the effect of generating the frame field is not necessarily better than other methods.However,when the mesh is dense and the feature precision is high enough,it can be found that the algorithm has a large improvement in the calculation speed,and the direction of the generated frame field and the curvature are closer.