MATLAB ASSIGNMENT

Matlab Assignment

Matlab Assignment

Introduction

Surface triangulation is an important milestone in 3D mesh generation as it forms the input for the tetrahedralisation of volumes which is a 3D analog of triangulation in 2D. Efficient methods like Delaunay triangulation cannot be applied to surfaces as the Delaunay criterion is not defined for surfaces like it is for planar domains (2D) or volumes (3D). The following two methods can be employed for surface triangulation:

1.Triangulation in Parametric Domain - One-to-one mapping of the surface component onto a 2D parametric domain is done. Graded triangulation is then applied to the parametric plane taking into consideration the actual edge lengths and curvature in 3D. The generated grid is then transformed back to 3D surface. Since the basic triangulation takes place in the 2D (u,v) plane, this method is just an extension of the 2D algorithm described earlier, and is expected to provide quality grids for various applications.

2.Direct Triangulation - Triangulation done in 2D can be extended to a surface by performing direct triangulation (i.e., attempting to triangulate the surface as it is) [NS95]. The graded triangulation as is cannot be applied here, as Delaunay criterion is not defined for surfaces, hence initial triangulation cannot be carried out as described.

Triangulated surface representation

This work develops an approach to interactively sculpting with variation free-form curves and surfaces. Because such shapes generally cannot be explicitly computed, we are concerned with approximating these surface shapes at interactive speeds. Our representation for the topology and approximate shape of a variation surface is a set of sample points in 3D with an associated surface triangulation (we will often refer to this simply as a mesh). These meshes are represented in our modeler as collections of nodes with radially ordered neighbor relations. Curves are approximated as piecewise-linear (PWL) sequences of nodes joined by edges.

We will sometimes need to operate on an embedded curve in a surface, such as a boundary or control curve. In this case the nodes and edges making up the curve must be contained in the mesh as well. No assumptions are made about the three-dimensional shape of the mesh. In particular, there are no ?atness assumptions, and we will not detect or prevent self-intersection. However, the triangulation itself must completely determine the topology of the surface that is being approximated. Regardless of the coarseness of the sampling, a triangular facet in the mesh should always correspond to a continuous triangular piece of the approximated surface, and a closed PWL boundary curve in the mesh should exist for each boundary curve in the smooth surface.

Approximating surface shapes

A fair amount of computational machinery goes towards maintaining variation shape approximations interactively (in spite of our initial minimalist intentions). We break the approximation problem into a number of reasonably simple pieces, most of them transcending our choice of a point-wise surface (as opposed to, say, smooth triangular patches) as the approximating representation. At the lowest level, local surface reconstruction lets us compute over a mesh as if it were a smooth surface, by ...