1. (c) 2002 University of Wisconsin
Last Time
Hermite Curves
Bezier Curves
11/21/02
(c) 2002 University of Wisconsin

2. (c) 2002 University of Wisconsin
Today
Bezier Continuity
B-spline Curves
11/21/02
(c) 2002 University of Wisconsin

3. (c) 2002 University of Wisconsin
Longer Curves
A single cubic Bezier or Hermite curve can only capture a small class of curves
At most 2 inflection points
One solution is to raise the degree
Allows more control, at the expense of more control points and higher degree polynomials
Control is not local, one control point influences entire curve
Alternate, most common solution is to join pieces of cubic curve together into piecewise cubic curves
Total curve can be broken into pieces, each of which is cubic
Local control: Each control point only influences a limited part of the curve
Interaction and design is much easier
11/21/02
(c) 2002 University of Wisconsin

4. Piecewise Bezier Curve
“knot”
P0,0
P1,3
P0,3
P1,0
P1,2
P1,1
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5. (c) 2002 University of Wisconsin
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
Question: How do we ensure that two Hermite curves are C1 across a knot?
Question: How do we ensure that two Bezier curves are C0, or C1, or C2 across a knot?
11/21/02
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6. (c) 2002 University of Wisconsin
Achieving Continuity
For Hermite curves, the user specifies the derivatives, so C1 is achieved simply by sharing points and derivatives across the knot
For Bezier curves:
They interpolate their endpoints, so C0 is achieved by sharing control points
The parametric derivative is a constant multiple of the vector joining the first/last 2 control points
So C1 is achieved by setting P0,3=P1,0=J, and making P0,2 and J and P1,1 collinear, with J-P0,2=P1,1-J
C2 comes from further constraints on P0,1 and P1,2
11/21/02
(c) 2002 University of Wisconsin

7. (c) 2002 University of Wisconsin
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.
11/21/02
(c) 2002 University of Wisconsin

8. (c) 2002 University of Wisconsin
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?
11/21/02
(c) 2002 University of Wisconsin

9. (c) 2002 University of Wisconsin
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?
11/21/02
(c) 2002 University of Wisconsin

10. Bezier Geometric Continuity
P0,1
P0,2
P0,0
P1,3 J
P1,1
P1,2
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(c) 2002 University of Wisconsin

11. Problem with Bezier Curves
To make a long continuous curve with Bezier segments requires using many segments
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
11/21/02
(c) 2002 University of Wisconsin

12. (c) 2002 University of Wisconsin
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
11/21/02
(c) 2002 University of Wisconsin

13. 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
11/21/02
(c) 2002 University of Wisconsin

14. (c) 2002 University of Wisconsin
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
11/21/02
(c) 2002 University of Wisconsin

15. 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:
11/21/02
(c) 2002 University of Wisconsin

16. Uniform Cubic B-splines on [0,m)
Curve:
n is the total number of control points
d is the order of the curves, 2  d  n+1
Bk,d are the uniform B-spline blending functions of degree d-1
Pk are the control points
Each Bk,d is only non-zero for a small range of t values, so the curve has local control
11/21/02
(c) 2002 University of Wisconsin

17. Uniform Cubic B-spline Blending Functions
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18. (c) 2002 University of Wisconsin
Computing the Curve
P1B1,4
P4B4,4
P0B0,4
P2B2,4
P6B6,4
P3B3,4
P5B5,4
The curve can’t start until there are 4 basis functions active
11/21/02
(c) 2002 University of Wisconsin

19. Using Uniform B-splines
At any point t along a piecewise uniform cubic B-spline, there are four non-zero blending functions
Each of these blending functions is a translation of B0,4
Consider the interval 0t<1
We pick up the 4th section of B0,4
We pick up the 3rd section of B1,4
We pick up the 2nd section of B2,4
We pick up the 1st section of B3,4
11/21/02
(c) 2002 University of Wisconsin

20. (c) 2002 University of Wisconsin
Demo
11/21/02
(c) 2002 University of Wisconsin

21. Uniform B-spline at Arbitrary t
The interval from an integer parameter value i to i+1 is essentially the same as the interval from 0 to 1
The parameter value is offset by i
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: i = floor(t)
Evaluate:
Valid parameter range: 0t<n-3, where n is the number of control points
11/21/02
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22. (c) 2002 University of Wisconsin
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
11/21/02
(c) 2002 University of Wisconsin

23. (c) 2002 University of Wisconsin
Demo
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(c) 2002 University of Wisconsin

24. 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
11/21/02
(c) 2002 University of Wisconsin

25. From B-spline to Bezier
Both B-spline and Bezier curves represent cubic curves, so either can be used to go from one to the other
Recall, a point on the curve can be represented by a matrix equation:
P is the column vector of control points
M depends on the representation: MB-spline and MBezier
T is the column vector containing: t3, t2, t, 1
By equating points generated by each representation, we can find a matrix MB-spline->Bezier that converts B-spline control points into Bezier control points
11/21/02
(c) 2002 University of Wisconsin

26. B-spline to Bezier Matrix
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27. 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
Non-uniform B-splines can have blending functions starting and stopping anywhere, and the blending functions are not all the same
11/21/02
(c) 2002 University of Wisconsin

28. (c) 2002 University of Wisconsin
B-Spline Knot Vectors
Knots: Define a sequence of parameter values at which the blending functions will be switched on and off
Knot values are increasing, and there are n+d+1 of them, forming a knot vector: (t0,t1,…,tn+d) with t0  t1  …  tn+d
Curve only defined for parameter values between td-1 and tn+1
These parameter values correspond to the places where the pieces of the curve meet
There is one control point for each value in the knot vector
The blending functions are recursively defined in terms of the knots and the curve degree
11/21/02
(c) 2002 University of Wisconsin

29. B-Spline Blending Functions
The recurrence relation starts with the 1st order B-splines, just boxes, and builds up successively higher orders
This algorithm is the Cox - de Boor algorithm
Carl de Boor is a professor here
11/21/02
(c) 2002 University of Wisconsin

30. Uniform Cubic B-splines
Uniform cubic B-splines arise when the knot vector is of the form (-3,-2,-1,0,1,…,n+1)
Each blending function is non-zero over a parameter interval of length 4
All of the blending functions are translations of each other
Each is shifted one unit across from the previous one
Bk,d(t)=Bk+1,d(t+1)
The blending functions are the result of convolving a box with itself d times, although we will not use this fact
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31. (c) 2002 University of Wisconsin
Bk,1
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32. (c) 2002 University of Wisconsin
Bk,2
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33. (c) 2002 University of Wisconsin
Bk,3
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34. (c) 2002 University of Wisconsin
B0,4
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35. (c) 2002 University of Wisconsin
B0,4
Note that the functions given on slides 5 and 6 are translates of this function obtained by using (t-1), (t-2) and (t-3) instead of just t, and then selecting only a sub-range of t values for each function
11/21/02
(c) 2002 University of Wisconsin

36. Interpolation and Continuity
The knot vector gives a user control over interpolation and continuity
If the first knot is repeated three times, the curve will interpolate the control point for that knot
Repeated knot example: (-3,-3,-3, -2, -1, 0, …)
If a knot is repeated, so is the corresponding control point
If an interior knot is repeated, continuity at that point goes down by 1
Interior points can be interpolated by repeating interior knots
A deep investigation of B-splines is beyond the scope of this class
11/21/02
(c) 2002 University of Wisconsin

37. (c) 2002 University of Wisconsin
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
11/21/02
(c) 2002 University of Wisconsin

38. Refining Uniform Cubic B-splines
Basic idea: Generate 2n-3 new control points:
Add a new control point in the middle of each curve segment: P’0,1, P’1,2, P’2,3 , …, P’n-2,n-1
Modify existing control points: P’1, P’2, …, P’n-2
Throw away the first and last control
Rules:
If the curve is a loop, generate 2n new control points by averaging across the loop
When drawing, don’t draw the control polygon, join the X(i) points
11/21/02
(c) 2002 University of Wisconsin

39. (c) 2002 University of Wisconsin
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 evaluating in screen space
Can perfectly represent conic sections: circles, ellipses, etc
Piecewise cubic curves cannot do this
11/21/02
(c) 2002 University of Wisconsin