1. 3D Modeling Topics Gerald Farin Computer Science PRISM: Partnership for Research In Spatial Modeling ASU

2. PRISM ’s expertises are well-suited to interdisciplinary research Data Acquisition Geometric Modeling Analysis Visualization Data Archiving Rapid Prototyping

3. Overview History Curvature Surface analysis Ruled surfaces Coons surfaces Continuity NURBS Subdivision surfaces

4. Lofting

5. before fairing

6. after fairing

7. History 1962C. de Boor: bicubic splines 1972J. Ferguson, Boeing: cubic splines 1974W. Gordon, R. Riesenfeld: B-splines 1964S. Coons: Coons patches, conics 1975K. Vesprille: NURBS

8. History 1958P. de Casteljau, Citroen: de Casteljau algorithm 1965P. Bezier, Renault: Bezier curves

9. Overview History Curvature Surface analysis Ruled surfaces Coons surfaces Continuity NURBS Subdivision surfaces

10. Osculating circle

11. Frenet frame Interactive demo

12. Helix curve fit cubic quintic

13. Helix curvature plot cubic quintic

14. Curvature plot, cubic B-spline

15. B-spline curve operations redundancy creation processes: Knot insertion Degree raising

16. Knot insertion quintic cubic

17. B-spline curve fairing “Invert” redundancy creation processes: Knot removal Degree lowering

18. Cubic spline Before fairing After fairing

19. Curvature & Beauty

20. Curvature & Cars

21. Overview History Curvature Surface analysis Ruled surfaces Coons surfaces Continuity NURBS Subdivision surfaces

22. Surface curvatures

24. Lines of curvature

25. Surface curvatures Gaussian = min curvature x max curvature Gaussian = 0: developable surface Gaussian curvature signs

26. First Gaussian image

27. Gaussian application

28. Gaussian curvature

29. Mean curvature

30. Max curvature

31. Reflection lines

33. Overview History Curvature Surface analysis Ruled surfaces Coons surfaces Continuity NURBS Subdivision surfaces

34. Ruled surface

35. Ruled, bilinear surface

36. Ruled surfaces

37. Overview History Curvature Surface analysis Ruled surfaces Coons surfaces Continuity NURBS Subdivision surfaces

38. Coons surface

39. + Ruled in u

40. + Ruled in v

41. - Bilinear

42. Coons surfaces

43. Overview History Curvature Surface analysis Ruled surfaces Coons surfaces Continuity NURBS Subdivision surfaces

44. G1 vs C1 surfaces C1: differentiable across common boundary of two surfaces; requires a common domain. G1: common tangent plane along common boundary; no common domain required.

45. Tangent ribbon

46. G1 condition

47. T-splines

49. G1 mixed patch types

50. History Curvature Surface analysis Ruled surfaces Coons surfaces Continuity NURBS Subdivision surfaces

51. NURBS

52. Conics: Pascal’s theorem

53. Homogeneous coordinates

54. Polynomial vs rational

55. Conics: ellipse; 0<w<1

56. Conics: parabola; w=1

57. Conics: hyperbola; w>1

58. NURB curve, all weights = 1

59. NURB curve, one weight = 0.2

60. NURB curve, one weight = 3

61. B-spline curves

62. Bicubic NURB surface

63. One weight changed

64. NURBS: torus

65. NURB hierarchies Conics: subset of rational Bezier Rational Bezier: subset of NURB curves Conics: not subset of splines Parabolas: subset of splines Quadrics: subset of NURB surfaces Conic lofted surfaces: not subset of NURB surfaces

66. History Curvature Surface analysis Ruled surfaces Coons surfaces Continuity NURBS Subdivision surfaces

67. Subdivision Movie introducing Catmull-Clark subdivision

68. Viking ships

69. Symmetry Analysis

70. Whole body scanner from Cyberware Art Meets Engineering Twister, Dan Collins 1995 Rendering in 3D Studio Max

71. Water Table & Pop Growth (1900 - 2000)