Curve ModelingNURBS

Dr. S.M. MalaekAssistant: M. Younesi

Motivation

Motivation

We need more controlmore controlover the curve

MotivationDesign Sketch

MotivationRendered

MotivationFinal Physical Product

MotivationDifferent weights of control points

MotivationIt is hard to product exact circle with B-spline.

Circles can only be represented with rationalrational function (i.e., functions that are quotients of two polynomials).

four closed B-spline curves with 8 control points :

Degree 2 Degree 3 Degree 5 Degree 10

Motivation Circles, ellipses and many other curves that cannot be represented by polynomials.

we need an extension to B-splinecurves.

Motivation Generalize B-splines to rational curvesusing homogeneous coordinateshomogeneous coordinates:

Non-Uniform Rational B-Splines(NURBS)

NURBS Different weights of Pi, to have different attraction factors.

W1=0.4

W3=3.0

Wi=1

NURBS Different weights of Pi, to have different attraction factors.

Wi=1 W0=W2=W3=W5=1, W1=W4=4

NURBS B-spline Curve: Given n+1 control points P0, P1, ..., Pnand knot vector U = { u0, u1, ..., um } of m+1 knots, the B-spline curve of degree p :

i

n

ipi uNu PC )()(

0,∑=

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

∑==

i

iw

ii

ii

n

ipi

w

wzwywxw

uNu0

, )()(C

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

1i

i

i

i zyx

P⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

i

iw

ii

ii

wi

wzwywxw

P

Multiply the coordinates of Piwith a weight w

B-Spline Curve in homogeneous coordinate

NURBS

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

∑==

i

iw

ii

ii

n

ipi

w

wzwywxw

uNu0

, )()(C

( )( )( )

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

∑∑∑∑

=

=

=

=

=

i

ni pi

ii

ni pi

ii

ni pi

ii

ni pi

w

wuNzwuNywuNxwuN

u

)()()()(

)(

0 ,

0 ,

0 ,

0 ,

C

NURBS Convert it back to Cartesian coordinate by dividing Cw(u) with the fourth coordinate:

∑

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

∑=

==

n

ii

i

i

nj ipj

ipi

zyx

wuNwuN

u0

0 ,

,

1)(

)()(C

∑∑ =

=

=n

iiipin

i ipi

wuNwuN

u0

,

0 ,

)()(

1)( PC

∑=

=n

iipi uRu

0, )()( PC∑ =

=nj jpj

ipipi

wuN

wuNuR

0 ,

,,

)(

)()(

TwoTwoImmediate Results

Two Immediate Results

1. If all weights are equal to 1, a NURBScurve reduces to a B-spline curve.

2. NURBS Curves are Rational.

∑=

=n

iipi uRu

0, )()( PC

∑ =

= nj jpj

ipipi

wuN

wuNuR

0 ,

,,

)(

)()(

Propertiesof

NURBS NURBS Basis Function

Properties of NURBS Basis Function1. Ri,p(u) is a degree p rational function in u

2. Nonnegativity -- For all i and p, Ri,p(u) is nonnegative

3. Local Support -- Ri,p(u) is a non-zero on [ui,ui+p+1)

4. On any knot span [ui, ui+1) at most p+1 degree p basis functions are non-zero.

5. Partition of Unity -- The sum of all non-zero degree pbasis functions on span [ui, ui+1) is 1:

Properties of NURBS Basis Function6. If the number of knots is m+1, the degree of the basis

functions is p, and the number of degree p basis functions is n+1, then m = n + p + 1 :

7. Basis function Ri,p(u) is a composite curve of degree prational functions with joining points at knots in [ui, ui+p+1 )

8. At a knot of multiplicity k, basis function Ri,p(u) is Cp-k

continuous.

9. If wi = c for all i, where c is a non-zero constant, Ri,p(u) = Ni,p(u)

Propertiesof

NURBS NURBS Curves

Properties of NURBS Curves1. NURBS curve C(u) is a piecewise curve with each

component a degree p rational curve.

2. Equality m = n + p + 1 must be satisfied .

3. A clamped NURBS curve C(u) passes through the two end control points P0 and Pn

4. Strong Convex Hull Property

5. Local Modification Scheme6. C(u) is Cp-k continuous at a knot of multiplicity k

7. B-spline Curves and Bézier Curves Are Special Cases of NURBS Curves

Properties of NURBS Curves8. Modifying Weights: increasing the value of wi will

pull the curve toward control point Pi

a NURBS curve of degree 6and its NURBS basis functions. The selected control point is P9.

Properties of NURBS Curves Modifying Weights: decreasing the value of wi will pushthe curve away from control point Pi

If a weight becomes zero, the coefficient of Pi is zero and, control point Pi has no impact on the computation of C(u) for any u (i.e., Pi is "disabled").

Properties of NURBS Curves

A NURBS curve of degree 6defined by 9 control points (n = 8) and 16 knots (m = 15) .The selected control point is P4. Since the coefficient of P4, N4,6(u), is non-zero on [u4, u4+6+1) = [0,1),changing w4 affects the entire curve!

Rational Bézier Curves

Rational Bézier Curves We have learned that projecting a 4-dimensional B-spline curve to hyperplane w=1 yields a 3-dimensional NURBS curve.

What if this B-spline curve is a Bézier curve?

The result is a Rational BRational Béézierzier curve!

Rational Bézier Curves Since a rational Bézier curve is a special case of NURBS curves, rational Bézier curves satisfy all important properties that NURBS curves have.

since there is no internal knots, rational Bézier curves do not have the local modification property, which means modifying a control point or its weight will cause a global change

∑=

=n

iini uRu

0, )()( PC ∑ =

= n

j jjn

inini

wuB

wuBuR

0 ,

,,

)(

)()(

Rational Bézier Curves

Modifying the weight of a control point will push or pull the curve away from

or toward the control point.

∑=

=n

iini uRu

0, )()( PC ∑ =

= n

j jjn

inini

wuB

wuBuR

0 ,

,,

)(

)()(

Rational Bézier Curves:Conic Sections

Rational Bézier Curves: Conic Sections we use rational Bézier curve of degree 2, the coefficients are :

22,2

1,2

20,2

)()1(2)(

)1()(

uuBuuuB

uuB

=−=−=

∑ =

= n

j jjn

inini

wuB

wuBuR

0 ,

,,

)(

)()(

∑=

=n

iini uRu

0, )()( PC

Rational Bézier Curves: Conic Sections

The equation of this rational Bézier curve of degree 2 is :

( )210 PPPC 2222 )1(2)1(

)1(2)1(1)( uuwuu

uuwuuu +−+−

+−+−=

10,1

<≤−

= rr

rw

r=0W=0

Straight line

r<1/2W<1Ellipse

r=1/2W=1Parabola

r>1/2W>1Hyperbola

P0=1P1=wP2=1

Rational Bézier Curves: Conic Sections

r=0W=0

Straight line

r<1/2W<1Ellipse r=1/2W=1Parabola r>1/2W>1Hyperbola

P0=1P1=wP2=1

Rational Bézier Curves: Conic Sections

Circular Arcs and Circles

)sin(aw = Weights w is equal to the sine of the half ngle at control point P1

Rational Bézier Curves: Conic Sections

Quarter Circle: P1=90º, a=45º,

22)45sin( == ow

Rational Bézier Curves: Conic Sections

1/3 Circle: P1=60º, a=30º,

21)30sin( == ow

Rational Bézier Curves: Conic Sections

Full Circle:

⎥⎦⎤

⎢⎣⎡= 1,1,1,

32,

32,

32,

31,

31,

31,0,0,0U ⎥⎦

⎤⎢⎣⎡= 1,1,1,

43,

43,

21,

21,

41,

41,0,0,0U