Properties of B-spline curves

B-spline curve

Human Centered CAD Lab.

1 2009-05-21

Properties of B-spline curves B-spline curve: Degree of curve is independent of number of control

points Bezier curve: global modification Modification of any one control point changes the curve

shape everywhere All the blending functions have non-zero value in the

whole interval 0≤u≤1

2

Bezier curve: global modification

Bezier curve of degree 3

3

Desired Blending Function

n

0=iii (u)=(u) BPP

Consider degree 1 blending functions, and n=4

uu1 u2 u3 u4

1B0(u)

uu1 u2 u3 u4

1B1(u)

uu1 u2 u3 u4

1B2(u)

uu1 u2 u3 u4

1B3(u)

uu1 u2 u3 u4

1B4(u)

4

1

10 )(

uuuuB

12

2

1

uuuu

uu

23

3

12

1

uuuuuuuu

34

4

23

2

uuuuuuuu

34

34 )(

uuuuuB

10 uu

10 uu

21 uuu 32 uuu

32 uuu

43 uuu

43 uuu

21 uuu

P0 has an effect only for 0≤u≤u1

P1 has an effect only for 0≤u≤u2

P2 has an effect only for u1≤u≤u3



)(1 uB )(2 uB

)(2 uB

Resulting Curve1

0 uu )()()()()(1

11

101100 u

uu

uuuuBu PPBPPP

For

… straight line from P0 to P1

21uuu For

… straight line from P1 to P2

Similarly

P0

P1P2

P3

P4

5

)()()()()(12

12

12

212211 uu

uuuuuuuuBu

PPBPPP

Blending Function

6

Consider degree 2 blending functions, and n=5

P0 has an effect only for 0≤u≤u1 P1 has an effect only for 0≤u≤u2P2 has an effect only for 0≤u≤u3 P3 has an effect only for u1≤u≤u4P4 has an effect only for u2≤u≤u4 P5 has an effect only for u3≤u≤u4

Resulting Curve

7

Ni,k : degree (k-1) of u, k : order (independent of number of control points n)

ti: knot values, boundary of non-zero range of each blending function

t0 (for i=0)to tn+k (for i=n) are needed (n+k+1 values)

→ At any value of u, there should be only one non-zero Ni,1(u)

otherwise

tut 01

(u)

000

tt(u),u)(t

tt(u),)t(u(u),

tut (u),=(u)

1iii,1

1iki

1k1iki

i1ki

1kiiki

1n1k

n

0=ikii

N

NNN

ΝPP

8

(a)

(b)

(c)

B-spline curve equation – cont’

B-spline curve equation – cont’ Only the differences in ti (i=0, ···, n+k) is important in (b) Can be shifted as a whole, parameter range should be shifted together A portion of B-spline curve is affected by a limited number of control

points For u in [tl, tl+1]

Control points associated with blending functions that are non-zero in [tl, tl+1] have effect

Nl,1(u) is nonzero in [tl, tl+1] among Ni,1(u) Substitute Nl,1(u) into the right-hand side of (b) Nl,2(u), Nl-1,2(u) can be non-zero Apply recursively From Nl,2(u), Nl,3(u) and Nl-1,3(u) can be non-zero From Nl-1,2(u), Nl-1,3(u) and Nl-2,3(u)can be non-zero

9

N i-2,3

N i,1

N i-1,2

N i-k+1,k

N i-k+2,k-11 N i-k+2,k

N i-1,3

N i,2 N i,3 N i,k-1

N i-1,k-1

N i,k

N i-1,k

10

B-spline curve equation – cont’

B-spline curve equation – cont’ Control points that have influence

in the region [tl, tl+1] are Pl-k+1, Pl-k+2, ···, Pl k control points

Control points modify: Example

11

B-spline curve - Knot Knot: t0, t1, ···, tn+k

Parameter range is determined by knots Periodic knots

ti= i-k 0≤ i≤ n+k Non-periodic knots

k times duplicatesknin2k-n

nik1k-ik times duplicateski00

ti

12

(d)

Periodic vs. Non-periodic

Periodic knot

Non-periodic knot

13

B-spline curve - Knot By duplicating knot k times at the ends Curve passes through first control point and last control

point Periodic knot First control point and last control point do not pass

through curve as other control points Non-periodic knots are used in most CAD systems

knot interval is uniform in (d) Uniform B-spline (vs. non-uniform B-spline)

During manipulation of curve shape, knots are added or removed Non-uniform knot → non-uniform B-spline curve

14

Example program Knot Insertion Example1

Example2

15

Expansion of curve equation Ex) K=3, P0, P1, P2 non-periodic uniform B-spline

t0=0, t1=0, t2=0, t3=1, t4=1, t5=10 ≤ u ≤ 1↑ ↑(=t2) (=t3)

16

Expansion of curve equation – cont’

17

otherwise01)(utut1

(u)

otherwise01)(utut1

(u)

otherwise01)u(0tut1

(u)

otherwise00)(utut1

(u)

otherwise00)(utut1

(u)

544,1

433,1

322,1

211,1

100,1

N

N

N

N

N

At u=0, select N2,1(u) to be non-zero among N0,1(0), N1,1(0), N2,1(0)

Selection of any one is O.K. At u=1, select N2,1(u) similarly Only N2,1(u) needs to be considered among blending

functions of order 1


18


192

2

10

2

2

35

3,25

24

2,22

2,3

24

2,24

13

1,21

1,3

21,2

13

1,23

02

0,20

0,3

2,1

34

3,14

23

2,12

2,2

2,1

23

2,13

12

1,11

1,2

uu)-2u(1u)-(1(u)

utt

u)(ttt)t(u(u)

u)2u(1u)u(1u)u(1tt

u)(ttt)t(u(u)

u)(11u)(1

ttu)(t

tt)t(u(u)

u1

utt

u)(ttt)t(u(u)

u)(11u)(1

ttu)(t

tt)t(u(u)

PPPP

NNN

NNN

NNNN

NNNN

NNNN

Consider Bezier curve defined by P0, P1, P2

Non-periodic B-spline curve having k (order) control points ends in Bezier curve

Bezier curve is a special case of B-spline curve

202

111

020 u)(1u

22

u)(1u12

u)(1u02

(u) PPPP

20


Example Ex)

K=3, P0, P1, P2, P3, P4, P5

t0=0, t1=0, t2=0, t3=1, t4=2, t5=3, t6=4, t7=4, t8=40≤u≤4

otherwise 04u3 1

(u)

otherwise 03u2 1

(u)

otherwise 02u1 1

(u)

otherwise 01u0 1

(u)

5,1

4,1

3,1

2,1

N

N

N

N

21

22

5,1

67

6,17

56

5,15

5,2

5.14,1

56

5,16

45

4,14

4,2

4.13,1

45

4,15

34

3,13

3,2

3,12,1

34

3,14

23

2,12

2,2

2,1

23

2,13

12

1,11

1,2

3)(utt

u)(ttt)t(u(u)

u)(42)(utt

u)(ttt)t(u(u)

u)(31)(utt

u)(ttt)t(u(u)

u)(2utt

u)(ttt)t(u(u)

u)(1tt

u)(ttt)t(u(u)

NNNN

NNNNN

NNNNN

NNNNN

NNNN

Example – cont’

2,21,224

2,24

13

1,211,3

2,121,213

1,23

02

0,2o0,3

2u2u

ttu)(t

tt)t(u(u)

u)(1u)(1tt

u)(ttt

tuu

NNNNN

NNNNN

3.1

2

2,1 2u)-(2+

2u)u-(2+u)-u(1= NN

3,22,235

3,25

24

2,222,3

2u3

2u

ttu)(t

tt)t(u(u) NNNNN

4,12

3,12,12

2u)(3

21)u)(u(3

2u)u(2

2u NNN

4,23,246

4,26

35

3,233,3

2u4

21u

ttu)(t

tt)t(u(u) NNNNN

5,1

2

4,13,1

2

2u)-(4+

22)-u)(u-(4+

2u)-1)(3-(u+

21)-(u= NNN

23

Example – cont’

5,24,2

57

5,27

46

4,24

4,3 u)(42

2utt

u)(ttt)t(u(u) NNNNN

5,14,1

2

3)u)(u(42

u)2)(4(u2

2)(u NN

5,12

5,2

68

6,28

57

5,25

5,3 3)(u3)(utt

u)(ttt)t(u(u) NNNNN

24

Example – cont’

25

Example – cont’

55,12

45,14,1

2

35,1

2

4,13,1

2

24,1

2

3,12,1

2

3)(u

3)u)(u(42

u)2)(4(u2

2)(u2u)(4

22)u)(u(4

2u)1)(3(u

21)(u

2u)(3

21)u)(u(3

2u)u(2

2u

P

P

P

P

N

NN

NNN

NNN

13,1

2

2,102,12

2u)(2

2u)u(2u)u(1u)(1(u) PPP

NNN

u1 2 3 4

1N0,3 :

u1 2 3 4

1N1,3 :

u1 2 3 4

1N2,3 :

u1 2 3 4

1N3,3 :

1 2 3 4

1

u

N4,3 :

1 2 3 4

1

u

N5,3 :

26

Example – cont’

For each knot interval, coefficients of certain control points = 0 Only subset of control points has influence

For 0≤u≤1, all Ni,1 except N2,1 are 0

27

Example – cont’

2

2

102

1

2u

2u)u(2u)u(1u)(1(u) PPPP

Similarly

Example – cont’

52

42

3

2

4

4

2

32

2

2

3

3

2

21

2

2

3)(u32)20u3u(21

2u)(4(u)

4u3

22)(u11)10u2u(

21

2u)(3(u)

3u2

21)(u

21)u)(u(3

2u)u(2

2u)(2(u)

2u1

PPPP

PPPP

PPPP

28

29

Example – cont’

C2 continuity is not satisfied. (∵k=3, degree 2) For curve of order k, neighboring curves have same

derivatives up to (k-2)-th derivative at the common knot Each curve segment is defined by k control points. Any one control point can influence up to maximum k

curve segments.

Example – cont’

30

continuiyC(3)(3)(2),(2)(1),(1) 1433221 PPPPPP

count curve segment including Pk-1

P0 P1 Pk-1 Pk Pn-k Pn-k+1 Pn-1 Pn

P1( )u

P2 ( )u

Pn-k+1( )u

Pn-k+2 ( )u

31

Example – cont’

Least squares curve approximation (Interpolation)

Determine parameter values corresponding to data points

k=1,…,m-1

32

Calculation of control points

mn00

i

QPQP

P )C(

,

]1,0[)(Ni

uuun

0i

m)thansmallerisn(

)(minimizing PPfor Look 2

1-n1

1-m

1k kkuf CQ

33

Least squares curve approximation – cont’


34


1n2,1,lf

l

,forminimumat0 P

35

Least squares curve approximation – cont’ One equation with n variables P1 ……, Pn-1. l can be 1, ……, n-1 n-1 equations with n-1 variables can be can be generated

(NTN) P = R

36


37

Least squares curve approximation – cont’ Determination of Knot vector (n+k+1) knots are needed for order k (n-k+1) internal knots need to be determined since k

multiple knots at 0, 0, … , 0 and 1,1, … ,1 already exist

38


jqii satisfyinginteger highest theis

39

Internal knots

Least squares curve approximation – cont’Example 1

5.3244

16,644

When

qmkn

For j = 1

40

Knot vector = {0,0,0,0,t4,1,1,1,1}

i=int(1·3.5)=3α=1·3.5-3=0.5T4-1+1= t4 =(1-0.5)u3-1+0.5u3

=0.5(u2+u3)


313

245112

,1245

When

q

mkn

41

Example 2


42

P(u) – Q(v) = 0 3 scalar equations, two unknowns Px(u) – Qx(v) = 0 Py(u) – Qy(v) = 0

Use Newton Raphson method Derivative of Px, Qx, Py, Qy need to be calculated f1(x1,…,xn) = 0

f2(x1,…,xn) = 0

fn(x1,…,xn) = 0

43

Intersection between curves

n

2

1

n

2

1

n

n

1

n

n

2

1

2

n

1

1

1

nn

n1

1

nn1nnn2211n

nn

11

1

1n11nn22111

f

f

f

xΔ

xΔ

xΔ

xf

xf

xf

xf

xf

xf

xΔxfxΔ

xf)x,,(xf)xΔx,,xΔx,xΔ(xf

xΔxfxΔ

xf)x,,(xf)xΔx,,xΔx,xΔ(xf

44

Intersection between curves – cont’

Intersection between curves – cont’ If initial values of u, v are too far from real solution, the

iteration diverges. Hard to find all the intersection points. Cannot handle the case of overlapping curves. Two curves are regarded to intersect each other if they lie

within numerical tolerance. Control polygons are approximated to the curve by

subdivision and initial values of u,v can be provided closely by intersecting control polygons

Better to detect special situation in advance before resorting to numerical solution.

Tuning tolerance values is necessary

45

Straight line vs. curve P(u) = P0 + u(P1 – P0) Q(v) = P0 + u(P1 – P0) (a) Apply dot product (P0×P1) to both sides of eq(a) gives

(P0×P1)·Q(v) = 0non-linear equation of v

46

Non-uniform Rational B-spline (NURBS) curve Use same Blending functions as B-spline Control points are given in homogeneous

coordinates

n

ikii

n

ikiii

n

ikiii

n

ikiii

iiiiiiiiii

uNhh

uNzhhz

uNyhhy

uNxhhx

hhzhyhxzyx

0,

0

,

0,

0,

)(

)()(

)()(

)()(

),,,(),,(

47

Non-uniform Rational B-spline (NURBS) curve – cont’

)(

)()(

0

,

0

,

n

ikii

n

ikiii

uNh

uNhu

Passes through the 1st and the last control points

( When non-periodic knots are used )

Numerator is B-spline with hiPi as control points

→ h0P0, hnPn at parameter boundary values

Similarly denominator has values of h0, hn at parameter boundary values

48

Non-uniform Rational B-spline (NURBS) curve – cont’

1)(1 ,

0

splineBuNh ki

n

ii

Directions of tangent vectors are P1-P0, Pn-Pn-1 at starting and ending points

B-spline curve is a special case of NURBS

49

Non-uniform Rational B-spline (NURBS) curve – cont’ Curve shape can be changed by changing weight(hi) Increasing weight has an effect of pulling curve

toward associated control point Example program

Conic curve can be represented exactly Reducing program coding effort

50

Control points of NURBS curve equivalent to a circular arc

Can be used when center angle is less than 180o.

Arc with a center angle bigger than 180o is split into two and combined later

51

Example

Ex.

P2 P1

P0P4

P3

θ ①②

52

Example – cont’

222111111000

1,2

1,1

)0,1(),1,1(),1,0(

)3,2(111000

12

145cos1

)1,0(),1,1(),0,1(

432

432

210

210

knot

hhh

knknot

hhh

Similarly,

53

Example – cont’

22211000

1,2

1

12

11

)0,1(),1,1(),1,0(),1,1(),0,1(

합성

43

210

43

210

knot

hh

hhh

Composition

54

Date post:	02-Dec-2021
Category:	Documents
Upload:	others
View:	3 times
Download:	0 times

Properties of B-spline curves

Documents