From Curves to Surfaces

• Bicubic surfaces work like spline curves

  • Use control points and basis matrices
  • Define 16 control parameters (points and/or derivatives)
  • Solve for coefficients (using a rank-4 tensor = 4x4x4x4 array!!!)

• Symmetry between u and v allows this tensor to be two copies of a regular matrix

  • B is the basis matrix, P the matrix of control values:

• Basis surface view

  • 4 basis cubics for u and 4 for v (may or may not be same basis)
  • 16 basis surfaces from multiplying each pair of cubics together
  • Bicubic surface is a linear combo (weighted sum) of basis surfaces

• Surface obtained by piecing together bicubic patches

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