1. © University of Wisconsin, CS559 Fall 2004
Last Time
Subdivision
Sphere
Fractals
Modified Butterfly scheme
Implicit Surfaces
12/2/2004
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2. © University of Wisconsin, CS559 Fall 2004
Today
Parametric curves
Hermite curves
Bezier curves
Homework 7 due Thursday Dec 9
Will be available for collection early in exam week
Best 5 of 7 homeworks are used to compute grade, each one equally weighted
12/2/2004
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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
12/2/2004
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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 Fall 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 Fall 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
12/2/2004
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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 Fall 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 Fall 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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11. © University of Wisconsin, CS559 Fall 2004
Hermite Spline (3)
Solving for the unknowns gives:
Rearranging gives: or
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12. © University of Wisconsin, CS559 Fall 2004
Basis Functions
A point on a Hermite curve is obtained by multiplying each control point by some function and summing
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13. © University of Wisconsin, CS559 Fall 2004
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 Fall 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 Fall 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
12/2/2004
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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
12/2/2004
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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 subdivided into two shorter curves that exactly cover the original
This suggests an 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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22. © University of Wisconsin, CS559 Fall 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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23. © University of Wisconsin, CS559 Fall 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
12/2/2004
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24. Piecewise Bezier Curve
“knot”
P0,0
P1,3
P0,3
P1,0
P1,2
P1,1
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25. © University of Wisconsin, CS559 Fall 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
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26. © University of Wisconsin, CS559 Fall 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.
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27. © University of Wisconsin, CS559 Fall 2004
Sketch of Proof for C1
Bezier curve equation:
Parametric derivative:
Evaluated at endpoint of curve (note proves tangent property):
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28. © University of Wisconsin, CS559 Fall 2004
Proof (cont)
P0,1
P0,2
P1,3
P0,0 J
P1,2
P1,1
C1 requires equal parametric derivatives:
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29. © University of Wisconsin, CS559 Fall 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?
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30. Bezier Geometric Continuity
P0,1
P0,2
P0,0
P1,3 J
P1,1
P1,2
for some k
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31. © 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
12/2/2004
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32. 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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33. © University of Wisconsin, CS559 Fall 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
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34. © University of Wisconsin, CS559 Fall 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)
12/2/2004
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