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Computer Graphics

Computer Graphics. Representing Curves and Surfaces. Review eq(11.5). Review eq(11.6/11.7). Review eq(11.8). Review eq(11.8). Review eq(11.9). Review eq(11.10). Review. Blending function (also called ‘Basis’ function). Hermite Curves. 以曲線端點 P 1 . P 4 以及端點斜率 R 1 . R 4 求曲線方程式.

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Computer Graphics

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  1. Computer Graphics Representing Curves and Surfaces

  2. Revieweq(11.5)

  3. Review eq(11.6/11.7)

  4. Review eq(11.8)

  5. Review eq(11.8)

  6. Review eq(11.9)

  7. Review eq(11.10)

  8. Review • Blending function (also called ‘Basis’ function)

  9. Hermite Curves • 以曲線端點P1.P4以及端點斜率R1.R4求曲線方程式

  10. Hermite Curveseq(11.12)

  11. Hermite Curveseq(11.13)

  12. Hermite Curveseq(11.14)

  13. Hermite Curveseq(11.15)

  14. Hermite Curves

  15. Hermite Curveseq(11.16)

  16. Hermite Curveseq(11.17)

  17. Hermite Curveseq(11.18)

  18. Hermite Curveseq(11.19)

  19. Hermite Curveseq(11.20)

  20. Hermite Curveseq(11.21)

  21. Hermite Curve

  22. Hermite Curveeq(11.22)

  23. Hermite Curveeq(11.23) • Reduce 6 multiplies and 3 additions to 3 multiplies and 3 additions.

  24. Bezier Curves • 以曲線端點P1.P4以及控制點P2.P3求曲線方程式 • 曲線端點斜率為

  25. Bezier Curveseq(11.24)

  26. Bezier Curveseq(11.25)

  27. Bezier Curveseq(11.26)

  28. Bezier Curveseq(11.27/11.28)

  29. Bezier Curveseq(11.29)

  30. Bezier Curves

  31. Bezier Curves • Define n as the order of Bezier curves. • Define i as control point.

  32. Bezier Curves

  33. Bezier Curves

  34. Bezier Curves • De Casteljau iterations

  35. Bezier Curves • Linear Bezier splinesControl points: P0, P1

  36. Bezier Curves • Quadratic Bezier splinesControl points: P0, P1, P2

  37. Bezier Curves • Quadratic Bezier splinesControl points: P0, P1, P2

  38. Bezier Curves • Cubic Bezier splinesControl points: P0, P1, P2, P3

  39. Bezier Curves • Cubic Bezier splinesControl points: P0, P1, P2, P3

  40. Bezier Curves • Cubic Bezier splinesControl points: P0, P1, P2, P3

  41. Bezier Curves

  42. Bezier Curves • http://www.ibiblio.org/e-notes/Splines/Bezier.htm

  43. Spline • Natural cubic spline • C0, C1, C2 continuous. • Interpolates(passes through) the control points. • Moving any one control point affects the entire curve.

  44. Spline • B-spline • Local control. • Moving a control point affects only a small part of a curve. • Do not interpolate their control points. • Sharing control points between segments.

  45. B-spline • m+1 control points P0, …, Pm, m≥3 • m-2 curve segments Q3, Q4, …, Qm • For each i≥4, there is a join point or knot between Qi-1 and Qi at the parameter value ti.

  46. B-splineeq(11.43/11.44)

  47. B-splineeq(11.44)

  48. Uniform Nonrational B-spline • ‘Uniform’ means that the knots are spaced at equal intervals of the parameter t. • ‘Nonrational’ is used to distinguish these splines from rational cubic polynomial curves, see Section 11.2.5 • We assume that t3=0 and the interval ti+1-ti=1

  49. Uniform Nonrational B-splineeq(11.32/11.33/11.34)

  50. Uniform Nonrational B-splineeq(11.35)

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