Smooth Connection Of Implicit Surface And Data Field Visualization

In the field of the Computer Aided Geometric Design(CAGD),surfaces can be classified into two categories:parametric surfaces and implicit surfaces.An implicit surface refers to the surface that is defined by the set of solutions of a real coefficient algebraic polynomial equation(f(x,y,z)=0);therefore we also call it the algebraic surface.The parameter surfaces have been at the center of research in geometric design for a long time due to their highly desirable properties such as the simple structure and easy computation. Compares with the parameter surface,the implicit surface has the following advantages: easily to judge a point in the surface,and the closure property under some geometric operations;for a parametric surface and an implicit surface,all results under geometric operations are denoted to implicit surfaces.Based on these advantages of implicit surfaces,it is very significant to study smoothly blending implicit algebraic curves and surfaces.Contour and iso-surface as common implicit algebraic curves and surfaces,its application value received more and more attention.Contour and iso-surface technology is widely used in visualization and many scalar field visualization problems can be solved by isosurfacing and rendering.Marching cubes algorithm is the most widely used iso-surface method till now.This paper introduces the basic principle of marching cubes algorithm,then gives out the definition,the disadvantages and advantages of the algorithms and explains how these problems are brought about.In the process of isosurfacing,the algorithm doesn't deal with the complexity of iso-surface in the voxel,and takes the triangles as iso-surfaces.However,an iso-surface should change its shape at some critical points and loops.Therefore,the iso-surface algorithm may encounter many problems in topology,precision and efficiency and can not satisfy many applications.In order to overcome the disadvantages of marching cubes algorithm,this paper incorporates spline function of smoothing cofactor into marching cubes algorithm.First,the data field is discretized into square or cube cells,and then the coordination conditions of element or voxel smooth connection are established.As a result,the smooth blending problems of implicit curves and surfaces are transformed into the problem to solve the coordination equations.By constructing a well-posed interpolation node group,the problem of multi-spline interpolation is successfully solved,and the specific expression of spline for unit element and voxel is obtained.And then using the marching cubes algorithm,the contour or iso-surface is extracted for a given value of the spline function.The C~1 smooth blending of quadratic implicit curve segments and the C~1 smooth blending of cubic implicit curve segments is achieved within element respectively,thereby the C~1 smooth blending of quadratic implicit surface patches and the C~1 smooth blending of cubic implicit surface patches is realized within voxel,which means that the contour and iso-surface reach the C~1 smooth continuous in the entire data fields.