curve modelling-b spline

8/11/2019 Curve Modelling-B Spline

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Curve Modeling

B-Spline Curves

Dr. S.M. MalaekAssistant: M. Younesi


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Motivation


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Consider designing the profile of a vase.

The left figure below is a Bzier curve of degree 11; but, it is difficult to

bend the "neck" toward the line segment P4P5.

The middle figure above uses this idea. It has three Bzier curvesegments of degree 3 with joining points marked with yellow rectangles.

The right figure above is a B-spline curveof degree 3defined by 8

control points .

B-Spline Basis: Motivation


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Those little dotssubdividethe B-spline curve into curve

segments.

One can move control pointsfor modifying the shape of thecurve just like what we do to Bzier curves.

We can also modifythe subdivision of the curve. Therefore,

B-spline curves have higher degree of freedomfor curve

design.


B S li B i M i i


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Subdividing the curve directly is difficult to do. Instead, we

subdivide the domain of the curve.

The domain of a curve is [0,1],this closed interval issubdividedby points called knots.

These knots be 0


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In summary : to design a B-spline curve,

we need a set of control points, a set of

knotsand a set of coefficients, one for each

control point, so that all curve segments are

joined together satisfying certain continuity

condition.


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The computation of the coefficients isperhaps the most complex step because they

must ensure certain continuity conditions.


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B-Spline Curves


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B-Spline Curves

(Two Advantages)

1. The degree of a B-spline polynmial canbe set independently of the number of

control points.

2. B-splines allow local control over the

shape of a spline curve (or surface)


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B-Spline Curves(Two Advantages)

A B-spline curve that is defined by 6 control point,

and shows the effect of varying the degree of thepolynomials (2,3, and 4)

Q3is defined by P0,P1,P2,P3

Q4 is defined by P1,P2,P3,P4 Q5 is defined by P2,P3,P4,P5

Each curve segment

shares control points.


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B-Spline Curves(Two Advantages)

The effect of changingtheposition of control

point P4 (locality property).


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B-Spline Curves

Bzier Curve B-Spline Curve


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B-SplineBasis Functions


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B-Spline Basis Functions

(Knots, Knot Vector) Let Ube a set of m+ 1non-decreasing

numbers, u0


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(Knots, Knot Vector)

The half-open interval [ui, ui+1)is the i-th knot

span. Some ui's may be equal, some knot spansmay

not exist.

muuuuU ,,,, 210


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(Knots)

If a knotuiappearsk times (i.e., ui= ui+1 = ... =

ui+k-1), where k > 1,uiis a multiple knotof

multiplicity k, written asui(k). If uiappears only once, it is a simple knot.

If theknotsare equally spaced(i.e., ui+1- uiis a

constantfor 0


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All B-spline basis functions are supposedto have their domain on [u0, um].

We use u0= 0and um= 1frequently so thatthe domainis the closed interval [0,1].


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To define B-spline basis functions, we need one

more parameter.

The degreeof these basis functions,p. The i-th B-

spline basis function of degreep, written as

Ni,p(u), is defined recursively as follows:

)()()(

otherwise0

if1)0(

1,1

11

1

1,,

1

0,

uNuu

uuuN

uu

uuuN

uuuN

pi

ipi

pi

pi

ipi

i

pi

ii

i

S i i i


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The above is usually referred to as the Cox-de Boor

recursion formula.

If thedegreeis zero(i.e.,p = 0), these basis functionsare all step functions.

basis functionNi,0(u)is 1if uis in the i-th knot span

[ui, ui+1).

)()()(

otherwise0

if1)0(

1,1

11

11,,

1

0,

uNuu

uuuN

uu

uuuN

uuuN

pi

ipi

pipi

ipi

ipi

ii

i

We have four knots u0

= 0, u1

= 1,

u2= 2 and u3= 3, knot spans 0, 1

and2are [0,1), [1,2), [2,3)and the

basis functionsof degree0are

N0,0(u) = 1on [0,1)and 0

elsewhere,N1,0(u) = 1on [1,2)and

0 elsewhere, andN2,0(u) = 1on[2,3)and 0elsewhere.


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To understand the way of computingNi,p(u)forp

greater than 0, we use the triangular computationscheme.

B S li B i F ti


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B-Spline Basis Functions To computeNi,1(u),Ni,0(u) andNi+1,0(u) are required. Therefore, we

can computeN0,1(u),N1,1(u),N2,1(u),N3,1(u) and so on. All of these

Ni,1(u)'s are written on the third column. Once allNi,1(u)'s have been

computed, we can computeNi,2(u)'s and put them on the fourthcolumn. This process continues until all requiredNi,p(u)'s are

computed.


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Since u0 = 0, u1 = 1 and u2 = 2, the above becomes

B S li B i F i


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uis in [0,1): In this case, onlyN0,1(u) contributes to the value ofN0,2(u). Sinc

N0,1(u) is u, we have

uis in [1,2) : In this case, bothN0,1(u) andN1,1(u) contribute toN0,2(u). Since

N0,1(u) = 2 - uandN1,1(u) = u- 1 on [1,2), we have

u is in [2,3): In this case, only N1,1(u) contributes to N0,2(u). Since N1,1(u) = 3

u on [2,3), we have


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Two ImportantObservation


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Basis function Ni ,p(u) is non-zero on

[ui, ui+p+1). Or, equivalently, Ni ,p(u) isnon-zero on p+1 knot spans [ui, ui+1),

[ui+1, ui+2), ..., [ui+p, ui+p+1).

Two Important Observation


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Two Important Observation

On any knot span [ui, ui+1), at most p+1

degree pbasis functions are non-zero,

namely: Ni-p,p(u), Ni-p+1,p(u), Ni-p+2,p(u), ...,

Ni-1,p(u) and Ni ,p(u),


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(Important Properties )


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)()()(

otherwise0if1)0(

1,1

11

1

1,,

1

0,

uNuu

uuuN

uu

uuuN

uuuN

pi

ipi

pi

pi

ipi

i

pi

ii

i

1. Ni,p(u)is a degreeppolynomial in u.

2. Nonnegativity-- For all i,pand u,Ni,p(u)is non-negative

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


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4. On any span [ui, ui+1),at mostp+1 degreepbasis

functionsare non-zero,namely:Ni-p,p(u), Ni-p+1,p(u), Ni-

p+2,p(u), ..., and Ni,p(u).

5. The sumof all non-zero degreep basis functionson span

[ui, ui+1)is 1.

6. If the numberof knotsis m+1, the degreeof the

basis functionsisp, and the numberof degreep

basis functionsis n+1, then m = n + p + 1


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7. Basis functionNi,p(u)is a composite curveofdegreeppolynomials withjoining pointsat knots

in [ui, ui+p+1)

8. At aknotof multiplicity k, basis functionNi,p(u)

is Cp-kcontinuous.

Increasing multiplicity decreases thelevel of continuity, and increasing

degree increases continuity.


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(Computation Examples)

Simple Knots Suppose the knot vector is U= { 0, 0.25, 0.5, 0.75, 1 }.

Basis functions of degree 0:N0,0(u),N1,0(u),N2,0(u) andN3,0(u) defined on knot span [0,0.25,), [0.25,0.5), [0.5,0.75)

and [0.75,1), respectively.


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All Ni,1(u)'s(U= { 0, 0.25, 0.5, 0.75, 1 }(:

5.025.0)21(2

25.004)(

1,0

uu

uuuN

for

for

75.05.0for3

5.025.0for14)(1,1uuuuuN

175.0for)1(4

75.05.0for)12(2)(1,2

uu

uuuN

Since the internal knots 0.25, 0.5 and 0.75 are all simple(i.e., k= 1)

andp= 1, there arep - k+ 1 = 1 non-zerobasis function and three

knots. Moreover,N0,1(u),N1,1(u) andN2,1(u) are C0continuous at knots

0.25, 0.5 and 0.75, respectively.

B S li B i F ti


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(Computation Examples) FromNi,1(u)'s, one can compute the basis functions of degree 2.

Since m= 4,p= 2, and m = n + p+ 1, we have n= 1 and there areonly two basis functions of degree 2:N0,2(u) andN1,2(u).(U= { 0,

0.25, 0.5, 0.75, 1 }(:

each basis function is a compositecurve of three degree 2curve

segments.

composite curve is ofC1

continuity

175.0for)1(8

75.05.0for885.1

5.025.0for845.0

)(

2

2

2

2,1

uu

uuu

uuu

uN

75.05.0for8125.45.025.0for16125.1

25.00for8

)( 2

2

2

2,0

uuuuuu

uu

uN


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B-Spline Basis Functions (Computation Examples)

Knots with Positive Multiplicity:

Suppose the knot vector is U = { 0, 0, 0, 0.3, 0.5, 0.5, 0.6, 1, 1, 1{

Since m= 9 andp= 0 (degree 0 basis functions), we have n=

m - p- 1 = 8. there are only fournon-zero basis functions of

degree 0:N2,0(u),N3,0(u),N5,0(u)andN6,0(u).


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Basis functions of degree 1:Sincepis 1, n= m - p- 1 = 7.The following table shows the result

asis Function Range Equation

N0,1(u)

all u 0

N1,1(u) [0, 0.3) 1 - (10/3)

u

N2,1(u)

[0, 0.3) (10/3)u

[0.3, 0.5) 2.5(1 - 2u)

N3,1(u)

[0.3, 0.5) 5u- 1.5

N4,1(u)

[0.5, 0.6) 6 - 10u

N5,1(u)

[0.5, 0.6) 10u- 5

[0.6, 1) 2.5(1 - u)

N6,1(u)

[0.6, 1) 2.5u- 1.5

N7,1(u)

all u 0


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Basis functions of degree 1:


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(Computation Examples) Sincep= 2, we have n= m - p- 1 = 6. The following table

contains allNi,2(u)'s:

Function Range Equation

N0,2

(u) [0, 0.3) (1 - (10/3)u)2

N

1,2

(u)[0, 0.3) (20/3)(u- (8/3)u2)

[0.3, 0.5) 2.5(1 - 2u)2

N2,2

(u)[0, 0.3) (20/3)u2

[0.3, 0.5) -3.75 + 25u- 35u2

N3,2

(u)[0.3, 0.5) (5u- 1.5)2

[0.5, 0.6) (6 - 10u)2

N4,2

(u)[0.5, 0.6) 20(-2 + 7u- 6u2)

[0.6, 1) 5(1 - u)2

N5,2

(u)[0.5, 0.6) 12.5(2u- 1)2

[0.6, 1) 2.5(-4 + 11.5u- 7.5u2)

N6,2

(u) [0.6, 1) 2.5(9 - 30u+ 25u2)

B Spline Basis Functions


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(Computation Examples) Basis functions of degree 2:

Since its multiplicity is 2 and the degree of thesebasis functionsis

2, basis functionN3,2(u)is C0continuous at 0.5(2). This is whyN3,2(u)has a sharp angle at0.5(2).

For knots not at the two ends, say 0.3 and0.6,C1continuity is

maintained since all of them are simple knots.

U = { 0, 0, 0, 0.3, 0.5, 0.5, 0.6, 1, 1, 1{


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B-SplineCurves

B Spline Curves


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B-Spline Curves

(Definition) Given n+ 1control points P0, P1, ..., Pnand a knot vector

U= { u0, u1, ..., um},the B-spline curve of degreepdefinedby these control pointsand knot vector Uis

Thepoint on the curvethat corresponds to a knot ui, C(ui), is

referred to as a knot point.

The knot pointsdividea B-spline curve into curve segments,

each of which is defined on a knot span.

1,)()( 00

,

nmpuuuuNu m

n

i

ipi pC

B Spline Curves


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B-Spline Curves

(Definition)

The degree of a B-spline basis function is an input.

To changetheshape of a B-spline curve, one can modifyone or more of these control parameters:

1. Thepositions of control points

2. Thepositions of knots

3. The degree of thecurve

1,)()( 00,

nmpuuuuNum

n

iipi pC

(Open Clamped & Closed B-Spline Curves)


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(Open, Clamped & Closed B-Spline Curves) Open B-spline curves:If the knot vector does not have any particular

structure, the generated curve will not touchthe firstand last legs of the

control polyline.

Clamped B-spline curve: If the first knotand the last knotmultiplicityp+1, curve istangentto thefirst and the lastlegsat the first and last

control polyline, as a Bzier curvedoes.

Closed B-spline curves:Byrepeating someknotsand control points, the

generated curve can be a closedone. In this case, the start and the endofthe generated curvejoin togetherforming a closed loop.

Open B-Spline Clamped B-Spline Closed B-Spline

control points (n=9) and p = 3. m must be 13 so that the knot vector has 14 knots. To have the

clamped effect, the firstp+1 = 4 and the last 4 knots must be identical. The remaining 14 - (4

+ 4) = 6 knots can be anywhere in the domain. In fact, the curve is generated with knotvector U= 0, 0, 0, 0, 0.14, 0.28, 0.42, 0.57, 0.71, 0.85, 1, 1, 1, 1 .


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OpenB-Spline Curves


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Open B-Spline Curves

Recall from the B-spline basis function

property that on a knot span [ui, ui+1),thereare at mostp+1non-zero basis functionsof

degreep.

For open B-spline curves, the

domain is [up, um-p].

Open B Spline Curves


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Open B-Spline CurvesExample 1:

knot vector U= { 0, 0.25, 0.5, 0.75, 1 }, where m= 4. If the

basis functions are of degree1 (i.e.,p= 1), there are threebasis functionsN0,1(u),N1,1(u)andN2,1(u).

Since this knot vector is not clamped, the first and the last

knot spans (i.e., [0, 0.25)and [0.75, 1)) have only one non-zerobasis functions while the second and third knot spans

(i.e., [0.25, 0.5)and [0.5, 0.75)) have two non-zerobasis

functions.

Open B Spline Curves


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Open B-Spline CurvesExample 2:


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ClampedB-Spline Curves

Clamped B-Spline Curves


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Clamped B Spline Curves We use an exampleuse to illustrate the change between an open

curve and a clamped one:

An open B-splinecurve of degree 4, n= 8and a uniform knot

vector { 0, 1/13, 2/13, 3/13, ..., 12/13, 1 }.

Multiplicity 5(i.e.,p+1),(second, third, fourth andfifth knotto

0)the curve not onlypassesthrough the first control pointbut alsoistangent to the first legof the control polyline.


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ClosedB-Spline Curves

Closed B-Spline Curves


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Closed B Spline Curves To construct a closed B-splinecurve C(u)of degreepdefined

by n+1 control points,the number ofknots is m+1, We must:

1. Designan uniform knot sequenceof m+1 knots: u0= 0, u

1=

1/m, u2= 2/m, ..., um= 1.Note that the domain of the curve is [up, un-p].

2. Wrapthe firstpand lastpcontrol points. More precisely, letP0 = Pn-p+1, P1= Pn-p+2, ..., Pp-2= Pn-1andPp-1= Pn.



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C osed Sp e Cu ves Example. Figure (a) shows an open B-splinecurve of degree

3defined by 10 (n= 9) control pointsand a uniform knot

vector.

In the figure, control point pairs0 and 7, Figure (b),1and 8,

Figure (c), and2and 9, Figure (d)are placed close to each

other to illustrate the construction.

a


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a b

c


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B-Spline CurvesImportant Properties

B-Spline Curves Important Properties


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B Spline Curves Important Properties1. B-spline curve C(u) is a piecewise curve with each

component a curve of degreep.

Example:where n= 10, m= 14andp= 3, the first four knotsand last four knotsareclamped and the 7 internalknots are

uniformly spaced. There are 8 knot spans, each of which

corresponds to a curve segment.

Clamped B-Spline Curve Bzier Curve (degree 10!)



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B Spline Curves Important Properties

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

3. Clamped B-splinecurve C(u)passes through the

two end control pointsP0and Pn.

4. Strong Convex Hull Property:A B-spline curve

is contained in the convex hull of its control

polyline.

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5. Local Modification Scheme:changing the position of

control point Pionly affects the curve C(u) on interval[ui, ui+p+1).

The right figure shows the result of moving P2to the lower right corner. Only

thefirst, second and thirdcurve segmentschangetheir shapesand all remaining

curve segments stay in their original place without any change.



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B-Spline Curves Important Properties A B-spline curve of degree 4 defined by 13 control points and 18 knots .

Move P6.

The coefficient of P6isN6,4(u), which is non-zero on [u6, u11). Thus, movingP6 affects curve segments 3, 4, 5, 6 and 7. Curve segments 1, 2, 8 and 9 are

not affected.

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6. C(u) is Cp-kcontinuous at a knot of multiplicity k

7. Affine Invariance

curve modelling-b spline

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