1. CS 450: Computer Graphics PARAMETRIC SPLINES AND SURFACES
Spring 2015
Dr. Michael J. Reale

2. INTRODUCTION
Mostly, we’ve concentrated on objects that are made up of triangles
Hardware is set up to render triangles
Objects and animation paths can have many different underlying geometric descriptions
Example: curves and curved surfaces can be described precisely by equations  equations can then be evaluated, and triangles can be created and rendered

3. ADVANTAGES OF CURVES AnD CURVED SURFACES
1) Compact representation
E.g., can define a curve precisely with 4 control points (instead of approximating it with a bunch of lines)
2) Scalable
I.e., can turn into 2 polygons or 2000 polygons, depending on how much detail you want
3) Smoother and more continuous primitives
4) Animation and collision detection may be simpler and faster
I.e., using underlying mathematical descriptions rather than checking each polygon with every other polygon
Bonus advantages for real-time rendering:
1) Less memory needed for representation
2) Fewer points to transform
3) If graphics hardware DOES handle curved surface descriptions directly, less to transfer from CPU to GPU

4. OVERVIEW We will touch on the following subjects:
Parametric Curves
Parametric Curved Surfaces
Although there are many kinds of curves/splines/patches out there, we will focus on Bézier curves and Bézier patches

5. PARAMETRIC CURVES

6. INTRODUCTION
Parametric curve = curve that is a function of a parameter t  p(t)
t = 0  start of the curve
t = 1  end of the curve
Increments of t  evenly-spaced distances along curve path  great for animations or camera paths!
When dealing with 2D or 3D points, each dimension is a separate function of t  x(t), y(t), z(t)
For the slides that follow, however, we will look at a single function p(t) that outputs a point (whatever its dimension)

7. LINEAR INTERPOLATION Linear interpolation = traces out straight line:
p0 = starting point, p1 = ending point
Problem: more than two points in your path  sudden jerks as you jump from one line segment to the next (at joints)  discontinuous at joints
There must be a better way…
What we get with linear
interpolation
 What we want…

8. INTRODUCING BEZIER CURVES
Bézier curves (pronounced beh-ze-eh)
Basic idea is repeatedly (and recursively) doing linear interpolation  called de Casteljau algorithm
Let’s say we have three control points: a, b, and c
To get a point along the curve (say, at t = 1/3) 
Get d = 1/3 of the way from a to b
Get e = 1/3 of the way from b to c
Get f = p(1/3) = 1/3 of the way from d to e
In general:

9. BEZIER CURVE DEGREES AND CONTROL POINTS
The maximum degree of this example is n = 2 (quadratic)
The number of control points is (n+1) = 3
Given a Bezier curve of degree n, the number of control points is (n+1)

10. BEZIER CURVES: DE CASTELJAU ALGORITHM
A given point on the curve is defined recursively
Control points: p1 , p2 , p3 , …
After applying linear interpolation k times  intermediate control points pik are obtained
Previous example:

11. BEZIER CURVES: DEFINITION
A Bezier curve with (n+1) initial control points can be described recursively:
A single point on the curve is defined by:

12. BEZIER CURVES: PREVIOUS EXAMPLE
So, how does this work? Let’s look at our previous example…
We already know our initial control points (k = 0):
Get the next interpolation level (k = 1):
Final level (k = 2 = n):

13. BEZIER CURVES: ANOTHER EXAMPLE
Let’s say we have a quartic Bezier curve
Degree n = 4
Number of control points (n+1) = 5
We know values of initial control points
Work our way up tree, interpolating as we go

14. BEZIER CURVES USING BERNSTEIN POLYNOMIALS
Remember how earlier we had turned our quadratic Bezier curve into a nice tidy algebraic formula?
Turns out we can do the same with EVERY Bezier curve WITHOUT having to do the repeated interpolation  use Bernstein form:
This form uses Bernstein polynomials (first term is binomial coefficient):

15. BEZIER CURVES USING BERNSTEIN POLYNOMIALS
Two properties of Bernstein polynomials:
Will stay between 0 and 1 if t is between 0 and 1 
All polynomials sum to 1 
Means curve will stay “close” to control points
In fact, a Bezier curve fits within convex hull of control points:

16. BEZIER CURVES USING BERNSTEIN POLYNOMIALS
Bernstein polynomials  also called blending functions or Bezier basis functions
n = 1
n = 2
n = 3

17. BEZIER CURVES in MATRIX FORM
It is comparatively easy to define a Bezier curve with a matrix
Create a matrix that contains the correct Bernstein polynomial values
Create another matrix for the control points
Reuse both of these matrices for each value of t you need
Example: a cubic Bezier curve in matrix form

18. DISADVANTAGES OF BEZIER CURVES
Do not pass through all control points (except endpoints)
Degree increases with number of control points  more expensive to evaluate
One solution to the latter problem: piecewise Bezier curves

19. PIECEWISE BEZIER CURVES
Let’s say we have two cubic Bezier curves:
q0 , q1 , q2 , q3
r0 , r1 , r2 , r3
We want to join these such that:
p(0) = q0
p(1) = q3 = r0
p(3) = r3
How do we do this?

20. PIECEWISE BEZIER CURVES
First of all, we have a t value that can be greater than 1:
If t is [0,1]  use first Bezier curve
If t > 1  scale and translate value and use t’ on second Bezier curve:

21. PIECEWISE BEZIER CURVES: C0 CONTINUITY
To connect the curves, we could just set q3 = r0
HOWEVER, there could be a sudden change in direction at joint  derivative not continuous
This gives us C0 continuity  no gaps in position

22. PIECEWISE BEZIER CURVES: G1 CONTINUITY
Tangent of curve at q0  parallel with (q1 – q0 )
Tangent of curve at q3  parallel with (q3 – q2 )
So, make sure (incoming tangent at joint) = (outgoing tangent at joint):
This gives us G1 continuity
Says nothing about the lengths of the tangents
Gn = geometric continuity  weaker than corresponding Cn continuity

23. PIECEWISE BEZIER CURVES: C1 CONTINUITY
In previous example, derivatives would be continuous if c = 1.0  implies t2 = 2.0
HOWEVER, we want t2 = 3.0
To get even better continuity, we can define c as follows:
This gives us C1 continuity  derivative is continuous (even at joints)
I.e., tangent vectors point in same direction and are of same length

24. C0 continuity (position)
G1 continuity
(tangent directions)
C1 continuity
(derivatives)

25. PARAMETRIC SURFACES

26. INTRODUCTION
Parametric surfaces = natural extension of parametric curves
Similar to the jump from lines to triangles/polygons
Often tessellated by evaluating surface at difference positions and then building triangles for rendering
Patch = surface with rectangular domain
Here we will discuss Bezier patches

27. BEZIER PATCHES Bezier patches = extends Bezier curves by:
Using two parameters (u,v)
Using bilinear interpolation
Also lies in convex hull of control points
Start with four points (instead of two): a, b, c, d
Interpolate a & b and c & d to get e and f:
Finally, linearly interpolate between e & f using v:

28. BEZIER PATCHES
More points can be added (i.e., degree can be increased)
Example: biquadratic Bezier surface with 9 control points (n = 2, # of control points = (n+1)*(n+1) = 3*3)

29. BEZIER PATCHES IN BERNSTEIN FORM
In Bernstein form, a Bezier patch is defined by:
m and n = degree of the surface
“Compound degree” = m x n

30. BEZIER PATCHES IN BERNSTEIN FORM: ANOTHER VIEW
If we reinterpret the previous equation 
We suddenly notice that qi(v) is a Bezier curve  so, basically, we are computing the points along each Bezier curve with a fixed v value, then computing the value from a Bezier curve made from THOSE points

31. BEZIER PATCH: EXAMPLE Generated quads 4 x 4 control points
Shaded quads

32. Move one control point
Most change near control point

33. PIECEWISE BEZIER PATCHES
Usually you will want to stitch patches together to get a complete surface
However, C0 continuity is going to look terrible
Can enforce C1 continuity by doing the following:
Take the two rows near boundary from each patch: a2j and b1j
For each j  a2j , b0j , and b1j :
Must be collinear (i.e., lie on line)
Must have same ratio (with k constant for all j):

34. PIECEWISE BEZIER PATCHES
Problem with previous approach: lose a lot of degrees of freedom (don’t have as much control over patches)
Even worse at corner where 4 patches meet  9 points all in same plane and evenly spaced
If you only want G1 continuity at corners (and only there), can relax to only make them coplanar

35. PIECEWISE BEZIER PATCHES
C1 continuity  required for good-looking texturing
G1 continuity  reasonable for reflections and shading, but C1 continuity is better

36. C0 continuity
(position)
C1 continuity
(derivative)