Triangular Patches

deCasteljau Revisited Barycentrically:

• Linear blend expressed in barycentric terms

  

• Higher powers and a symmetric form of the Bernstein polynomials:

  

• Examples

  

Surfaces - Barycentric Blends on Triangles:

• Formulas:

  

Triangular deCasteljau:

• Join adjacently indexed by triangles
• Find r:s:t barycentric point in each triangle
• Join adjacent points by triangles
• Repeat
  • Final point is the surface point P(r,s,t)
  • Final triangle is tangent to the surface at P(r,s,t)
• Triangle up/down schemes become tretrahedral up/down schemes

Properties:

• Each boundary curve is a Bézier curve
• Patches will be joined smoothly if pairs of boundary triangles are affine images of domain triangles