What should be the out put of Delauny Mesh Triangulation algorithm for 3-Dimension?

Question

If this are the input points:

to the Delauny Mesh Triangulation algorithm , then what will be output of Delauny Mesh Triangulation algorithm for 3-Dimension ?

A. This one:

or B. This one [ConvexHull of all given input points]

What is your answer? A or B

Answer 1

Strictly speaking, your result B is correct, and A is a "trimmed down" version without edges longer than a certain threshold.

But: check out alpha shapes . They generalize the concept to fit both your results! They simply become the results with different alpha values.

However, alpha shapes are mostly a theoretical concept. One of the most popular algorithms that constructs surfaces from points is ball-pivoting , which will generate the surface of a specific alpha-shape.

Answer 2

You don't get a surface but a "volume" made of tetrahedrons with B as surface.

Most of the faces of the tetrahedrons have to be removed to get a clean surface as in A, but for this you have to decide which tetrahedrons are inside and which are outside.

In 2D the same problem would be to get a line from the triangle mesh. What may allow to choose the right line is the density of points which aligns with the surface when data comes from some sensor. For random points there is no "good" solution.

Answer 3

The output in A is obtained by surface reconstruction from points . There are several methods for this, many of them based on Delaunay triangulation. See for instance http://cgal.inria.fr/Reconstruction/ and http://www.cse.ohio-state.edu/~tamaldey/surfrecon.htm .

Answer 4

The Delaunay triangulation is equivalent to computing the convex hull of the points

SOURCE: http://www.exaflop.org/docs/cgafaq/cga6.html#Subject 6.01: Where can I get source for Voronoi/Delaunay triangulation?