1. Parametric Curves & Surfaces
Introduction to Computer Graphics
CSE 470/598
Arizona State University
Dianne Hansford

2. Overview What is a parametric curve/surface?
Why use parametric curves & surfaces?
Bézier curves & surfaces
NURBS
Trimmed surfaces
OpenGL library

3. What is a parametric curve?
Recall functions from calculus ...
Example: y = 2x – 2x2
To illustrate, we
plot graph of function x y x
2x – 2x2 =
Parametric curves
give us more flexibility

4. What is a parametric curve?
2D parametric curve takes the form x y
f(t)
g(t)
Where f(t) and g(t)
are functions of t =
Example: Line thru points a and b x y
(1-t) ax + t bx
(1-t) ay+ t by =
Mapping of the real line to 2D: here t in [0,1]  line segment a,b

5. What is a parametric curve?
3D curves defined similarly x y z
f(t)
g(t)
h(t) =
Example: helix x y z
cos(t)
sin(t) t =

6. Bézier Curves Polynomial parametric curves
f(t), g(t), h(t) are polynomial functions
Bézier curve b(t)
Bézier control points bi
Bézier polygon
Curve mimics shape of polygon
t in [0,1] maps to curve “between” polygon
b(0) = b0 and b(1) = bn
figure: degree n=3 (cubic)

7. Bézier Curves Examples linear: b(t) = (1-t) b0 + t b1 n=1
quadratic: b(t) = (1-t)2 b0 + 2(1-t)t b1 + t2 b2
n=2
cubic: b(t) = (1-t)3 b0 + 3(1-t)2 t b1
+ 3(1-t)t2 b2 + t3 b3
n=3
Bernstein basis Bin (t) = {n!/(n-i)! i!} (1-t)n-i ti

8. Bézier Curves

9. Bézier Curves Bézier points and Bernstein basis
Nice, intuitive method to create curves
Variable display resolution
Minimal storage needs

10. Bézier Curves Bézier points and Bernstein basis Compare to
nice, intuitive method to create curves
Compare to
Monomial basis: 1, t, t2, t3 ,....
ex: quadratic a(t) = a0 + t a1 + t2 a2
a0 is point on curve
a1 is first derivative vector
a2 is second derivative vector
Not very practical to design curves with!
at t=0

11. Bézier Curves local and global parameter intervals
Piecewise Bézier curves  global parameter u
e.g., time
Each curve evaluated for t in [0,1]
[u0,u1]
If specify u in global space
then must find t in local space
[u1,u2]
t = (u-u0) / (u1-u0)
figure: 2 quadratic curves

12. Bézier Curves Piecewise Bézier curves
Conditions to create a smooth transition
Filled squares are “junction” Bezier points -- start/endpoint of a curve

13. Bézier Curves in OGL Basic steps:
Define curve by specifying degree, control points and parameter space [u0,u1]
Enable evaluator
Call evaluator with parameter u in [u0, u1]
glMap1*()
Autocreate uniformly spaced u: glMapGrid1*()
glEvalMesh1()
Specify each u:
glEvalCoord1*() or
Color and texture available too!

14. Bézier Curve Evaluation
de Casteljau algorithm
another example of repeated subdivision
On each polygon leg,
construct a point in the ratio t : (1-t)
bn0(t) is point on curve
figure: n=3

15. What is a parametric surface?
3D parametric surface takes the form
Where f,g,h are bivariate functions of u and v x y z
f(u,v)
g(u,v)
h(u,v) =
Example:
x(u,v) = u v
u2 + v2
mapping u,v-space to 3-space;
this happens to be a function too

16. Bézier Surface (Patch)
Polynomial parametric surface
f(u,v), g(u,v), h(u,v) are polynomial functions
written in the Bernstein basis
Bézier surface b(u,v)
Bézier control points bij
Bézier control net

17. Bézier Surface
Structure
b33
b03
(1,1)
b30 v v u
(0,0)
b00 u

18. Bézier Surface Properties boundary curves lie on surface defined by
boundary polygons

19. Bézier Surface Properties Nice, intuitive method for creating surfaces
Variable display resolution
Minimal storage

20. Bézier Surface Multiple patches connected smoothly
Conditions on control net similar to curves …
difficult to do manually

21. Bézier Surface Display shaded wireframe choose direction
OGL: triangles & normals
created for you
choose direction
isoparametric curves
OGL: glMap2*, glEvalCoord2*
glMapGrid2, glEvalMesh2

22. NURBS Non-uniform Rational B-splines
B-splines are piecewise polynomials
One or more Bezier curves /surfaces
One control polygon
Rational: let’s us represent circles exactly
GLU NURBS utility

23. Trimmed Surfaces
Parametric surface with parts of the domain “invisible”
Surf
Lab
domain
Jorg Peters’ UFL group
GLU Trimmed NURBS utility

24. References
The Essentials of CAGD by Gerald Farin & DCH, AK Peters
Ken Joy’s CAGD notes (UC Davis)
Jorg Peters’ UFL SurfLab group
OpenGL Red Book – Chapter 12