2 coen 290 - computer graphics i evening’s goals n discuss types of algebraic curves and surfaces...

2COEN 290 - Computer Graphics I

Evening’s Goals

Discuss types of algebraic curves and surfaces

Develop an understanding of curve basis and blending functions

Introduce Non-Uniform Rational B-Splines• also known as NURBS


Problem #1

We want to create a curve (surface) which passes through as set of data points


Problem #2

We want to represent a curve (surface) for modeling objects• compact form• easy to share and manipulate• doesn’t necessarily pass through points


Types of Algebraic Curves (Surfaces)

Interpolating• curve passes through points• useful of scientific visualization and data

analysis Approximating

• curve’s shape controlled by points• useful for geometric modeling


Representing Curves

We’ll generally use parametric cubic polynomials• easy to work with• nice continuity properties

3

0

33

2210)(

i

iiuc

ucucuccup


Parametric Equations

Compute values based on a parameter• parameter generally defined over a limited

space• for our examples, let

For example

1,0u

)2sin()2cos()( uuuf


What’s Required for Determining a Curve

Enough data points to determine coefficients• four points required for a cubic curve

For a smooth curve, want to match• continuity• derivatives

Good news is that this has all been figured out


Geometry Matrices

We can identify curve types by their geometry matrix• another 4x4 matrix

Used to define the polynomial coefficients for a curves blending functions

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COEN 290 - Computer Graphics I

Interpolating Curves

We can use the interpolating geometry matrix to determine coefficients

Given four points in n-dimensional space , compute

3

2

1

0

3

2

1

0

5.45.135.135.4

5.4185.229

15.495.5

0001

p

p

p

p

c

c

c

c

ip

ic

11


Recasting the Matrix Form as Polynomials

We can rewrite things a little

cuup

)( 321 uuuu

pMc g

pub

pMuup g

)(

)(

12


Blending Functions

Computing static coefficients is alright, but we’d like a more general approach

Recast the problem to finding simple polynomials which can be used as coefficients

3

0

)()(i

ii pubup

13


Blending Functions ( cont. )

Here are the blending functions for the interpolation curve

323

322

321

320

5.45.41)(

5.13185.4)(

5.135.229)(

5.495.51)(

uuuub

uuuub

uuuub

uuuub

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Blending Functions ( cont. )

-0.5

-0.25

0

0.25

0.5

0.75

1

1.250.00

0.09

0.18

0.27

0.36

0.45

0.54

0.63

0.72

0.81

0.90

0.99

b0(u)

b1(u)

b2(u)

b3(u)

Blending Functions for the Interpolation case

15


That’s nice … but

Interpolating curves have their problems• need both data values and coefficients to share• hard to control curvature (derivatives)

Like a less data dependent solution

16


Approximating Curves

Control the shape of the curve by positioning control points

Multiples types available• Bezier• B-Splines• NURBS (Non-Uniform Rational B-Splines)

Also available for surfaces

17


Bezier Curves

Developed by a car designer at Renault Advantages

• curve contained to convex hull• easy to manipulate• easy to render

Disadvantages• bad continuity at endpoints

– tough to join multiple Bezier splines

18


Bezier Curves ( cont. )

Bezier geometry matrix

1331

0363

0033

0001

gM

19


Bernstein Polynomials

Bezier blending functions are a special set of called the Bernstein Polynomials• basis of OpenGL’s curve evaluators

knknk uu

k

nub

)1()(

)!(!

!

knk

n

k

n

20


Bezier Blending Functions

0

0.25

0.5

0.75

1

1.25

0.00

0.09

0.18

0.27

0.36

0.45

0.54

0.63

0.72

0.81

0.90

0.99

b0(u)

b1(u)

b2(u)

b3(u)

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Cubic B-Splines

B-Splines provide the one thing which Bezier curves don’t -- continuity of derivatives

However, they’re more difficult to model with• curve doesn’t intersect any of the control

vertices

22


Curve Continuity

Like all the splines we’ve seen, the curve is only defined over four control points

How do we match the curve’s derivatives?

2ip

1ipip

1ip

2ip

23


Cubic B-Splines ( cont. )

B-Spline Geometry Matrix

1331

0363

0303

0141

61

gM

24


B-Spline Blending Functions

-0

1/6

1/3

1/2

2/3

5/6

1

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

b0(u)

b1(u)

b2(u)

b3(u)

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B-Spline Blending Functions ( cont. )

Notice something about the blending functions

Additionally, note that

3

0

0.1)(i

i ub

)`1()0(

)1()0(

)1()0(

32

21

10

bb

bb

bb

26


Basis Functions

The blending functions for B-splines form a basis set.• The “B” in B-spline is for “basis”

The basis functions for B-splines comes from the following recursion relation

otherwise0

1)( 10 kk

k

uuuuB

kB

)()()( 11

1

11uBuBuB d

kuuuud

kuuuud

k kdk

dk

kdk

k

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Rendering Curves - Method 1

Evaluate the polynomial explicitly

33

2210)( ucucuccup

float px( float u ) { float v = c[0].x; v += c[1].x * u; v += c[2].x * pow( u, 2 ); v += c[3].x * pow( u, 3 ); return v;}

28


Evaluating Polynomials

That’s about the worst way you can compute a polynomial• very inefficient

–pow( u, n );

This is better, but still not great

float px( float u ) { float v = c[0].x; v += c[1].x * u; v += c[2].x * u*u; v += c[3].x * u*u*u; return v;}

29


Horner’s Method

We do a lot more work than necessary• computing u2 and u3 is wasteful

uuu

uuuu

uuu

3

2

30


Horner’s Method ( cont. )

Rewrite the polynomial

uccucuc

ucucuccup

3210

33

2210)(

float px( float u ) { return c[0].x + u*(c[1].x + u*(c[2].x + u*c[3].x)); }

31


Rendering Curves - Method 1 ( cont. )

Even with Horner’s Method, this isn’t the best way to render• equal sized steps in parameter doesn’t

necessarily mean equal sized steps in world space

glBegin( GL_LINE_STRIP );for ( u = 0; u <= 1; u += du ) glVertex2f( px(u), py(u) );glEnd();

32



Use subdivision and recursion to render curves better

Bezier curves subdivide easily

0p

1p 2p

3p

p

1021 ppp

33



Three subdivision steps required

212

11102

10

3221

22121

11021

0

pppppp

ppppppppp

32108

1

1021

0

33 pppp

ppp

34



void drawCurve( Point p[] ) { if ( length( p ) < MAX_LENGTH ) draw( p ); Point p01 = 0.5*( p[0] + p[1] ); Point p12 = 0.5*( p[1] + p[2] ); Point p23 = 0.5*( p[2] + p[3] ); Point p012 = 0.5*( p01 + p12 ); Point p123 = 0.5*( p12 + p13 ); Point m = 0.5*( p012 + p123 ); drawCurve( p[0], p01, p012, m ); drawCurve( m, p123, p23, p[3] );}

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2 coen 290 - computer graphics i evening’s goals n discuss types of algebraic curves and surfaces...

Documents

coefficients n

computer graphics

polynomials n

bezier curves n

n interpolating curves

cubic curve n

4x4 matrix n

geometry matrices n