B-Splines Properties 1. Convex hull property 2. Curve follows the - - PowerPoint PPT Presentation

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Feb 22, 2024 257 likes •458 views

B-Splines Properties 1. Convex hull property 2. Curve follows the shape of the defining polygon 3. Maximum order = number of control points 4. Invariance to affine transformation 5. Variation diminishing property B-Splines Control Handles

SLIDE 1

B-Splines

Properties

1. Convex hull property
2. Curve follows the shape of the defining polygon
3. Maximum order = number of control points
4. Invariance to affine transformation
5. Variation diminishing property

SLIDE 2

B-Splines

Control Handles

Knot vector (uniform, open uniform, non-

uniform)

Order k
Number and position of control point

SLIDE 3

B-Splines

B1 B2 B3 B4

k=2 k=3 k=4

Open-uniform knot vector Control polygon B1 B2 B3 B4

Order k

SLIDE 4

B-Splines

B2 B4 B3 B1

Multiple points

Open uniform knot vector k=4

Control polygon B1 B2 B3 B4

SLIDE 5

B-Splines

Multiple points

Open uniform knot vector k=4

Control polygon B1 B2 B2 B3 B4

B2 B2 B4 B3 B1

SLIDE 6

B-Splines

Multiple points

Open uniform knot vector k=4

Control polygon B1 B2 B2 B2 B3 B4

B2 B2 B2 B4 B3 B1

SLIDE 7

B-Splines

Local Control

B2 B1 B3 B4 B5 B5’ B5’ ’ B7 B6 B8

Open uniform knot vector k=4

SLIDE 8

B-Splines

Rational B-Splines

∑

+ =

=

1 1

) ( ) (

n i ik h i

t N B t P

Projection of a non-rational (polynomial) B-Spline in 4D into 3D

SLIDE 9

B-Splines

Rational B-Splines

∑ ∑ ∑ ∑

+ = + = + = + =

= = =

1 1 1 1 1 1 1 1

) ( ) ( ) ( ) ( ) ( ) ( ) (

n i ik i ik i ik n i ik i n i ik i n i i ik i

t N h t N h t R t R B t N h h t N B t P

SLIDE 10

B-Splines

Rational B-Splines

Generalization of non-rational B-Spline basis functions

and curve

Most properties and characteristics of non-rational

B-spline are carry forwarded

Additional control handle of hi
Enables construction of common analytical shapes

conic sections

SLIDE 11

B-Splines

Rational B-Splines

Invariant to projective transformation The curve is invariant with respect to the projective transformation applied to the defining vertices of the control polygon

SLIDE 12

B-Splines

B2 B1 B3 B4 B5 0.5 1 5

n+1 = 5, k = 3 X = [ 0 0 0 1 2 3 3 3 ] H = [ 1 1 h3 1 1 ]

Rational B-Splines

SLIDE 13

B-Splines

NURBS : Non Uniform Rational B-Spline Most generalized form of B-Spline Non Uniform Knot Vector Rational Basis Functions

Rational B-Splines

SLIDE 14

B-Splines

Curve Fitting Find a B-Spline passing through D1, D2 … Dj

D1 D2 D3

SLIDE 15

B-Splines

Curve Fitting

1 , 1 2 , 2 1 , 1 1 2 , 1 2 2 , 2 1 2 , 1 2 2 1 1 , 1 2 1 , 2 1 1 , 1 1 1 1 1 ,

) ( ... ) ( ) ( ) ( ) ( ... ) ( ) ( ) ( ) ( ... ) ( ) ( ) ( ) ( ) (

+ + + + + + + =

+ + + = + + + = + + + = = ∑

n j k n j k j k j j n k n k k n k n k k n i k i i

B t N B t N B t N t D B t N B t N B t N t D B t N B t N B t N t D t N B t P

SLIDE 16

B-Splines

When j = n+1 => N square matrix

Curve Fitting [ ] [ ] [ ] [ ] [ ] [ ] [ ]

D N B t N t N t N t N N B N D

j k n j k k n k 1 , 1 , 1 1 , 1 1 , 1

) ( ) ( ) ( ) (

− + +

= ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = = … …

…

SLIDE 17

B-Splines

Curve Fitting

[ ] [ ] [ ] [ ] [ ] [ ] [ ][ ] [ ] [ ] [ ]

[ ] [ ] [ ]

D N N N B B N N D N B N D

T T T T 1 −

= = =

SLIDE 18

B-Splines

Parameter value tj

For lth data point

2

2 1 2 1 max

≥ − − =

∑ ∑

= − = −

l D D D D t t

j s s s l s s s l

Curve Fitting

t1 = 0, tmax = maximum value of the knot vector

SLIDE 19

B-Splines

Parameter value tj

3 4 2 3 1 2 1 2 max 2

D D D D D D D D t t − + − + − − =

D1 D2 D3