Parametrizing Triangulated Meshes

Chalana Bezawada

Kernel Group

PRISM

3DK3DK – – September 15, 2000

)) , ( ), , ( ), , ( ( ) , (v u z v u y v u x v u xParametrization

Given a set of data points and their triangulations in 3D , each data point in the triangulation is assigned a unique pair of values (u, v) in the 2D domain by means of projection.

3DK3DK – – September 15, 2000

Projection of data points

Used to find parametrization of (u,v) of the domain

Methods:

-Plane projection

-Spherical projection

-Cylindrical projection

3DK3DK – – September 15, 2000

Good parametrizationShould preserve the geometry of the original mesh.Should not produce folds (or overlapping triangles) in the domain.

Classic Methods:

-Uniform parametrization–Chord length parametrization–Centripetal parametrization

3DK3DK – – September 15, 2000

Projection of data points

Used to find parametrization of (u,v) of the domain

Methods:

-Plane projection

-Spherical projection

-Cylindrical projection

3DK3DK – – September 15, 2000

Floater’s technique (M. S. Floater - CAGD/1997)

Floater's method finds parametrization based on the concepts of graph theory. Considering points P0, P1, …. , Pn-1 to be the internal nodes and points Pn, Pn+1, …. , Pm-1 to be the boundary nodes of the original mesh, floater obtains the parametrization as follows:

• Choose parameters corresponding to the boundary nodes to be the vertices of any (m-n) sided convex polygon in an anti-clockwise sequence.

• Write each internal node as a convex combination of its neighboring nodes.

One drawback of this approach is that the boundary points are initially mapped onto a closed convex polygon in the domain, irrespective of the geometry of the original mesh boundary.

3DK3DK – – September 15, 2000

New parametrization technique (Dr. Farin)

Based on the geometry of the original mesh, we consider every pair of neighboring triangles in the original mesh.

Find the point x1 on l1 that is closest to l2, and point x2 on l2 that is closest to l1.

The computed points x1 and x2 will be identical only if the two neighboring triangles are coplanar.

Even in the case when l1 and l2 do not actually intersect, an equation can be formed by forcing the two points x1 and x2 to be identical.

3DK3DK – – September 15, 2000

With this approach one might get overlapping triangles in the domain if the angle between the planes containing the two neighboring triangles is very small, and if either

A possible solution to this problem is to get the values of α and β close to the interval [0,1].

This can be achieved by replacing the points Pi1 and Pi4 with the centroids of the respective triangles for calculating the values of α and β.

Special case

3DK3DK – – September 15, 2000

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Original mesh

Simple plane projection

Parameter domain Resulting surface

Our adaptive technique

Parameter domain Resulting surface

More results

Without trimming of parameter domain

With trimming of parameter domain