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Tensor-Product Surfaces

Explore the principles, properties, and algorithms of Lagrange and Bezier surfaces, from curve interpolation to derivatives. Learn to work with matrix forms and apply deCasteljau's algorithm for precise surface modeling.

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Tensor-Product Surfaces

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  1. Tensor-Product Surfaces Dr. Scott Schaefer

  2. Smooth Surfaces • Lagrange Surfaces • Interpolating sets of curves • Bezier Surfaces • B-spline Surfaces

  3. Lagrange Surfaces

  4. Lagrange Surfaces

  5. Lagrange Surfaces

  6. Lagrange Surfaces

  7. Lagrange Surfaces

  8. Lagrange Surfaces

  9. Lagrange Surfaces

  10. Lagrange Surfaces

  11. Lagrange Surfaces

  12. Lagrange Surfaces

  13. Lagrange Surfaces

  14. Lagrange Surfaces

  15. Lagrange Surfaces

  16. Lagrange Surfaces

  17. Lagrange Surfaces – Properties • Surface interpolates all control points • The boundaries of the surface are Lagrange curves defined by the control points on the boundary

  18. Interpolating Sets of Curves • Given a set of parametric curves p0(t), p1(t), …, pn(t) , build a surface that interpolates them

  19. Interpolating Sets of Curves • Given a set of parametric curves p0(t), p1(t), …, pn(t) , build a surface that interpolates them • Evaluate each curve at parameter value t, then use these points as the control points for a Lagrange curve of degree n • Evaluate this new curve at parameter value s

  20. Bezier Surfaces

  21. Bezier Surfaces

  22. Bezier Surfaces

  23. Bezier Surfaces

  24. Bezier Surfaces

  25. Bezier Surfaces

  26. Bezier Surfaces

  27. Bezier Surfaces

  28. Bezier Surfaces

  29. Bezier Surfaces

  30. Bezier Surfaces

  31. Bezier Surfaces – Properties • Surface lies in convex hull of control points • Surface interpolates the four corner control points • Boundary curves are Bezier curves defined only by control points on boundary

  32. General Tensor Product Surfaces

  33. General Tensor Product Surfaces

  34. Properties • Curve properties/algorithms apply to surfaces too • Convex hull

  35. Properties • Curve properties/algorithms apply to surfaces too • Convex hull • Degree elevation

  36. Properties • Curve properties/algorithms apply to surfaces too • Convex hull • Degree elevation • Evaluation algorithms

  37. Properties • Curve properties/algorithms apply to surfaces too • Convex hull • Degree elevation • Evaluation algorithms • …. • Analog of variation diminishing does not apply!!!

  38. Matrix Form of Quadrilateral Bezier Patches

  39. Matrix Form of Quadrilateral Bezier Patches

  40. deCasteljau Algorithm for Bezier Surfaces

  41. deCasteljau Algorithm for Bezier Surfaces

  42. deCasteljau Algorithm for Bezier Surfaces

  43. deCasteljau Algorithm for Bezier Surfaces

  44. deCasteljau Algorithm for Bezier Surfaces

  45. deCasteljau Algorithm for Bezier Surfaces

  46. deCasteljau Algorithm for Bezier Surfaces

  47. deCasteljau Algorithm for Bezier Surfaces

  48. deCasteljau Algorithm for Bezier Surfaces

  49. Derivatives of Bezier Surfaces • Exact evaluate in the s-direction and use those control points to compute derivative in t-direction • Exact evaluate in the t-direction and use those control points to compute derivative in s-direction • Use a pyramid algorithm to compute derivatives

  50. Derivatives using deCasteljau’s algorithm

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