1. © University of Wisconsin, CS559 Spring 2004
Last Time
Subdivision
Sphere
Fractals
Modified Butterfly scheme
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2. © University of Wisconsin, CS559 Spring 2004
Today
Parametric curves
Hermite curves
Bezier curves
Homework 6 due
Homework 7 available Friday, due May 4
No lecture April 22 (Thursday)
4/20/04
© University of Wisconsin, CS559 Spring 2004

3. Subdivision Shortcomings
Subdivision surfaces suffer from parameterization problems
How are texture coordinates handled?
Surface are 2D in nature, how do we attached a 2D space to subdivision surfaces?
Subdivision surfaces can be difficult to model with
Still may take complex underlying surface to get desired shape
Effects of changes to underlying mesh are not always obvious
Evaluation is non-trivial
Methods exist for taking a point in the underlying mesh and figuring out where it will go on the surface, but it isn’t easy
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© University of Wisconsin, CS559 Spring 2004

4. What are Parametric Curves?
Define a parameter space
1D for curves
2D for surfaces
Define a mapping from parameter space to 3D points
A function that takes parameter values and gives back 3D points
The result is a parametric curve or surface
(Fx(t), Fy(t))
Mapping:F:t→(x,y) 1 1 t
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5. © University of Wisconsin, CS559 Spring 2004
Why Parametric Curves?
Parametric curves are intended to provide the generality of polygon meshes but with fewer parameters for smooth surfaces
Polygon meshes have as many parameters as there are vertices (at least)
Fewer parameters makes it faster to create a curve, and easier to edit an existing curve
Normal vectors and texture coordinates can be easily defined everywhere
Parametric curves are easier to animate than polygon meshes
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6. © University of Wisconsin, CS559 Spring 2004
Parametric Curves
We have seen the parametric form for a line:
Note that x, y and z are each given by an equation that involves:
The parameter t
Some user specified control points, x0 and x1
This is an example of a parametric curve
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© University of Wisconsin, CS559 Spring 2004

7. Basis Functions (first sighting)
A line is the sum of two functions multiplied by vectors:
A linear combination of basis functions
t and 1-t are the basis functions
The weights are called control points
(x0,y0,z0) and (x1,y1,z1) are the control points
They control the shape and position of the curve
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8. © University of Wisconsin, CS559 Spring 2004
Hermite Spline
A spline is a parametric curve defined by control points
The term spline dates from engineering drawing, where a spline was a piece of flexible wood used to draw smooth curves
The control points are adjusted by the user to control the shape of the curve
A Hermite spline is a curve for which the user provides:
The endpoints of the curve
The parametric derivatives of the curve at the endpoints (tangents with length)
The parametric derivatives are dx/dt, dy/dt, dz/dt
That is enough to define a cubic Hermite spline
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9. Control Point Interpretation
End Tangent
Start Tangent
End Point
Start Point
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10. © University of Wisconsin, CS559 Spring 2004
Hermite Spline (2)
Say the user provides
A cubic spline has degree 3, and is of the form:
For some constants a, b, c and d derived from the control points, but how?
We have constraints:
The curve must pass through x0 when t=0
The derivative must be x’0 when t=0
The curve must pass through x1 when t=1
The derivative must be x’1 when t=1
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Hermite Spline (3)
Solving for the unknowns gives:
Rearranging gives: or
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Basis Functions
A point on a Hermite curve is obtained by multiplying each control point by some function and summing
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Splines in 2D and 3D
For higher dimensions, define the control points in higher dimensions (that is, as vectors)
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14. © University of Wisconsin, CS559 Spring 2004
Bezier Curves (1)
Different choices of basis functions give different curves
Choice of basis determines how the control points influence the curve
In Hermite case, two control points define endpoints, and two more define parametric derivatives
For Bezier curves, two control points define endpoints, and two control the tangents at the endpoints in a geometric way
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15. Control Point Interpretation
Point along start tangent
End Point
Point along end Tangent
Start Point
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16. © University of Wisconsin, CS559 Spring 2004
Bezier Curves (2)
The user supplies d control points, pi
Write the curve as:
The functions Bid are the Bernstein polynomials of degree d
Where else have you seen them?
This equation can be written as a matrix equation also
There is a matrix to take Hermite control points to Bezier control points
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17. Bezier Basis Functions for d=3
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18. Bezier Curves of Varying Degree
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19. Bezier Curve Properties
The first and last control points are interpolated
The tangent to the curve at the first control point is along the line joining the first and second control points
The tangent at the last control point is along the line joining the second last and last control points
The curve lies entirely within the convex hull of its control points
The Bernstein polynomials (the basis functions) sum to 1 and are everywhere positive
They can be rendered in many ways
E.g.: Convert to line segments with a subdivision algorithm
4/20/04
© University of Wisconsin, CS559 Spring 2004

20. Rendering Bezier Curves (1)
Evaluate the curve at a fixed set of parameter values and join the points with straight lines
Advantage: Very simple
Disadvantages:
Expensive to evaluate the curve at many points
No easy way of knowing how fine to sample points, and maybe sampling rate must be different along curve
No easy way to adapt. In particular, it is hard to measure the deviation of a line segment from the exact curve
4/20/04
© University of Wisconsin, CS559 Spring 2004

21. Rendering Bezier Curves (2)
Recall that a Bezier curve lies entirely within the convex hull of its control vertices
If the control vertices are nearly collinear, then the convex hull is a good approximation to the curve
Also, a cubic Bezier curve can be broken into two shorter cubic Bezier curves that exactly cover the original curve
This suggests a rendering algorithm:
Keep breaking the curve into sub-curves
Stop when the control points of each sub-curve are nearly collinear
Draw the control polygon - the polygon formed by the control points
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© University of Wisconsin, CS559 Spring 2004

22. Sub-Dividing Bezier Curves
Step 1: Find the midpoints of the lines joining the original control vertices. Call them M01, M12, M23
Step 2: Find the midpoints of the lines joining M01, M12 and M12, M23. Call them M012, M123
Step 3: Find the midpoint of the line joining M012, M123. Call it M0123
The curve with control points P0, M01, M012 and M0123 exactly follows the original curve from the point with t=0 to the point with t=0.5
The curve with control points M0123 , M123 , M23 and P3 exactly follows the original curve from the point with t=0.5 to the point with t=1
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23. Sub-Dividing Bezier Curves
M12 P1 P2
M012
M0123
M123
M01
M23 P0 P3
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24. Sub-Dividing Bezier CurvesP1 P2 P0 P3
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25. de Casteljau’s Algorithm
You can find the point on a Bezier curve for any parameter value t with a similar algorithm
Say you want t=0.25, instead of taking midpoints take points 0.25 of the way
M12 P2 P1
M23
t=0.25
M01 P0 P3
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26. © University of Wisconsin, CS559 Spring 2004
Invariance
Translational invariance means that translating the control points and then evaluating the curve is the same as evaluating and then translating the curve
Rotational invariance means that rotating the control points and then evaluating the curve is the same as evaluating and then rotating the curve
These properties are essential for parametric curves used in graphics
It is easy to prove that Bezier curves, Hermite curves and everything else we will study are translation and rotation invariant
Some forms of curves, rational splines, are also perspective invariant
Can do perspective transform of control points and then evaluate the curve
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27. © University of Wisconsin, CS559 Spring 2004
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
4/20/04
© University of Wisconsin, CS559 Spring 2004

28. Piecewise Bezier Curve
“knot”
P0,0
P1,3
P0,3
P1,0
P1,2
P1,1
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29. © 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
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?
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30. © University of Wisconsin, CS559 Spring 2004
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
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31. © 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.
4/20/04
© University of Wisconsin, CS559 Spring 2004