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Smooth Curves

Learn how to create and control smooth curves through interpolation methods, including Lagrange interpolation, Bezier curves, and B-spline curves. Understand the intuitive control of curves using control points and explore polynomial interpolation for precise curve shaping.

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Smooth Curves

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  1. Smooth Curves Dr. Scott Schaefer

  2. Smooth Curves • Interpolation • Interpolation through Linear Algebra • Lagrange interpolation • Bezier curves • B-spline curves

  3. Smooth Curves • How do we create smooth curves?

  4. Smooth Curves • How do we create smooth curves? • Parametric curves with polynomials

  5. Smooth Curves • Controlling the shape of the curve

  6. Smooth Curves • Controlling the shape of the curve

  7. Smooth Curves • Controlling the shape of the curve

  8. Smooth Curves • Controlling the shape of the curve

  9. Smooth Curves • Controlling the shape of the curve

  10. Smooth Curves • Controlling the shape of the curve

  11. Smooth Curves • Controlling the shape of the curve

  12. Smooth Curves • Controlling the shape of the curve

  13. Smooth Curves • Controlling the shape of the curve

  14. Smooth Curves • Controlling the shape of the curve Power-basis coefficients not intuitive for controlling shape of curve!!!

  15. Interpolation • Find a polynomial that passes through specified values

  16. Interpolation • Find a polynomial that passes through specified values

  17. Interpolation • Find a polynomial that passes through specified values

  18. Interpolation • Find a polynomial that passes through specified values

  19. Interpolation • Find a polynomial that passes through specified values Intuitive control of curve using “control points”!!!

  20. Interpolation • Perform interpolation for each component separately • Combine result to obtain parametric curve

  21. Interpolation • Perform interpolation for each component separately • Combine result to obtain parametric curve

  22. Interpolation • Perform interpolation for each component separately • Combine result to obtain parametric curve

  23. Lagrange Interpolation • Identical to matrix method but uses a geometric construction

  24. Lagrange Interpolation • Identical to matrix method but uses a geometric construction

  25. Lagrange Interpolation • Identical to matrix method but uses a geometric construction

  26. Lagrange Interpolation • Identical to matrix method but uses a geometric construction

  27. Lagrange Interpolation • Identical to matrix method but uses a geometric construction

  28. Lagrange Interpolation • Identical to matrix method but uses a geometric construction

  29. Lagrange Interpolation • Identical to matrix method but uses a geometric construction

  30. Lagrange Interpolation • Identical to matrix method but uses a geometric construction

  31. Lagrange Interpolation • Identical to matrix method but uses a geometric construction

  32. Lagrange Interpolation • Identical to matrix method but uses a geometric construction

  33. Lagrange Interpolation • Identical to matrix method but uses a geometric construction

  34. Lagrange Interpolation • Identical to matrix method but uses a geometric construction

  35. Lagrange Interpolation • Identical to matrix method but uses a geometric construction

  36. Uniform Lagrange Interpolation • Base Case • Linear interpolation between two points • Inductive Step • Assume we have points yi, …, yn+1+i • Build interpolating polynomials f(t), g(t) of degree n for yi, …, yn+i and yi+1, …, yn+1+i • h(t) interpolates all points yi, …, yn+1+i and is of degree n+1 • Moreover, h(t) is unique

  37. Lagrange Evaluation Pseudocode Lagrange(pts, i, t) n = length(pts)-2 if n is 0 return pts[0](i+1-t)+pts[1](t-i) f=Lagrange(pts without last element, i, t) g=Lagrange(pts without first element, i+1, t) return ((n+1+i-t)f+(t-i)g)/(n+1)

  38. Problems with Interpolation

  39. Problems with Interpolation

  40. Bezier Curves • Polynomial curves that seek to approximate rather than to interpolate

  41. Bernstein Polynomials • Degree 1: (1-t), t • Degree 2: (1-t)2, 2(1-t)t, t2 • Degree 3: (1-t)3, 3(1-t)2t, 3(1-t)t2, t3

  42. Bernstein Polynomials • Degree 1: (1-t), t • Degree 2: (1-t)2, 2(1-t)t, t2 • Degree 3: (1-t)3, 3(1-t)2t, 3(1-t)t2, t3 • Degree 4: (1-t)4, 4(1-t)3t, 6(1-t)2t2, 4(1-t)t3,t4

  43. Bernstein Polynomials • Degree 1: (1-t), t • Degree 2: (1-t)2, 2(1-t)t, t2 • Degree 3: (1-t)3, 3(1-t)2t, 3(1-t)t2, t3 • Degree 4: (1-t)4, 4(1-t)3t, 6(1-t)2t2, 4(1-t)t3,t4 • Degree 5: (1-t)5, 5(1-t)4t, 10(1-t)3t2, 10(1-t)2t3 ,5(1-t)t4,t5 • … • Degree n: for

  44. Bezier Curves

  45. Bezier Curves

  46. Bezier Curves

  47. Bezier Curve Properties • Interpolate end-points

  48. Bezier Curve Properties • Interpolate end-points

  49. Bezier Curve Properties • Interpolate end-points

  50. Bezier Curve Properties • Interpolate end-points

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