1. MIT EECS 6.837, Durand and Cutler Curves & Surfaces

2. MIT EECS 6.837, Durand and Cutler Schedule Sunday October 5 th, * 3-5 PM *, Room TBA: Review Session for Quiz 1 Extra Office Hours on Monday (NE43 Graphics Lab) Tuesday October 7 th : Quiz 1: In class 1 hand-written 8.5x11 sheet of notes allowed Wednesday October 15th: Assignment 4 (Grid Acceleration) due

3. MIT EECS 6.837, Durand and Cutler Last Time: Acceleration Data Structures

4. MIT EECS 6.837, Durand and Cutler Questions?

5. MIT EECS 6.837, Durand and Cutler Today Review Motivation –Limitations of Polygonal Models –Phong Normal Interpolation –Some Modeling Tools & Definitions Curves Surfaces / Patches Subdivision Surfaces Procedural Texturing

6. MIT EECS 6.837, Durand and Cutler Limitations of Polygonal Meshes planar facets fixed resolution deformation is difficult no natural parameterization

7. MIT EECS 6.837, Durand and Cutler Can We Disguise the Facets?

8. MIT EECS 6.837, Durand and Cutler Phong Normal Interpolation Not Phong Shading from Assignment 3 Instead of using the normal of the triangle, interpolate an averaged normal at each vertex across the face Must be renormalized

9. MIT EECS 6.837, Durand and Cutler 10K facets 1K facets 1K smooth 10K smooth

10. MIT EECS 6.837, Durand and Cutler Better, but not always good enough Still low resolution (missing fine details) Still have polygonal silhouettes Intersection depth is planar Collisions in a simulation Solid Texturing...

11. MIT EECS 6.837, Durand and Cutler Some Non-Polygonal Modeling Tools Extrusion Spline Surfaces/Patches Surface of Revolution Quadrics and other implicit polynomials

12. MIT EECS 6.837, Durand and Cutler Continuity definitions: C 0 continuous –curve/surface has no breaks/gaps/holes –"watertight" C 1 continuous –curve/surface derivative is continuous –"looks smooth, no facets" C 2 continuous –curve/surface 2 nd derivative is continuous –Actually important for shading

13. MIT EECS 6.837, Durand and Cutler Questions?

14. MIT EECS 6.837, Durand and Cutler Today Review Motivation Curves –What's a Spline? –Linear Interpolation –Interpolation Curves vs. Approximation Curves –Bézier –BSpline (NURBS) Surfaces / Patches Subdivision Surfaces Procedural Texturing

15. MIT EECS 6.837, Durand and Cutler BSpline (approximation) Definition: What's a Spline? Smooth curve defined by some control points Moving the control points changes the curve InterpolationBézier (approximation)

16. MIT EECS 6.837, Durand and Cutler Interpolation Curves / Splines www.abm.org

17. MIT EECS 6.837, Durand and Cutler Linear Interpolation Simplest "curve" between two points

18. MIT EECS 6.837, Durand and Cutler Interpolation Curves Curve is constrained to pass through all control points Given points P 0, P 1,... P n, find lowest degree polynomial which passes through the points x(t) = a n-1 t n-1 +.... + a 2 t 2 + a 1 t + a 0 y(t) = b n-1 t n-1 +.... + b 2 t 2 + b 1 t + b 0

19. MIT EECS 6.837, Durand and Cutler Interpolation vs. Approximation Curves Interpolation curve must pass through control points Approximation curve is influenced by control points

20. MIT EECS 6.837, Durand and Cutler Interpolation vs. Approximation Curves Interpolation Curve – over constrained → lots of (undesirable?) oscillations Approximation Curve – more reasonable?

21. MIT EECS 6.837, Durand and Cutler Cubic Bézier Curve 4 control points Curve passes through first & last control point Curve is tangent at P 0 to (P 0 -P 1 ) and at P 4 to (P 4 -P 3 )

22. MIT EECS 6.837, Durand and Cutler Cubic Bézier Curve de Casteljau's algorithm for constructing Bézier curves t t t t t t

23. MIT EECS 6.837, Durand and Cutler Cubic Bézier Curve

24. MIT EECS 6.837, Durand and Cutler Connecting Cubic Bézier Curves How can we guarantee C0 continuity (no gaps)? How can we guarantee C1 continuity (tangent vectors match)? Asymmetric: Curve goes through some control points but misses others

25. MIT EECS 6.837, Durand and Cutler Higher-Order Bézier Curves > 4 control points Bernstein Polynomials as the basis functions Every control point affects the entire curve –Not simply a local effect –More difficult to control for modeling

26. MIT EECS 6.837, Durand and Cutler Cubic BSplines ≥ 4 control points Locally cubic Curve is not constrained to pass through any control points

27. MIT EECS 6.837, Durand and Cutler Cubic BSplines Iterative method for constructing BSplines Shirley, Fundamentals of Computer Graphics

28. MIT EECS 6.837, Durand and Cutler Cubic BSplines

29. MIT EECS 6.837, Durand and Cutler Cubic BSplines can be chained together better control locally (windowing)

30. MIT EECS 6.837, Durand and Cutler Bézier is not the same as BSpline Relationship to the control points is different Bézier BSpline

31. MIT EECS 6.837, Durand and Cutler Bezier is not the same as Bspline But we can convert between the curves using the basis functions:

32. MIT EECS 6.837, Durand and Cutler NURBS (generalized BSplines) BSpline: uniform cubic BSpline NURBS: Non-Uniform Rational BSpline –non-uniform = different spacing between the blending functions, a.k.a. knots –rational = ratio of polynomials (instead of cubic)

33. MIT EECS 6.837, Durand and Cutler Questions?

34. MIT EECS 6.837, Durand and Cutler Today Review Motivation Spline Curves Spline Surfaces / Patches –Tensor Product –Bilinear Patches –Bezier Patches Subdivision Surfaces Procedural Texturing

35. MIT EECS 6.837, Durand and Cutler Tensor Product Of two vectors: Similarly, we can define a surface as the tensor product of two curves.... Farin, Curves and Surfaces for Computer Aided Geometric Design

36. MIT EECS 6.837, Durand and Cutler Bilinear Patch

37. MIT EECS 6.837, Durand and Cutler Bilinear Patch Smooth version of quadrilateral with non-planar vertices... –But will this help us model smooth surfaces? –Do we have control of the derivative at the edges?

38. MIT EECS 6.837, Durand and Cutler Bicubic Bezier Patch

39. MIT EECS 6.837, Durand and Cutler Editing Bicubic Bezier Patches Curve Basis Functions Surface Basis Functions

40. MIT EECS 6.837, Durand and Cutler Modeling with Bicubic Bezier Patches Original Teapot specified with Bezier Patches

41. MIT EECS 6.837, Durand and Cutler Modeling Headaches Original Teapot model is not "watertight": intersecting surfaces at spout & handle, no bottom, a hole at the spout tip, a gap between lid & base

42. MIT EECS 6.837, Durand and Cutler Trimming Curves for Patches Shirley, Fundamentals of Computer Graphics

43. MIT EECS 6.837, Durand and Cutler Questions?

44. MIT EECS 6.837, Durand and Cutler Today Review Motivation Spline Curves Spline Surfaces / Patches Subdivision Surfaces Procedural Texturing

45. MIT EECS 6.837, Durand and Cutler Chaikin's Algorithm

46. MIT EECS 6.837, Durand and Cutler Doo-Sabin Subdivision

47. MIT EECS 6.837, Durand and Cutler Doo-Sabin Subdivision http://www.ke.ics.saitama-u.ac.jp/xuz/pic/doo-sabin.gif

48. MIT EECS 6.837, Durand and Cutler Loop Subdivision Shirley, Fundamentals of Computer Graphics

49. MIT EECS 6.837, Durand and Cutler Loop Subdivision Some edges can be specified as crease edges http://grail.cs.washington.edu/projects/subdivision/

50. MIT EECS 6.837, Durand and Cutler Weird Subdivision Surface Models Justin Legakis

51. MIT EECS 6.837, Durand and Cutler Questions?

52. MIT EECS 6.837, Durand and Cutler Today Review Motivation Spline Curves Spline Surfaces / Patches Procedural Texturing

53. MIT EECS 6.837, Durand and Cutler Procedural Textures Image by Turner Whitted f (x,y,z) → color

54. MIT EECS 6.837, Durand and Cutler Procedural Solid Textures Noise Turbulence Ken Perlin Justin Legakis

55. MIT EECS 6.837, Durand and Cutler Questions?

56. MIT EECS 6.837, Durand and Cutler Next Thursday: Animation I: Keyframing