8. Q about GL to render polygon
glBegin(GL_TRIANGLES)
glVertex3f(0,0,0);
glVertex3f(1,1,0);
glVertex3f(1,0,0);
glVertex3f(0,1,0);
glEnd();
(B)
glBegin(GL_QUADS)
(C)
glBegin(GL_TRIANGLES)
glVertex3f(0,0,0);
glVertex3f(1,1,0);
glVertex3f(1,0,0);
glVertex3f(0,1,0);
glEnd();
(D)
glBegin(GL_QUADS)
(E) I just really don’t know

10. Turn on all pixels inside the triangle
But what about pixels on the edge?

12. Top-Left Rasterization Rule
[Rasterization Rules (Direct3D 10) – MSDN]

13. Q About Rasterization

15. Q about Normals What is the per-polygon normal shown? 1,1,1 0,0,1
1,0,0
0,1,0
Don’t know

16. Q about Normals What is the per-vertex normal at point A? 1,1,1 1,1,-1
1/sqrt(3), 1/sqrt(3), 1/sqrt(3)
- 1/sqrt(3), 1/sqrt(3),1/sqrt(3)
Don’t know

21. Limitations of Polygonal Meshes
Planar facets (& silhouettes)
Fixed resolution
Deformation is difficult
No natural parameterization (for texture mapping)

22. Can We Disguise the Facets?

23. Some Non-Polygonal Modeling Tools
Extrusion
Surface of Revolution
Spline Surfaces/Patches
Quadrics and other implicit polynomials

24. Continuity definitions:
C0 continuous
curve/surface has no breaks/gaps/holes
G1 continuous
tangent at joint has same direction
C1 continuous
curve/surface derivative is continuous
tangent at join has same direction and magnitude
Cn continuous
curve/surface through nth derivative is continuous
important for shading

25. Definition: What's a Spline?
Smooth curve defined by some control points
Moving the control points changes the curve
Interpolation
Bézier (approximation)
BSpline (approximation)

26. Interpolation Curves / Splines

27. Linear Interpolation Simplest "curve" between two points Q(t) =
Spline Basis Functions
a.k.a. Blending Functions

29. [bilinear interpolation]

30. Q about bi-linear interpolation
What is the value at point A? 1 2 3 4
Don’t know

31. Interpolation is not just about surfaces, used heavily in images
Suppose you start with the smallest image and need a big one?

32. Nearest, bilinear, b-spline interpolation

33. Interpolation Curves
Curve is constrained to pass through all control points
Given points P0, P1, ... Pn, find lowest degree polynomial which passes through the points x(t) = an-1tn a2t2 + a1t + a0 y(t) = bn-1tn b2t2 + b1t + b0

34. Interpolation vs. Approximation Curves
curve must pass through control points
Approximation
curve is influenced by control points

35. Interpolation vs. Approximation Curves
Interpolation Curve – over constrained → lots of (undesirable?) oscillations
Approximation Curve – more reasonable?

36. Cubic Bézier Curve 4 control points
Curve passes through first & last control point
Curve is tangent at P0 to (P0-P1) and at P4 to (P4-P3)
A Bézier curve is bounded by the convex hull of its control points.

37. Linear Interpolation Simplest "curve" between two points Q(t) =
Spline Basis Functions
a.k.a. Blending Functions

38. Bernstein Polynomials
Cubic Bézier Curve
Bernstein Polynomials

39. Connecting Cubic Bézier Curves
Asymmetric: Curve goes through some control points but misses others
How can we guarantee C0 continuity?
How can we guarantee G1 continuity?
How can we guarantee C1 continuity?
Can’t guarantee higher C2 or higher continuity

40. Connecting Cubic Bézier Curves
Where is this curve
C0 continuous?
G1 continuous?
C1 continuous?
What’s the relationship between:
the # of control points, and
the # of cubic Bézier subcurves?

41. Cubic BSplines ≥ 4 control points Locally cubic
Curve is not constrained to pass through any control points
A BSpline curve is also bounded by the convex hull of its control points.

42. Cubic BSplines

43. Cubic BSplines Can be chained together
Better control locally (windowing)

44. Bézier is not the same as BSpline

45. Bézier is not the same as BSpline
Relationship to the control points is different
Bézier
BSpline

46. Converting between Bézier & BSpline
original control points as Bézier
new BSpline control points to match Bézier
new Bézier control points to match BSpline
original control points as BSpline

47. Converting between Bézier & BSpline
Using the basis functions:

48. 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)

49. Bicubic Bezier Patch

50. Editing Bicubic Bezier Patches
Curve Basis Functions
Surface Basis Functions

51. Modeling with Bicubic Bezier Patches
Original Teapot specified with Bezier Patches

52. 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

53. Cubic Bézier Curve t t t t t t
de Casteljau's algorithm for constructing Bézier curves t t t t t t

54. Subdivision surfaces

55. Doo-Sabin Subdivision

56. Doo-Sabin Subdivision

57. Bezier Patches? or
Triangle Mesh?
Henrik Wann Jensen

58. Shirley, Fundamentals of Computer Graphics
Loop Subdivision
Shirley, Fundamentals of Computer Graphics

59. Loop Subdivision Some edges can be specified as crease edges