Curves and Surfaces

C d S f

Curves and Surfaces

Intro. to Computer Graphics Spring 2008 Y G Shin Spring 2008, Y. G. Shin

Representation of Curves and Representation of Curves and Surfaces

„

Key words: surface modeling,

„

Key words: surface modeling, parametric surface, continuity,

control points, basis functions, Bezier

control points, basis functions, Bezier

curve, B-spline curve

Why we need surface models?

„ All shapes can be described in terms of points.

y

But, it is impractical to enumerate the points that comprise a shape

„ We define shape indirectly through

expressions that relate certain properties of points that comprise them.

Intrinsic and extrinsic properties

„

Intrinsic properties

„ B has four sides y

„ B has four sides

„ All four sides have equal length g

„ All four angles are 90^o, ...

„

Extrinsic properties

„ two horizontal sides x

„ two vertical sides vertices of B are at P0, P1, P2 and P3

x

P3

Intrinsic and extrinsic properties

„ Shape definitions that use extrinsic properties of the shape are dependent on the coordinate of the shape are dependent on the coordinate system used.

„ a line: y = 3, 2 ≤ x ≤ 7y , Í Axis dependency

3

2 7 xx

Axis Independence

A mathematical representation of a line/curve is axis

i d d t if it h d d l th

independent if its shape depends on only the relative position of the points defining its

characteristic vectors and is independent of the characteristic vectors and is independent of the coordinate system used.

Axis-independent shape definition

„ Shape definitions that use intrinsic properties of the shape a e a is independent

shape are axis-independent.

) 1

( +

=

x

) 1

(

) 1

(

2 1

2 1

+

−

=

+

−

=

y y

x x

tp p

t y

tp p

t x

3

) 3 , 2 ( )

, (

1 0

1 1

1

= =

≤

≤

y

x p

p p

t

2 7 x

) 3 , 7 ( ) ,

(

2

=

=

y

p p

p

Curve & Surface Models

„ Explicit/implicit

P i / i

„ Parametric/non-parametric

„ Approximation

l h ll ti f d ti

„ polygon mesh : a collection of edges, vertices, and polygons

Nonparametric explicit Nonparametric explicit representation

„

x = x y f(x) y = f(x)

„

successive values of y can be obtained

b l l f

by plugging in successive values of x.

„

easy to generate polygons or line segments

„

single-valued function single valued function

2

2 x

r

y = −

2

2 x

r y

y

−

−

=

Nonparametric implicit Nonparametric implicit representation

„

f ( x , y , z ) = 0

„ Define curves as solution of equation system

„ E.g., a circle:g

2 2

2

+ y r r

x + =

Nonparametric implicit Nonparametric implicit representation

„

algebraic quadric surfaces

f i l i l f d 2

: f is a polynomial of degree <= 2

2 2

2 )

( x y z ax

+ by

+ cz

+ dxy + eyz + fxz f

0 2

2 2

2 2

2 )

, , (

= +

+ +

+

+ +

+ +

+

=

k jz

hy gx

fxz eyz

dxy cz

by ax

z y x f

0 1

:

0 1

:

2 2

2 2

2

+

=

− +

+

y x

cylinder

z y

x sphere

0 :

0 1

:

2 2

2

+ − =

=

− +

z y

x corn

y x

cylinder

0

: x

+ y

+ z =

paraboloid

Nonparametric implicit Nonparametric implicit representation

„ Coefficients determine geometric properties

„ Hard to render (have to solve non-linear equation system)

C l d l l d

„ Can represent closed or multi-valued curves

„ Easy to classify point-membership

0 1

: x² + y² + z² − = spherep y

Parametric Curve

2 3

10 2 11

3 12

)

( t a t a t a t a

x = + + +

30 31

2 32 3

33

20 21

2 22 3

23

) (

) (

a t

a t

a t

a t

z

a t

a t

a t

a t

y

+ +

+

=

+ +

+

=

30 31

32

)

(

Parametric Curve (Example)

to

from

line P

= (x

,y

) P

= (x

,y

)

1 0

) 1

(

) 1

(

t ty

y t y

tx x

t x

≤

≤ +

=

+

−

=

points control

: ,

1 0

)

1 (

2 1

2 1

P P

t ty

y t

y = − + ≤ ≤

circle unit

functions blending

: 1

,

t − t

)) 2

sin(

), 2

(cos(

) (

Q u = u π u π

Parametric Curve Characteristics

„ Simple to render

„ evaluate parameter functionp

„ Hard to check whether a point lies on curve

„ have to compute the inverse mapping from (x, y) to t

„ Can represent closed or multi-valuedCan represent closed or multi valued curves

„ Curve or surface can be easily t l t d t t d

translated or rotated

„ Composite curves and surfaces can be formed by piecewise descriptions

formed by piecewise descriptions

Parametric Curve Characteristics

„ No infinite slope problem

) 0 2

/ ) 2 cos(

2 / ) 2 sin(

( )

( '

: form parametric

u u

u

Q

π π π π

0

, 0 1

: form implicit

) 0 , 2 / ) 2 cos(

, 2 / ) 2 sin(

( )

( '

2

2

+ − = =

−

=

z y

x

u u

u

Q

π π π π

=

= y

) 0 2

/ 1 0 ( i

d i i i

h

, 0 ,

1 at

∞

⇒

−

=

y x

f

' ( , , ) /

form implicit

) 0 , 2 / 1 , 0 ( is derivative parametric

the π

Parametric Curve Characteristics

„ Not unit form

(e.g.) a circle with radius 1 centered at the origin

⎞

⎛

⎞

⎛ ⎟⎟

⎠

⎜⎜ ⎞

⎝

⎛

=

≡ =

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎛

=

=

t

- 1 y

sin

cos x t

y x

θ θ

⎠

⎝

⎠

⎝ y y

Tangent line to a curve g

„ The straight line that gives the curve's slope at a point

point

„ Deduced from the derivative of the curve at the point

X(t), y(t)

derivative vector derivative vector

(x’(t),y’(t))

Piecewise Polynomial Curves

„ Cut curve into segments and represent each segment as a polynomial curve

segment as a polynomial curve

„ But how do we guarantee smoothness at the joints? ( continuity problem)

joints? ( continuity problem)

Continuityy

„

Implies a notion of smoothness at the connection points

connection points

„

Parametric continuity

W i th f f ti

„ We view the curve or surface as a function rather than a shape.

„ Matching the parametric derivatives of

„ Matching the parametric derivatives of adjoining curve sections at their common boundary

„ You need a parameterization

Parametric Continuity

drawn be

can it

if continuous is

curve a

: C

agree curves

two the

of values -

) z y x (

paper.

the from

pencil the

lifting ut

witho

, continuous also

is curve derivative

the :

C

agree.

curves two

the of

values )

z , y , x (

1

junction.

at their agree

) dt / dz , dt / dy , dt / dx ( i.e.,

[ ^{Q ( t )} ] ^{l t th j i i t}

dt d

of magnitude and

direction the

: C

2 2

2

[ ^{Q ( t )} ] ^{are equal at the join point}

dt d

Geometric Continuity

„

Geometric continuity is defined using only the shape of the curve

the shape of the curve

„

Geometric smoothness independent of t i ti

parametrization

point end

common a

at segments two

joining

0

: (

)

G

=

d ) h

l b

l (d

ly continuous changes

direction tangent

s curve' a

:

point end

common a

at segments

two joining

1

) (

:

C

G

=

magnitude) the

y necessaril but

equal, (direction

The order of polynomial The order of polynomial curves

k) degree (

1 k

order of

polynomial

a + ≡

k k 2

2 1

0

c u c u c u

c )

u ( P

) deg ee (

o de o

po y o a a

+ +

+ +

= L

„

In computer graphics, usually degree = 3

„ Sufficient flexibility w/o much cost

„ The cubic is the lowest degree polynomial that g p y gives

C

C

Curve models

„ Curve fitting techniques (interpolation techniques)

„ pass through each and every data point

„ linear approximation, natural cubic spline

„ Curve fairing techniques (approximation techniques)

„ few if any points on the curve pass through

h d d t i t

each and every data points

„ Hermite curve, Bezier curve, B-spline curve

Natural Spline Curves

„ Motivated by loftman’s spline

L t i f d l ti

„ Long narrow strip of wood or plastic

„ Shaped by lead weights (called ducks)

„ a cubic spline curve q(u) composed of cubic

„ a cubic spline curve, q(u), composed of cubic

polynomials that interpolate the points P₀, P₁, …, P_n

„ C^n-1 continuity can be achieved from splines of

„ C continuity can be achieved from splines of degree n

Natural Cubic Splines

„ divide the interval [a,b] into n intervals [u_i,u_i+1], for i=0 to n-1 . The numbers u are called knots

The numbers u_i are called knots.

„ The vector [u₀,u₁, …., u_i-1] is called a knot vector for the spline.

If the knots are equidistantly distributed in the interval [a,b], we If the knots are equidistantly distributed in the interval [a,b], we say the spline is uniform, otherwise we say it is non-uniform.

P1 P₃

P0

Pn

P2

u0 u₁ u₂ u₃ u_n

Natural Cubic Splines

„ each cubic spline curve is determined by the position vectors, tangent vectors and parameter values

vectors, tangent vectors and parameter values

3 3 i 2

2 i 1

i 0

i

i

( u ) c c u c u c u

q = + + +

1 to

1 for

) ( ' )

( '

to 1 for

)

( and

) (

1 1 1

−

=

=

=

=

=

+

−

−

n i

u q

u q

n i

p u

q p

u q

i i

i i

i i

i i

i i

P1 P₃

1

to 1 for

)

( '' )

( ''

1 to

1 for

)

( )

(

1 1

−

=

= ₊

+

n i

u q

u q

n i

u q

u q

i i

i i

i i

i i

P0

Pn

P2

u0 u₁ u₂ u₃ u_n

Cubic Splines

„ the polynomial coefficients of a cubic spline are dependent on all n control points

dependent on all n control points

→ a change in any one segment affects the entire curve

cu e

„ It is inconvenient to represent the curve directly using the coefficients

c

← the relationship between the shape of the curve and the coefficients is not clear or intuitive

⇒ th l i l f i t t l i t

⇒ rearrange the polynomial form into control points and basis functions (GEOMETRIC FORM)

Specifying Curves

„ Control Points

„ A set of points that influence the curve's shape p

„ Knots

„ Points that lie on the curve

„ Subinterval endpoints

Hermite Curves

„ Parametric curves

„ Defined by two end points with the derivative of the curve at these points

) 1 ( ' ) P

1 ( P

) 0 ( P

) 0 ( ' P

Hermite Curves(cubic polynomial)

2 3

10 11

2 12 3

13t a t a t a

a )

t (

x = + + +

30 31

2 32 3

33

20 21

2 22 3

23

a t a t

a t

a )

t ( z

a t a t

a t

a ) t ( y

+ +

+

=

+ +

+

=

t t

t ]

) t ( z ) t ( y ) t ( x [ ) t ( P

= =a₃ ³ +a₂ ² +a₁ +a₀

) 0 ( P

) 0 ( P

) a , a , a (

where

0 0

i 3 i 2 i 1 i

=

=

= a

a

a

) 1 ( ' P ) 0 ( ' P 2 ) 1 ( P 3 )

0 ( P 3

) 0 ( ' P

) 0 ( P

) 1 ( P

2 2

1

−

− +

−

=

=

= +

+ +

=

a a

a a

a a

a

1

0 1

2 3

) 1 ( P ) 0 ( ' P ) 1 ( P 2 ) 0 ( P 2

3 ) 1 ( ' P

) ( )

0 ( )

( 3 )

0 ( 3 )

0 (

3 2

+ +

−

= +

+

= a₃ 2a₂ a₁ a

1

) 1 ( ' ) P

1 ( P

Hermite Curves

) 0 ( ' P

2 3

2 3

4 3

2

1( ) (0) ( ) (1) '(0) '(1)

)

(t B t P B t P B P B P

P = + + +

) (

2 3

4 2

3 3

2 3

2 2

3 1

) ( 2

) (

3 2

) ( 1

3 2

) (

t t

t B t

t t

t B

t t

t B t

t t

B

−

= +

−

=

+

−

= +

−

=

) (t

B_ii

( )

) 1 ( ' P ), 0 ( ' P ), 1 ( P ), 0 ( P

: blending functions

: geometric coefficients

Hermite Curves

1 P 1 2

2

⎥ ⎤

⎢ ⎡

⎥ ⎤

⎢ ⎡ −

[ ]

' P P 0

1 0

0

1 2

3 1 3

t t

t )

t (

P

⎥ ⎥

⎥

⎢ ⎢

⎢

⎥ ⎥

⎥

⎢ ⎢

⎢ − − −

=

' P P 0

0 0 1

0 1 0 0

1 0

⎥ ⎥

⎢ ⎦

⎢

⎥ ⎣

⎥

⎢ ⎦

⎢

⎣

) G B

( G M

T

= ⋅

⋅

= ⋅

H H

G

M

:Hermite geometry vector

H

Represent Polynomials with basis Represent Polynomials with basis functions

„ Polynomials including degree k forms a vector space P^k+1 f

„ Specify a curve P(u) as a position in the vector space P^k+1 via the coordinate (p₀,L,p_k)

and the basis (1,t,t²,L,t^k )

≤

≤

=

i

p p

k i

t t

b

points control

:

functions basis

: 0

, )

( L

∑

=

⇒

k

t b p t

Q ,p ,

p

) ( )

(

points control

L :

∑

⇒

i

i i

b t p

t Q

0

) ( )

(

Properties shared Properties shared

by most useful bases

„ Convex hull property

if and basis functions are not negative over k

the interval they are defined then any point on the

1 ) (

0

∑

= k

i

i t b

the interval they are defined then any point on the curve is a weighted average of its control points.

⇒ no points on the curve lies outside the polygon formed by joining the control points by joining the control points together

⇒ inexpensive means for l l ti th b d f calculating the bound of a curve or surface in space

Properties shared Properties shared

by most useful bases

„ Affine invariance - any linear transformation or translation of the control points defines a new curve that is the just the

the control points defines a new curve that is the just the

transformation or translation of the original curve. (Perspective transform is not affine.) )

„ Variation diminishing - no straight line intersects a curve more times than it intersects the curve's control polyline. It

l h h l ( d ) f h

implies that the complexity (i.e., turning and twisting) of the curve is no more complex than the control polyline.

Bezier Curves

ƒ Developed by Pierre Bézier in the 1970's for

CAD/CAM operations (PostScript drawing model)

n

CAD/CAM operations. (PostScript drawing model)

ƒ Represent a polynomial segment as

p₃ p₀

i n i

n

0 i

i , n i

) 1

( C )

( J

1 t

0 , ) t ( J p )

t ( P

=

≤

≤

=

∑

p₁

are the Bernstein functions

i n i

i n i

,

n (t) C t (1 t)

J = − ⁻

) t ( J

p₁

p₂ are the Bernstein functions

„ basis or blending function of degree n used to scale or blend the control points

) t ( J

„ used to scale or blend the control points

Bezier blending functions

property hull

convex

1 ) t ( J that

Note

n

0 i

i ,

n = →

∑

Bezier Curves (example)

3 2

1

0(1,1), (2,3), (4,3) and (3,1),

Given p p p p

1 0

) ( )

(

curve.

Bezier the

find

t t

J t

P

n ≤ ≤

∑

0 ,

3.

n ces, four verti

are there

Since

1 0

), ( )

(

P t J t t

i

i n

=

→

≤

≤

=

∑

=

3 2

2 1

, 3 3

0 , 3

) ( )

1 ( 3 )

(

) 1

( 3 )

(

)

1 ( ) (

,

J J

t t

t J

t t

J = − = −

3 , 3 3 2

, 3 2 1

, 3 1 0

, 3 0

3 3

, 3 2

2 , 3

) ( Thus,

) (

) 1

( 3 )

(

J p J

p J

p J

p t

P

t t

J t

t t

J

+ +

+

=

=

−

=

3 3 2

2 1

2 0

3 3 (1 ) 3 (1 )

) 1

(

= −t p + t −t p + t −t p +t p

Bezier Curves (Matrix Form)

⋅

=

⋅

⋅

= T M

G B G

) t ( P

[ ]

[ ]

=

B

J J

J B

p p

p G

where )

(

[ L ]

⎤

⎤ ⎡

⎡ − −

=

0 n

, n 1

, n 0

, n

1 p 3

3 1

J J

J B

L

[ ] _⎥ ^{⎥ ⎥}

⎤

⎢ ⎢

⎢ ⎡

⎥ ⎥

⎥ ⎤

⎢ ⎢

⎢ ⎡

= −

0 2

3

p

p 0

0 3

3

0 3 6

1 3 t t

t )

t (

P [ ]

⎥ ⎥

⎢ ⎦

⎢

⎥ ⎣

⎥

⎢ ⎦

⎢

⎣

−

3 2

p p 0

0 0 1

0 0 3 ) 3

(

⎦

⎦ ⎣

⎣

Bezier Curves

„ The Bezier curve of order n+1 (degree n) has n+1 control points

control points.

„ We can think a Bezier curve as a weighted average of all of its control points

all of its control points

1

0 tP

t)P - (1

P(t) = +

Linear (n=1) :

] tP t)P

- t[(1 ]

tP t)P

- t)[(1 -

(1

P(t) = ₀ + ₁ + ₁ + ₂

Quadratic (n=2) :

2

2P 2(1 ) t P

t) - (1

P(t) = + −t tP +

Cubic (n=3) :

2 1

0 2(1 ) t P

P t) - (1

P(t) = + −t tP +

3 3 2

2 1

2 0

3 3 (1 ) 3 (1 )

) 1 (

P(t) = −t p + t −t p + t −t p +t p

Bezier Curves

Bezier Curves

„ A curve that is made of several Bézier curves is called a composite Bézier curve or a Bézier spline curve

composite Bézier curve or a Bézier spline curve.

„ Tangential continuity between Bezier segments :

)

( )

( Q

− Q

= k R

− R

„ Continuity conditions create restrictions on control points

)

( )

( Q

Q

C¹ continuity C continuity

) 0 ( ' )

1 (

' R

Q =

) (

) (

2 0

3 1

0 1

2 3

Q R

Q R

R R

Q Q

− +

=

⇒

−

=

−

⇒

) (

= Q₃ + Q₃ −Q₂

Bezier Spline Curves

„ C² continuous two cubic Bezier segments V(t) and W(t) with the control points (V₀ ,V₁ ,V₂ ,V₃) and

ƒ For cubic Bezier spline:

V'(0) = 3(V₁ - V₀) , V'(1) = 3(V₃ – V₂) ,

(W₀ ,W₁ ,W₂ ,W₃).

V (0) 3(V₁ V₀) , V (1) 3(V₃ V₂) ,

V"(0) = 6(V₀ – 2V₁ + V₂) , V"(1) = 6(V₁ – 2V₂ + V₃)

ƒ Continuity at the junction point Î W₀ = V₃. C ti it f th fi t d i ti '(0) '(1)

ƒ Continuity of the first derivative W'(0) = V'(1) Î W₁ - W₀ = V₃ - V₂ => W₁ = 2V₃ - V₂

i.e. W₁₁ depends on V₂ & V₃₃

ƒ Continuity of the second derivative W"(0) = V"(1) Î W₀ - 2W₁ + W₂ = V₁ - 2V₂ + V₃

Î W₂ = 2W₁- (2V₂ - V₁) Î W₂ 2W₁ (2V₂ V₁)

Only one control point W₃ of the Bezier curve W(t) is really free.

Characteristics of Characteristics of Bezier Curves

„ Convex hull

„ Affine invariance

„ Affine invariance

„ Variation diminishing

„ The degree of the polynomial defining the curve

i l h h b f d fi i

segment is one less than the number of defining control points.

„ In CAGD applications, a curve may have a so

„ In CAGD applications, a curve may have a so

complicated shape that it cannot be represented by a single Bézier cubic curve

„ Global control (disadv ) : change a control point affects

„ Global control (disadv.) : change a control point affects the continuity of the curve.

The de Casteljau Algorithm

„ Evaluation of the Bezier curve function Repeated linear interpolation

„ Repeated linear interpolation

„ Example of a quadratic (degree 2) Bezier curve

b0 b0

b1 b2 b1 b2

3 l i i l 0 2

3 control points interpolate t = 0.2

The de Casteljau Algorithm

b0 b0

the point on the curve repeating the procedure

b1 b2 b1 b2

the point on the curve p g p Degree=3 and t=0.25

b0 b3b3

b0

b1 b2

Parametric Surface

„

Extend 2D parametric representation

i th b f t f t

„ increase the number of parameters from one to two, (s,t) in order to address each point in the 2D spaces.p

„ express the 3D structure of the curved 2D surface by introducing a parameter z coordinate, z(s,t), i.e., a patchh

) (

) (

)

(s t y f s t z f s t f

x

1 ,

0

).

, (

), , (

), , (

≤

≤

=

=

=

t s

t s f z

t s f y

t s f

x _{x y z}

Bicubic Bezier Surface

„ Bezier patch:16 control points define one patch

„ ease of interactivity & representationy p

j i j j n n

j i i n n

i

t j t

s n i s

t n

s _,

0 0

) 1 ( )

1 ( )

,

( P

P ⁻ ⎟⎟ − ⁻

⎠

⎜⎜ ⎞

⎝

− ⎛

⎟⎟⎠

⎜⎜ ⎞

⎝

=

∑

∑

[ ][ ]

v

2 3

T 2

3 ⎥⎥⎤

⎢⎢

⎡

j

i₌₀ ⎝ i ⎠ ₌₀⎝ j⎠

[ ][ ]

1 v BPB v

1 u u

u v)

P(u,

^{3 2 T}

⎥⎥

⎥⎥

⎢ ⎦

⎢⎢

⎢

⎣

=

p p p p

p p p p 0 P

3 6 3

1 3 - 3 1 B

where ^{10 11 12 13}

03 02 01 00

⎥⎥

⎥⎤

⎢⎢

⎢⎡

⎥ =

⎥⎥

⎤

⎢⎢

⎢⎡

−

−

=

⎦

⎣

p p p p

p p p P p

0

1 4 1

0 3 0 B 3

where

33 32 31 30

23 22 21

20 ⎥⎥

⎢ ⎦

⎢

⎥ ⎣

⎥

⎢ ⎦

⎢

⎣

−

B-Splines Curves p

∑

=

P

B

( u ) )

u (

Q ∑

= 0 k

d , k

k

B ( u ) P

) u ( Q

P_k : an input set of n+1 control points P_k : an input set of n+1 control points B_k,d : blending function of degree d-1

„ The polynomial curve has degree d-1 and C^d-2 continuity over the range of

u

Fo +1 cont ol points the c e is desc ibed ith +1

„ For n+1 control points, the curve is described with n+1 blending functions

„ The range of

u

„ The range of

u

B-Splines Curves p

A bi b li hi h i t f

A cubic b-spline which consists of three curve segments

Q₃ Q₄ Q₅ n=5, d=4

P2

P1 P₃ P₄

P0 P₅ 5+1 control points

4 ,

B0 B_1,4 B_2,4 B_3,4 B_4,4 B_5,4 5+1 blending functions with degree 4-1

u

u

^u

u

Cubic B-Splines

„ Each control point is associated with a unique bl di f ti

blending function.

⇒ (Local control) Each control point affects the shape of a

l f t l d

curve only over a range of a parameter values, d curve sections, where its associated basis function is nonzero.

Q Q

P2

P1 P₃ P₄

P0 P₅

Q₃ Q₄ Q₅

4 ,

B0 B_1,4 B_2,4 B_3,4 B_4,4 B_5,4

u0 u₁ u₉

B-Splines Curves p

„ Knot vector : a set of subinterval endpoints in non- decreasing sequence

decreasing sequence

} ,...,

,

{ u

u

u

U =

☞ uniform, open uniform, nonuniform B-splines.

4 ,

B0 B_1,4 B_2,4 B_3,4 B_4,4 B_5,4

u0 u₁ u₂ u₉

B-Splines Basis Functions

„

Cox-deBoor Algorithm

:generate the basis functions recursively u

u u

if 1

) u (

B ≤ ≤

otherwise

, 0

u u

u if ,

1 )

u (

B

=

≤ ≤

) u ( u B

u

u ) u

u (

B

k 1

d k

k d

,

k −

+

+

−

= −

) u ( u B

u

u u

k 1

d k

− + +

+ +

+ −

u

u

−

Uniform cubic B-spline basis Uniform cubic B-spline basis functions

„ Knots are spaced at equal intervals of parameter.

e g {0 1 2 3 4 5 6 7 8 9}

e.g., {0,1,2,3,4,5,6,7,8,9}

„ Bell-shaped basis function

Each blending function B is defined over four

„ Each blending function B_k,4 is defined over four subintervals starting at knot value

u

4

B0_0,4 B_{1 4} B_{2 4} B_{3 4} B_{4 4} B_{5 4} B B_1,4 B_2,4 B_3,4 B_4,4 B_5,4

u0 u₁ ^u₉

Parametric range of curve

Basis functions of Uniform Basis functions of Uniform Cubic B-splines

1

0 ≤ ≤

we get basis functions by b tit ti

In u _i

≤

≤

,

. 1 0 ≤ u ≤

substituting

3

0 (1 u )

6 ) 1

u (

B = −

2 3

1 (3u 6 u 4 )

6 ) 1

u ( B

6

+

−

=

2 3

2 ( u 3 u 3u 1)

6 ) 1

u ( B

6

+ +

+

−

=

3

3 u

6 ) 1

u (

B =

4 ,

B0 B_1,4 B_2,4 B_3,4 B_4,4 B_5,4

u0 u₁ u₉

Uniform Cubic B-splines

„

ith cubic segment

∑

= ³

0

3

3 ( )

) (

k i k i k

i

u p B u

Q

k : local control point index u : local control parameter

k = 0

1

0 ≤ ≤

u : local control parameter,

A cubic B-spline is a series of m-2 curve

h i i f

1 0 ≤ u ≤

segments, that approximate a series of m+1 control points _{P0,P1,L,Pm , m ≥ 3}

, ,

, , ₄

3 Q Qm

Q _L

Uniform Cubic B-splines

3 2 1 0 3

2 1 0

3 is defined P P P P which are scaled by B B B B Q

5 4 3 2 5

4 3 2 5

4 3 2 1 4

3 2 1 4

by scaled

are which

defined is

by scaled

are which

defined is

B B B B P

P P P Q

B B B B P

P P P Q

Uniform Quadratic B-splines

„ Let d=n=3, we need n+d+1=7 knot values:

{0 1 2 3 4 5 6}

{0,1,2,3,4,5,6}.

„ Get blending functions using Cox-deBoor Algorithm

3 ( )

2 ) 3

2 ( )

( _{0,2 1,2}

3 ,

0 u B u

u u B

u

B −

+

=

Read text book!!

∑

= ²

0

3 , 2

2 ( )

) (

k

k i

k i

i u p B u

Q