The computation of simplicial approximations of implicitly defined two- dimensional manifolds
Computation of a triangular approximation to the two-dimensional manifold \(M\) of solutions \(F(x)= 0\), where \(F: \mathbb{R}^ n\to \mathbb{R}^{n- 2}\), is explained. The described algorithm produces a data structure containing points approximately on the manifold, as well as adjacency information describing the triangles; it is asserted that manipulation of this structure is relatively efficient. This structure is generated with a procedure that is reminiscent of predictor-corrector methods for path following: Predictor points are first computed in tangent planes to the manifold, then a corrector step projects these points back onto the manifold. The tesselation proceeds as a ``frontal'' method; special techniques are described for filling out the tesselation uniformly in the neighborhood of a point on the manifold (essentially by subdividing the circular angle around the point), while care is taken to preserve the orientation of the manifold and its components. An adjustment technique is used when two fronts overlap (at seams). The paper contains a clear description of both the mathematical constructs and data structures. Although the paper reflects a FORTRAN 77 program, detailed pseudo-code adequate for implementation appears in the paper. Pictures corresponding to several numerical experiments, including an implicitly-defined sphere in \(\mathbb{R}^ 3\), the solution of equations describing deformation of a circular arc, and another special test problem, appear.
scientific article; zbMATH DE number 701650
scientific article