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Parametrizing Triangulated Meshes
Chalana Bezawada
Kernel Group
PRISM
3DK – September 15, 2000
Parametrization
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.
x(u, v )  ( x(u, v ), y (u, v ), z (u, v ))
3DK – September 15, 2000
Projection of data points
Used to find parametrization of (u,v) of the
domain
Methods:
-Plane projection
-Spherical projection
-Cylindrical projection
3DK – September 15, 2000
Good parametrization
Should 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
3DK – September 15, 2000
Projection of data points
Used to find parametrization of (u,v) of the
domain
Methods:
-Plane projection
-Spherical projection
-Cylindrical projection
3DK – 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.
3DK – 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.
3DK – September 15, 2000
Special case
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
,   0
or
1  , 
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.
3DK – September 15, 2000
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