Parametric Curves & Surfaces

Introduction to Computer GraphicsCSE 470/598

Arizona State University

Dianne Hansford

Overview

• What is a parametric curve/surface?• Why use parametric curves &

surfaces?• Bézier curves & surfaces• NURBS• Trimmed surfaces• OpenGL library

What is a parametric curve?

Recall functions from calculus ...

Example: y = 2x – 2x2

Parametric curvesgive us more flexibility

xy

=x2x – 2x2

To illustrate, weplot graph of function

What is a parametric curve?

2D parametric curve takes the form

xy

f(t)g(t)

Where f(t) and g(t)are functions of t

=

Example: Line thru points a and b

xy

(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

What is a parametric curve?

3D curves defined similarly

xyz

f(t)g(t)h(t)

=

Example: helix

xyz

cos(t)sin(t)t

=

Bézier Curves

Polynomial parametric curvesf(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)

Bézier CurvesExamples

linear: b(t) = (1-t) b0 + t b1

quadratic: b(t) = (1-t)2 b0 + 2(1-t)t b1 + t2

b2cubic: b(t) = (1-t)3 b0 + 3(1-t)2 t b1

+ 3(1-t)t2 b2 + t3 b3

Bernstein basis Bin (t) = {n!/(n-i)! i!} (1-t)n-i ti

n=1

n=2

n=3

Bézier Curves

Bézier Curves

Bézier points and Bernstein basis

Nice, intuitive method to create curves Variable display resolutionMinimal storage needs

Bézier Curves

Bézier points and Bernstein basis

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!

nice, intuitive method to create curves

Compare to

at t=0

Bézier Curves

local and global parameter intervals

Piecewise Bézier curves global parameter u e.g., time

[u0,u1]

[u1,u2]

Each curve evaluated for t in [0,1]

If specify u in global spacethen must find t in local space

t = (u-u0) / (u1-u0)figure: 2 quadratic curves

Bézier Curves

Piecewise Bézier curves

Conditions to create a smooth transition

Filled squares are “junction” Bezier points-- start/endpoint of a curve

Bézier Curves in OGL

Basic steps:

Define curve by specifying degree, control points and parameter space [u0,u1]

Enable evaluatorCall evaluator with parameter u in [u0, u1]

Specify each u:glEvalCoord1*()

Autocreate uniformly spaced u:glMapGrid1*()glEvalMesh1()

glMap1*()

or

Color and texture available too!

Bézier Curve Evaluation

de Casteljau algorithmanother example of repeated subdivision

On each polygon leg,construct a point in theratio t : (1-t)

bn0(t) is point on

curvefigure: n=3

What is a parametric surface?

3D parametric surface takes the form

xyz

f(u,v)g(u,v)h(u,v)

Where f,g,h are bivariate functions of u and v

=

mapping u,v-space to 3-space;this happens to be a function too

Example: x(u,v) =

uv

u2 + v2

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

Bézier Surface

Structure

v

(0,0)

u

(1,1)

b00

b33

b30

b03

uv

Bézier Surface

Properties

boundary curveslie on surface

boundary curvesdefined by boundary polygons

Bézier Surface

Properties

Nice, intuitive method for creating surfaces

Variable display resolution

Minimal storage

Bézier Surface

Multiple patches connected smoothly

Conditions on control netsimilar to curves …difficult to do manually

Bézier Surface

Display

wireframe shaded

choose directionisoparametric curves

OGL: glMap2*, glEvalCoord2*

glMapGrid2, glEvalMesh2

OGL: triangles & normalscreated for you

NURBS

Non-uniform Rational B-splines

B-splines are piecewise polynomialsOne or more Bezier curves /surfacesOne control polygon

Rational: let’s us represent circles exactly

GLU NURBS utility

Trimmed SurfacesParametric surface with parts of the

domain “invisible”

Jorg Peters’ UFL group

GLU Trimmed NURBS utility

Surf

Lab

domain

References

The Essentials of CAGDby Gerald Farin & DCH, AK Petershttp://eros.cagd.eas.asu.edu/%7Efarin/essbook/essbook.html

Ken Joy’s CAGD notes (UC Davis)http://graphics.cs.ucdavis.edu/CAGDNotes/homepage.html

Jorg Peters’ UFL SurfLab grouphttp://www.cise.ufl.edu/research/SurfLab/index.html

OpenGL Red Book – Chapter 12