1. © University of Wisconsin, CS559 Spring 2004
Last Time
Introduction to parametric curves
Hermite curves
Bezier curves
Continuity
4/27/04
© University of Wisconsin, CS559 Spring 2004

2. © University of Wisconsin, CS559 Spring 2004
Today
Geometric continuity
Parametric surfaces
General Tensor product surfaces
Bezier surfaces
BSplines
Homework 7 available, due Thursday May 6 in class
4/27/04
© University of Wisconsin, CS559 Spring 2004

3. © University of Wisconsin, CS559 Spring 2004
Continuity
When two curves are joined, we typically want some degree of continuity across the boundary (the knot)
C0, “C-zero”, point-wise continuous, curves share the same point where they join
C1, “C-one”, continuous derivatives, curves share the same parametric derivatives where they join
C2, “C-two”, continuous second derivatives, curves share the same parametric second derivatives where they join
Higher orders possible
4/27/04
© University of Wisconsin, CS559 Spring 2004

4. © University of Wisconsin, CS559 Spring 2004
Bezier Continuity
P0,1
P0,2
P0,0
P1,3 J
P1,2
P1,1
Disclaimer: PowerPoint curves are not Bezier curves, they are interpolating piecewise quadratic curves! This diagram is an approximation.
4/27/04
© University of Wisconsin, CS559 Spring 2004

5. © University of Wisconsin, CS559 Spring 2004
Sketch of Proof for C1
Bezier curve equation:
Parametric derivative:
Evaluated at endpoint of curve (note proves tangent property):
4/27/04
© University of Wisconsin, CS559 Spring 2004

6. © University of Wisconsin, CS559 Spring 2004
Proof (cont)
P0,1
P0,2
P1,3
P0,0 J
P1,2
P1,1
C1 requires equal parametric derivatives:
4/27/04
© University of Wisconsin, CS559 Spring 2004

7. © University of Wisconsin, CS559 Spring 2004
DOF and Locality
The number of degrees of freedom (DOF) can be thought of as the number of things a user gets to specify
If we have n piecewise Bezier curves joined with C0 continuity, how many DOF does the user have?
If we have n piecewise Bezier curves joined with C1 continuity, how many DOF does the user have?
Locality refers to the number of curve segments affected by a change in a control point
Local change affects fewer segments
How many segments of a piecewise cubic Bezier curve are affected by each control point if the curve has C1 continuity?
What about C2?
4/27/04
© University of Wisconsin, CS559 Spring 2004

8. © University of Wisconsin, CS559 Spring 2004
Geometric Continuity
Derivative continuity is important for animation
If an object moves along the curve with constant parametric speed, there should be no sudden jump at the knots
For other applications, tangent continuity might be enough
Requires that the tangents point in the same direction
Referred to as G1 geometric continuity
Curves could be made C1 with a re-parameterization: u=f(t)
The geometric version of C2 is G2, based on curves having the same radius of curvature across the knot
What is the tangent continuity constraint for a Bezier curve?
4/27/04
© University of Wisconsin, CS559 Spring 2004

9. Bezier Geometric Continuity
P0,1
P0,2
P0,0
P1,3 J
P1,1
P1,2
for some k
4/27/04
© University of Wisconsin, CS559 Spring 2004

10. © University of Wisconsin, CS559 Spring 2004
Parametric Surfaces
Define points on the surface in terms of two parameters
Simplest case: bilinear interpolation
x(s,1) s
P1,1
P0,1
x(s,t) t
P0,0 s
x(s,0)
P1,0
4/27/04
© University of Wisconsin, CS559 Spring 2004

11. Tensor Product Surface Patches
Defined over a rectangular domain
Valid parameter values come from within a rectangular region in parameter space: 0s<1, 0t<1
Use a rectangular grid of control points to specify the surface
4 points in the bi-linear case on the previous slide, more in other cases
Surface takes the form:
For some functions Fi,s and Fj,t
4/27/04
© University of Wisconsin, CS559 Spring 2004

12. © University of Wisconsin, CS559 Spring 2004
Bezier Patches
As with Bezier curves, Bin(s) and Bjm(t) are the Bernstein polynomials of degree n and m respectively
Most frequently, use n=m=3: cubic Bezier patch
Need 4x4=16 control points, Pi,j
4/27/04
© University of Wisconsin, CS559 Spring 2004

13. © University of Wisconsin, CS559 Spring 2004
Bezier Patches (2)
Edge curves are Bezier curves
Any curve of constant s or t is a Bezier curve
One way to think about it:
Each row of 4 control points defines a Bezier curve in s
Evaluating each of these curves at the same s provides 4 virtual control points
The virtual control points define a Bezier curve in t
Evaluating this curve at t gives the point x(s,t)
x(s,t)
4/27/04
© University of Wisconsin, CS559 Spring 2004

14. Properties of Bezier Patches
Which vertices, if any, does the patch interpolate? Why?
What can you say about the tangent plane at each corner? Why?
Does the patch lie within the convex hull of its control vertices?
4/27/04
© University of Wisconsin, CS559 Spring 2004

15. Properties of Bezier Patches
The patch interpolates its corner points
Comes from the interpolation property of the underlying curves
The tangent plane at each corner interpolates the corner vertex and the two neighboring edge vertices
The tangent plane is the plane that is perpendicular to the normal vector at a point
The tangent plane property derives from the curve tangent properties and the way to compute normal vectors
The patch lies within the convex hull of its control vertices
The basis functions sum to one and are positive everywhere
4/27/04
© University of Wisconsin, CS559 Spring 2004

16. Bezier Patch Matrix Form
Note that the 3 matrices stay the same if the control points do not change
The middle product can be pre-computed, leaving only:
4/27/04
© University of Wisconsin, CS559 Spring 2004

17. © University of Wisconsin, CS559 Spring 2004
Bezier Patch Meshes
A patch mesh is just many patches joined together along their edges
Patches meet along complete edges
Each patch must be a quadrilateral OK
Not OK
Not OK OK
4/27/04
© University of Wisconsin, CS559 Spring 2004

18. Bezier Mesh Continuity
Just like curves, the control points must satisfy constraints to ensure parametric continuity
How do we ensure C0 continuity along an edge?
How do we ensure C1 continuity along an edge?
How do we ensure C2 continuity along an edge?
For geometric continuity, constraints are less rigid
What can you say about the vertices around a corner if there must be C1 continuity at the corner point?
4/27/04
© University of Wisconsin, CS559 Spring 2004

19. Bezier Mesh Continuity
Just like curves, the control points must satisfy rigid constraints to ensure parametric continuity
C0 continuity along an edge? Share control points at the edge
C1 continuity along an edge? Control points across edge are collinear and equally spaced
C2 continuity along an edge? Constraints extent to points farther from the edge
For geometric continuity, constraints are less rigid
Still collinear for G1, but can be anywhere along the line
What can you say about the vertices around a corner if there must be C1 continuity at the corner point?
They are co-planar (not the interior points, just corner and edge)
4/27/04
© University of Wisconsin, CS559 Spring 2004

20. Rendering Bezier Patches
Option 1: Evaluate at fixed set of parameter values and join up with triangles
Can’t use quadrilaterals because points may not be co-planar
Ideal situation for triangle strips
Advantage: Simple, and OpenGL has commands to do it for you
Disadvantage: No easy way to control quality of appearance
Option 2: Subdivision
Allows control of error in the triangle approximation
Extend 1D curve subdivision to do surfaces
Similar to other subdivision schemes we’ve seen
4/27/04
© University of Wisconsin, CS559 Spring 2004

21. © University of Wisconsin, CS559 Spring 2004
Midpoint Subdivision
Repeatedly join midpoints to find new control vertices
Do it first for each row of original control points: 4x4 -> 4x7
Then do it for each column of new control points:4x7 -> 7x7
4/27/04
© University of Wisconsin, CS559 Spring 2004

22. © University of Wisconsin, CS559 Spring 2004
A Potential Problem
One (good) way to subdivide, is:
If a control mesh is flat enough – draw it
Else, subdivide into 4 sub-patches and recurse on each
Problem: Neighboring patches may not be subdivided to the same level
Cracks can appear because join edges have different control meshes
This can be fixed by adding extra edges
Crack
4/27/04
© University of Wisconsin, CS559 Spring 2004

23. Computing Normal Vectors
The partial derivative in the s direction is one tangent vector
The partial derivative in the t direction is another
Take their cross product, and normalize, to get the surface normal vector
4/27/04
© University of Wisconsin, CS559 Spring 2004

24. Bezier Curve/Surface Problems
To make a long continuous curve with Bezier segments requires using many segments
Same for large surface
Maintaining continuity requires constraints on the control point positions
The user cannot arbitrarily move control vertices and automatically maintain continuity
The constraints must be explicitly maintained
It is not intuitive to have control points that are not free
4/27/04
© University of Wisconsin, CS559 Spring 2004