1. Introduction to Parametric Curve and Surface Modeling

2. Motivation
Parametric curves and surfaces 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 fields can be properly defined everywhere
Parametric curves are easier to animate than polygon meshes

3. What is it? Mapping F :t → (x, y, z) 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
Mapping F :t → (x, y, z) 1 t

4. 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

5. 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
The parametric derivatives are dx/dt, dy/dt, dz/dt
That is enough to define a cubic Hermite spline, more derivatives are required for higher order curves

6. 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

7. Hermite Spline (3) Solving for the unknowns gives: Rearranging gives:

8. Basis Functions
A point on a Hermite curve is obtained by multiplying each control point by some function and summing
The functions are called basis functions

9. Hermite curves in 2D and 3D
We can treat x in the mapping as a vector. Its components can be explicitly written as

10. 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

11. Some Bezier Curves

12. Bezier Curves (2) The user supplies d control points, pi
Write the curve as:
The functions Bid are the Bernstein polynomials of degree d
This equation can be written as a matrix equation also
There is a matrix to take Hermite control points to Bezier control points

13. Bezier Basis Functions for d=3

14. Bezier Curve Matrix
Convert Bezier control points to Hermite control points

15. 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

16. Angel: Interactive Computer Graphics 4E © Addison-Wesley 2005
Convex Hull Property
The properties of the Bernstein polynomials ensure that all Bezier curves lie in the convex hull of their control points
Hence, even though we do not interpolate all the data, we cannot be too far away p1 p2
convex hull
Bezier curve p3 p0
Angel: Interactive Computer Graphics 4E © Addison-Wesley 2005

17. 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

18. 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

19. Sub-Dividing Bezier Curves
M12 P1 P2
M012
M0123
M123
M01
M23 P0 P3

20. Sub-Dividing Bezier CurvesP1 P2 P0 P3

21. 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

22. 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

23. 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

24. 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

25. Piecewise Bezier Curve
“knot”
P0,0
P1,3
P0,3
P1,0
P1,2
P1,1

26. 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?

27. 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

28. 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.

29. 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

30. 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

31. 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

32. 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

33. 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:

34. Angel: Interactive Computer Graphics 4E © Addison-Wesley 2005
Blending Functions
convex hull property
Angel: Interactive Computer Graphics 4E © Addison-Wesley 2005

35. Angel: Interactive Computer Graphics 4E © Addison-Wesley 2005
Splines and Basis
If we examine the cubic B-spline from the perspective of each control (data) point, each interior point contributes (through the blending functions) to four segments
We can rewrite p(u) in terms of the data points as
defining the basis functions {Bi(u)}
Angel: Interactive Computer Graphics 4E © Addison-Wesley 2005

36. Basis Cubic B-Spline Functions
In terms of the blending polynomials
Angel: Interactive Computer Graphics 4E © Addison-Wesley 2005

37. Uniform Cubic B-splines on [0,m)
Curve:
n+1 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

38. Uniform Cubic B-spline Blending Functions

39. 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

40. 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

41. 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

42. Closed curve
To create a closed B-spline curve, repeat the control points P0, P1, P2 at the end of the sequence
P0, P1, …., Pn, P0, P1, P2

43. 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
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

44. 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

45. Bk,1

46. Bk,2

47. Bk,3

48. B0,4

49. 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

50. 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

51. B-spline to Bezier Matrix

52. Parametric Surfaces x(s,1) s P1,1 P0,1 x(s,t) t P0,0 s x(s,0) P1,0
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

53. 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

54. 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

55. Bezier Patches (2) x(s,t) 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)

56. 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?

57. 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

58. 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:

59. 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

60. B-Spline Patches Defined just like Bezier surfaces:
Continuity is automatically obtained everywhere
defined over only 1/9 of region

61. Rational Curves Each point is the ratio of two curves Advantages:
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

62. OpenGL and Parametric Curves
OpenGL defines evaluators that evaluate a Bezier curve at a point
You give it the control points:
glMap1f(…)
Access a specific point:
glEvalCoord(param)
Access a sequence:
glMapGrid1f(n, t1,t2)
glEvalMesh1f(mode, p1, p2)
These functions evaluate the curve at equally spaced points - the poor way of drawing!

63. OpenGL and NURBS NURBS: Non-uniform Rational B-splines
The curved surface of choice in CAD packages
Support routines are part of the GLu utility library
Allows you to specify how they are rendered:
Can use points constantly spaced in parametric space
Can use various error tolerances - the good way!
Allows you to get back the lines that would be drawn
Allows you to specify trim curves
Only for surfaces
Cut out parts of the surface - in parametric space

64. 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)