CS 284: CAGD
Lecture #16 -- Mo 10/23, 2000.

Preparation:

Some Additional Comments on Cubic Curve Interpolation

Some Additional Comments on Homework #6

Collect Paper Selection Forms

Triangle Patch Surfaces

Triangular Bezier Patches (CH.17 in G. Farin's book)

• Motivation for triangular patches.
• Barycentric interpolation.
• Control point structure of triangular Bezier patches.
• DeCasteljau evaluation of triangular patches.

Triangular Bezier Patches for Data Interpolation

• This is a simpler task, before we tackle arbitrary free-form surfaces;
  - - we only have to deal with one component -- the height of the data function.
• The problem set-up:
  - - Irregularly placed data samples above the x-y-plane;
  - - Form a Delaunay triangulation to connect nearest neighbor points with edges;
  - - Data values are heights all of these sample points;
  - - We want to form a smooth interpolating height function through all these points.
• Approach
  - - For each domain triangle we will form one (or more) triangular Bezier patch(es);
  - - We may also have some derivative or curvature information for each data point;
  - - If the scheme requires such information, and it is not given, we will need to use a heuristic to come up with reasonable values for these parameters;
  - - There are many different schemes:

• Nine-parameter interpolant:
  - - Use a cubic patch;
  - - Divide edges into 3 equal parts;
  - - Chose heights of edge points from tangent planes at nearest vertex;
  - - Implicitly this will do a cubic Hermite interpolation along the edges;
  - - Choose b111 to give a "quadratic precision" -- if samples of other 9 points lie on a quadratic surface, then the central point b111 will also lie on this surface;
  - - This scheme does not yield C1-continuity across the edges.

• Quintic C1 interpolant:
  - - Use a quintic patch;
  - - Divide edges into 5 equal parts;
  - - Chose heights of first edge points from tangent planes at nearest vertex;
  - - Chose heights of second nearest points to vertices from curvature information;
  - - Implicitly this will do a quintic Hermite interpolation along the edges;
  - - Choose b122, B212, and B221 to give C1 continuity across the 3 edges -- tedious!
  - - This scheme uses C2 information but only yields C1-continuity.

• Clough-Tocher interpolant:
  - - Split the domain triangle to its centroid into 3 mini-triangles;
  - - This gives us 7 internal points rather than six, and in more convenient locations;
  - - Make each mini-triangle cubic and place its control points at a regular 1/3 subdivision;
  - - Find heights of the nearest neighbors to original data vertices from the tangent planes;
  - - Implicitly do cubic Hermite interpolation along the edges;
  - - The 3 points in the second row along an edge control the behavior of the cross-boundary tangent;
  - - This can be a quadratic function;
  - - The vectors at the end are already given by the tangent planes;
  - - There is one degree of freedom to choose the CB tangent in the middle of each edge,
  - - conveniently this is done by averaging the vectors at the end of this edge;
  - - The height of the third ring of control points is found by averaging the heights of their outer three nearest neighbors;
  - - The height of the central, shared, interpolated control point is found by averaging the heights of its three nearest neighbors;
  - - This will yield C2 continuity at this central point;
  - - This scheme yields C1-continuity across the outer edges.

• Powell-Sabin interpolant:
  - - Split the domain triangle to its incenter into 6 mini-triangles;
  - - Split the three edges by joining adjacent incenters;
  - - Make each mini-triangle quadratic and place its control points at a regular 1/2 subdivision;
  - - This gives us 7 internal points and 3 unevenly spaced points on each edge;
  - - Find heights of the nearest neighbors to original data vertices from the tangent planes;
  - - Find heights of the central edge points by linear interpolation from its neighbors on the edge;
  - - The central, shared, interpolated control point and all its nearest neighbors must lie in a plane,
  - - this plane is determined by the corners of this inner triangle;
  - - The resulting function is piecewise quadratic and everywhere C1-continuous.

Current Homework Assignment:

Reading Assignment:

Handout: "Local surface interpolation with shape parameters between adjoining Gregory patches," by Shirman and Sequin.