1. © University of Wisconsin, CS559 Fall 2004
Last Time
Parametric Curves (Splines)
Hermite curves
Bezier curves
Brief mentions of rational curves, transforming between representation, subdivision for Bezier
Homework 7 due Thursday
12/7/2004
© University of Wisconsin, CS559 Fall 2004

2. © University of Wisconsin, CS559 Fall 2004
Today
Parametric surface patches
Bsplines
12/7/2004
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3. © University of Wisconsin, CS559 Fall 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
12/7/2004
© University of Wisconsin, CS559 Fall 2004

4. 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
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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
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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)
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7. 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?
12/7/2004
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8. 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:
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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
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10. 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?
12/7/2004
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11. 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
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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
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13. 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
12/7/2004
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14. 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
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B-splines
B-splines automatically take care of continuity, with exactly one control vertex per curve segment
Many types of B-splines: degree may be different (linear, quadratic, cubic,…) and they may be uniform or non-uniform
We will only look closely at uniform B-splines
With uniform B-splines, continuity is always one degree lower than the degree of each curve piece
Linear B-splines have C0 continuity, cubic have C2, etc
12/7/2004
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16. Uniform Cubic B-spline on [0,1)
Four control points are required to define the curve for 0t<1 (t is the parameter)
Not surprising for a cubic curve with 4 degrees of freedom
The equation looks just like a Bezier curve, but with different basis functions
Also called blending functions - they describe how to blend the control points to make the curve
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Basis Functions on [0,1)
Does the curve interpolate its endpoints?
Does it lie inside its convex hull?
B1,4
B2,4
B0,4
B3,4
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18. Uniform Cubic B-spline on [0,1)
The blending functions sum to one, and are positive everywhere
The curve lies inside its convex hull
The curve does not interpolate its endpoints
Requires hacks or non-uniform B-splines
There is also a matrix form for the curve:
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Demo
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20. Uniform B-spline at Arbitrary t
The interval from an integer parameter value v to v+1 is essentially the same as the interval from 0 to 1
The parameter value is offset by v
A different set of control points is needed
To evaluate a uniform cubic B-spline at an arbitrary parameter value t:
Find the greatest integer less than or equal to t: v = floor(t)
Evaluate:
Valid parameter range: 0t<n-3, where n is the number of control points
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Loops
To create a loop, use control points from the start of the curve when computing values at the end of the curve:
Any parameter value is now valid
Although for numerical reasons it is sensible to keep it within a small multiple of n
12/7/2004
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22. © University of Wisconsin, CS559 Fall 2004
Demo
12/7/2004
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23. B-splines and Interpolation, Continuity
Uniform B-splines do not interpolate control points, unless:
You repeat a control point three times
But then all derivatives also vanish (=0) at that point
To do interpolation with non-zero derivatives you must use non-uniform B-splines with repeated knots
To align tangents, use double control vertices
Then tangent aligns similar to Bezier curve
Uniform B-splines are automatically C2
All the blending functions are C2, so sum of blending functions is C2
Provides an alternate way to define blending functions
To reduce continuity, must use non-uniform B-splines with repeated knots
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Rendering B-splines
Same basic options as for Bezier curves
Evaluate at a set of parameter values and join with lines
Hard to know where to evaluate, and how pts to use
Use a subdivision rule to break the curve into small pieces, and then join control points
What is the subdivision rule for B-splines?
Instead of subdivision, view splitting as refinement:
Inserting additional control points, and knots, between the existing points
Useful not just for rendering - also a user interface tool
Defined for uniform and non-uniform B-splines by the Oslo algorithm
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25. Non-Uniform B-Splines
Uniform B-splines are a special case of B-splines
Each blending function is the same
A blending functions starts at t=-3, t=-2, t=-1,…
Each blending function is non-zero for 4 units of the parameter (if cubic)
Non-uniform B-splines
Blending functions not all the same
Blending functions can star or stop anywhere
Fine control of continuity, without vanishing derivatives
A deep investigation of B-splines is beyond the scope of this class
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Rational Curves
Each point is the ratio of two curves
Just like homogeneous coordinates:
NURBS: x(t), y(t), z(t) and w(t) are non-uniform B-splines
Advantages:
Perspective invariant, so can be evaluated in screen space
Can perfectly represent conic sections: circles, ellipses, etc
Piecewise cubic curves cannot do this
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B-Spline Surfaces
Defined just like Bezier surfaces:
Continuity is automatically obtained everywhere
BUT, the control points must be in a rectangular grid OK
Not OK
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How to Choose a Spline
Hermite curves are good for single segments where you know the parametric derivative or want easy control of it
Bezier curves are good for single segments or patches where a user controls the points
B-splines are good for large continuous curves and surfaces
NURBS are the most general, and are good when that generality is useful, or when conic sections must be accurately represented (CAD)
12/7/2004
© University of Wisconsin, CS559 Fall 2004