1. Curves and Surfaces

2. Limitations of Polygonal Meshes
planar facets

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

4. Continuity definitions:
C0 continuous
curve/surface has no breaks/gaps/holes
"watertight"
C1 continuous
curve/surface derivative is continuous
"looks smooth, no facets"
C2 continuous
curve/surface 2nd derivative is continuous
Actually important for shading

5. Splines... Classic problem: How to draw smooth curves?
Spline curve: smooth curve that is defined by a
sequence of points
Requirements: …
H&B 8-8:

6. Splines 1 Spline curve: smooth curve that is defined by a
sequence of points
Approximating spline
Interpolating spline
H&B 8-8:

7. Splines 2 Convex hull: Smallest polygon that encloses all points
Approximating spline
Interpolating spline
H&B 8-8:

8. Splines 3 Control graph: polyline through sequence of points
Approximating spline
Interpolating spline
H&B 8-8:

9. Splines 4 Splines in computer graphics: Piecewise cubic splines
Segments
H&B 8-8:

10. Splines 5 Segments have to match ‘nicely’.
Given two segments P(u) en Q(v).
We consider the transition of P(1) to Q(0).
Zero-order parametric continuity
C0: P(1) = Q(0).
Endpoint of P(u) coincides with startpoint Q(v).
P(u)
Q(v)
H&B 8-8:

11. Splines 6 Segments have to match ‘nicely’.
Given two segments P(u) en Q(v).
We consider the transition of P(1) to Q(0).
First order parametric continuity
C1: dP(1)/du = dQ(0)/dv.
Direction of P(1) coincides with direction of Q(0).
P(u)
Q(v)
H&B 8-8:

12. Splines 7 First order parametric continuity gives a smooth
curve. Sometimes good enough, sometimes not.
line segment
circle arc
Suppose that you are bicycling over the curve.
What to do with the steering rod at the transition?
Turn around! Discontinuity in the curvature!
H&B 8-8:

13. Splines 8 Given two segments P(u) and Q(v).
We consider the transition of P(1) to Q(0).
Second order parametric continuity
C2: d2P(1)/du2 = d2Q(0)/dv2.
Curvatures in P(1) and Q(0) are equal.
P(u)
Q(v)
H&B 8-8:

14. Splines 9 So far: considered parametric continuity.
Here the vectors are exactly equal.
It suffices to require that the directions are the same:
geometric continuity.
Q(v)
P(u)
Q(v)
P(u)
H&B 8-8:

15. Splines 10 Given two segments P(u) en Q(v).
We consider the transition of P(1) to Q(0).
First order geometric continuity:
G1: dP(1)/du =  dQ(0)/dv with  >0.
Direction of P(1) coincides with direction Q(0).
P(u)
Q(v)
H&B 8-8:

16. Representation cubic spline 1
H&B 8-8:
U: Powers of u C: Coefficient matrix

17. Representation cubic spline 2
Control points or
Control vectors
H&B 8-8:
Matrix Mspline :’translates’ geometric info to coefficients

18. Representation cubic spline 3
H&B 8-8:

19. Representation cubic spline 4
H&B 8-8:

20. Representatie cubic spline 5
Puzzle: Describe a line segment between the points P0 en P1
with those three variants. P1
u=1 P0 u
u=0
H&B 8-8:

21. Representation cubic spline 6
H&B 8-8:

22. Representation cubic spline 7
H&B 8-8:

23. Representation cubic spline 8
H&B 8-8:

24. Spline surface 1
P33
P03
P30
P20
P10
H&B 8-8:
P00

25. Spline surface 2
H&B 8-8:

26. Spline surface 2
P33
P03
P30
P20
P10
H&B 8-8:
P00

27. Spline surface 3 v u
H&B 8-8:

28. Spline surface 4 v du := 1/nu; // nu: #facets u-direction u
dv := 1/nv; // nv: #facets v-direction
for i := 0 to nu1 do
u := i*du;
for j := 0 to nv  1 do
v := j*dv;
DrawQuad(P(u,v), P(u+du, v), P(u+du, v+dv), P(u, v+dv)) u
H&B 8-8:

29. Spline surface 5 // Alternative: calculate points first
for i := 0 to nu do
for j := 0 to nv do
Q[i, j] := P(i/nu, j/nv);
for i := 0 to nu  1 do
for j := 0 to nv  1 do
DrawQuad(Q[i, j], Q[i+1, j], Q[i+1, j+1], Q[i, j+1]) v u
H&B 8-8:

30. Spline surface 6 // Alternative: calculate points first,
// triangle version
for i := 0 to nu do
for j := 0 to nv do
Q[i, j] := P(i/nu, j/nv);
for i := 0 to nu  1 do
for j := 0 to nv  1 do
DrawTriangle(Q[i, j], Q[i+1, j], Q[i+1, j+1]);
DrawTriangle(Q[i, j], Q[i+1, j+1], Q[i, j+1]); v u
H&B 8-8:

31. Spline surface 7 // Alternative: calculate points first,
// triangle variant, triangle strip
for i := 0 to nu do
for j := 0 to nv do
Q[i, j] := P(i/nu, j/nv);
for i := 0 to nu  1 do
glBegin(GL_TRIANGLE_STRIP);
for j := 0 to nv  1 do
glVertex(Q[i, j]);
glVertex(Q[i, j+1]);
glEnd; v u
H&B 8-8:

32. Bézier spline curves 1
H&B 8-10:

33. Bézier spline curves 2 P1 P0
H&B 8-10:

34. Bézier spline curves 3 P1 P2 P0
H&B 8-10:

35. Bézier spline curves 4 P2 P3 P1 P0
H&B 8-10:

36. Bézier spline curves 5 P2 P3 P1 P0
H&B 8-10:

37. Bézier spline curves 6 P2 P3 P1 P0
H&B 8-10:

38. Bézier spline curves 7 P2 P3 P1 P0
H&B 8-10:

39. Bézier spline curves 8 P2 P3 Q1 P1 Q0 Q2 Q3 P0
H&B 8-10:

40. Bézier spline curves 8 P2 P3 Q1 P1 Q0 Q2 Q3 P0
H&B 8-10:

41. Bézier spline curves 9 P2 P3 Q1 P1 Q0 Q2 Q3 P0
H&B 8-10:

42. Bézier spline curves 9 P2 P3 P1 Q1 Q0 Q2 Q3 P0
H&B 8-10:

43. Bézier spline curves 10
H&B 8-10:

44. Bézier surface 1
P33
P03
P30
P20
P10
P00
H&B 8-10:

45. Bézier surface 2
P33
P30
Q33
P03
P00
Q30
Q03
H&B 8-10:
Q00

46. Bézier surface 2
P33
P30
Q33
P03
P00
Q03
Q30
H&B 8-10:
Q00

47. Bézier surface 3
P33
P30
P03
P00
Q30
H&B 8-10:
Q00

48. Bézier surface 3
P33
P30
P03
P00
Q30
H&B 8-10:
Q00

49. Bézier surface 3
P33
P30
P03
P00
Q30
H&B 8-10:
Q00