B-Spline Curves

7/17/2019 B-Spline Curves

http://slidepdf.com/reader/full/b-spline-curves 1/22

S. Suryakumar, ME, IITH ME3040/ME5090: Mathematical Elements for Geometrical Modelling 1

ME3040/ME5090: Mathematical Elements for Geometrical Modelling

B-Spline Curves




• Because B-spline can offer something more than HCC

and Bezier curve.

• Bezier and Hermite splines have global control only.

– Moving one control point affects the entire curve

• B-spline consists of a basis that allows an extra

degree of freedom which is the degree of the curve.

Degree of the curve can be independent of thenumber of control points.

• Hence Local control is available

Why more spline types?




B-Spline vs. Bezier Curve

Bezier B-Spline

BasicB-spline is a powerful generalization of Bezier

curve

Approximating curve Both interpolating and approximating possible

Curve is defined by n+1 control points Curve is defined by n+1 control points

Order of the curve is n Order of the curve is k-1

Global propagation Local propagation

0<u<1 0<u<(n-k+2)

Berstein Polynomials B-Spline basis functions




• Bezier Curve:

• B-Spline Curve:

B-Spline vs. Bezier Curve

=

= (1)− 0 ≤ ≤ 1

=

=

,() 0 ≤ ≤ + 2




• Non-uniform B-Spline Curve (n=6, k=3)

Non-uniform B-Spline Curve




• Bezier Curve (n=5):

• B-Spline Curve (n=5, k=3):

Definition

(1 ) + 5(1 ) + 10(1 )3+10(1)33 + 5(1 ) +

0 ≤ ≤ 1

,3 + ,3 + ,3 + 3,3 3 + ,3 + ,3()0 ≤ ≤ 4


http://slidepdf.com/reader/full/b-spline-curves 7/22S. Suryakumar, ME, IITH ME3040/ME5090: Mathematical Elements for Geometrical Modelling 7

• The curve is C(k-2) continuous

• The curve is made up of (n-k+2) segments

• Only k control points affect any segment of the curve

• A given control point affects 1 or 2 or … k curvesegments

Properties



B-Spline Basis Function

, 1 1, 1,

1 1

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

i i k i k i k

i k

i k i i k i

u t N u t u N u N ut t t t

1

,

(0 ) are called knot values0 if

i-k+1 if

2 if

1 if( )

0

i

i

i i

i k

t i n k

i k

t k i n

n k i n

t u t N u

otherwise



• Consider a cubic B-Spline for five control points.

• P(u)=P0N0,4(u)+P1N1,4(u)+P2N2,4(u)+P3N3,4(u)+P4N4,4(u)

• Optical illusion notwithstanding, this picture shows that one additional basis function

needs to be evaluated at each lower level using the recursive relation except the last

level Ni,1 which is evaluated using the unit step function part of basis function

equations.

Recursive & Non-Recursive parts of Basis Function

N0,4(u) N1,4(u) N2,4(u) N3,4(u) N4,4(u)

N0,3(u) N1,3(u) N2,3(u) N3,3(u) N4,3(u) N5,3(u)

N0,2(u) N1,2(u) N2,2(u) N3,2(u) N4,2(u) N5,2(u) N6,2(u)

N0,1(u) N1,1(u) N2,1(u) N3,1(u) N4,1(u) N5,1(u) N6,1(u) N7,1(u)



n=5 k=3



• n=5 and k=3

B-Spline Basis Function

(0 8) are called knot values

0,0,0,1,2,3,4,4,4

i

i

t i

t



Effect of Degree: k = 2





















• Non-Uniform B-Spline Curve

• Non-Uniform Rational B-Spline Curve

NURBS

=

=

ℎ,() =

=

ℎ,()

=

=,() 0 ≤ ≤ + 2





• Plot the b-spine curve with n=7 (8points) and k=3

• In Matlab/C++

• The 8 points are given by the user or embedded into

the code – does not matter

Assignment -3



Have a nice day!

Documents

B-Spline Curves