7 curvesbspline rev 2

8/2/2019 7 CurvesBSpline Rev 2

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B Spline Curve

ME C382: COMPUTER AIDED DESIGN


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Splines - History

Draftsman

A Duck (weight)

Ducks trace out curve

knot

strips of wood (splines)


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Why more spline lectures?

Because B-spline can offer somethingmore 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 basisthat allows anextra degree of freedom which is thedegree of the curve. Degree of the curve

can be independent of the number ofcontrol points.

Hence Local control is available


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B-Spline VersusBezier curve

B-Spline is a powerful generalization of theBezier curve

B-Spline can be both interpolating type andapproximating type

B-Spline provides local control thus better shapecontrol is possible

-depedent on the number control points; thuswith four control points, quadratic as well ascubic B-Spline curves are possible

In general, with (n+1) control points, B-Splinecurves of any degree upto n are possible.


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Parametric representation of B-Spline Curves

functionsbasisSpline-B)(

0)(

,

0

max,

=

==

uN

uuuN(u)

ki

n

i

kiiiiiPPPPPPPP

parametertheofvalueMaximum

ParametercurveSpline-BtheofDegree)1(

max =

=

=

u

uk


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Special characteristics of B-Spline equation

The control points (called de-Boor points)form the vertices of the control polygon(called de-Boor polygon)

The second parameter in the basisfunctions, k, controls the degree (k-1) ofthe resulting B-Spline curve and is

n epen en o e n+ ; n y e upperlimit of the degree is decided by the (n+1)

The maximum limit of the parameter is no

longer unity (that is 1); it is umax chosen byfixing the (n+1) and (k-1)


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Properties of B-Spline Curve

Partition of Unity: Enables invariance under affinetransformations.

1)(0

, ==

n

i

ki uN

Positivity: Enables the curve to lie entirelyin its convex hull.

0)(, uN ki


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Properties of B-Spline Curve

Local support: Enables local control.

],[if0)( 10

, ++

=

=kii

n

i

ki uuuuN

Continuity: Enables continuity thecontinuity of different levels, C0, C1, C2 etc.

able.differentiycontinousl2)-(kis0)(, uN ki


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Meaning of these characteristics

Partition of unity ensures that therelationship between the curve and itsdefining control points is invariant under

affine transformations. affine transformation - a transformation

h i m in i n f in l

transformations such as translation orrotation or reflection on an axis

Transformation - a function that changesthe position or direction of the axes of acoordinate system


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Positivity property guarantees that the

curve segment lies completely within theconvex hull of Pi.

Local support property indicates that each

segment of a B-Spline curve is influencedby only k control points or each control

.

Continuity property tells us about the fact

that B-Spline curve segment is continuousupto Ck-2 or Cn-1 (where (n+1)=number ofcontrol points) within the segment itself.


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-

s.knot valueorknotsparametriccalledareThe

zero.becomersdenominatoif000Choose

function.stepunitaisThis,0

,1

)(

)()(

)(

)()()(

1

1,

1

1,1

1

1,

,

=

=

+

=

+

++

+

+

+

u

otherwise

uuuN

where

uu

uNuu

uu

uNuuuN

i

ii

i

iki

ki

ki

iki

ki

iki

ertyofB-Spline

)2(0isofrangetheand

,2

,1,0

curve,openanFor

.(periodic)closed

orperiodic)-(nonopeniscurveSpline-Btheon whetherdependtheofvaluesThe

+

>+

+

0

n+1 + 1-k >0

(k-1) < (n+1)

(k-1) n

This relation says that a minimum of two, threeand four control points are required to define alinear, quadratic and cubic B-Spline curverespectively.


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Properties of B-spline Curves:O en Curves


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Since Ni,1(u) is a constant for k=1, a generalvalue of k produces a polynomial in u of

degree (k-1) and therefore a curve of degree(k-1).

The knot values are essential for obtaining the

parametric equation of degree (k-1) for thecurve. Once the parametric equation isdetermined, the point on the curve

max,

be determined by substitution. Thus it is veryessential to understand the purpose of the knotvalues (which are discrete and finite in

number, precisely n+k+1) and the variation ofthe parameter (which is continuous betweenthe limits of 0 and umax along the curve).


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The local control of B-Spline

The local control of B-Spline curve can beobtained by three ways:

1)By changing the position of the controlpoint(s). Moving any one control point willaffect only k number of curve portions

symmetrically around it.2)Using multiple control points by placing

several points at the same location

3)By choosing a different degree, (k-1).


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changing the position of the control point

The curve is cubic


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Increasing the degree of the curve tightens it.The lesser the degree of the curve the closer the

curve gets to the control points.(k 1) = 0 -> Zero degree curve (control

points themselves)

(k 1) = 1 -> First degree curve (controlpolygonal segments themselves)

(k 1) = 2 -> Second degree curve

qua rat c curve .(k 1) = 3 -> Third degree curve (cubic

curve)

(k 1) = 4 -> Fourth degree curve (quarticcurve)

(k 1) = 5 -> Fifth degree curve (quinticcurve)


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The quadratic curve is always tangent to

the midpoints of all the internal polygonsegments. This is not the case for otherdegrees. (k-1)=2

(k-1)=3

(k-1)=5


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Equivalence of B-Spline and Bezier curves

If k equals the number of control points (n+1),then the resulting B-Spline curve becomes aBezier curve. As expected, u varies from 0 to 1in this case.

Multiple control points induce regions of highcurvature of a B-Spline curve

Wh n m l i li i n in i l k-1 h

curve has a cusp or a sharp corner. When themultiplicity at any control point is k, then the B-Spline curve will pass through it (why? see next

slide). But Ck-2

continuity is maintained in bothcases.

Multiplicity of points maintains the degree of thecurve but the knot vector is changed.


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Changing one control point affects only k

curve portions symmetrically around it (wedefine one portion of curve as equivalentlength on the curve to one control

polygonal segment. We have reserved theterm curve segment to mean one entires line .

In other words, each portion (or a point init) is influenced by only k number of controlpoints surrounding it (This is the key

behind the procedure of finding the convexhull of a B-Spline curve).


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Finding convex hull for a B-SplineSince each portion (and any point on it) of

the curve is influenced by k number ofcontrol points the convex hull of a B-Splinefor specified degree (k-1) is obtained byfirst joining k number of control points ata time repeatedly from first to the last in

the control point set. This results in a setof convex hulls. Then, finally the convexhull of the entire B-Spline curve is

obtained as the union of these miniconvex hulls.


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When k-1=2, k=3 andhence take 3 points at atime


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A non-periodic B-Spline curve passesthrough the first and last control points Poand P

n

and is tangent to the first (P1

- Po

)and last (Pn Pn-1) control polygonalsegments, similar to the Bezier curve.


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THE RECURSIVE AND NON-RECURSIVE PARTS OF BASIS FUNCTION EQUATIONS

Consider a cubic B-Spline for five control points.

P(u)=P0

N0,4

(u)+P1

N1,4

(u)+P2

N2,4

(u)+P3

N3,4

(u)+P4

N4,4

(u)

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)

0,2 u 1,2 u 2,2 u 3,2 u 4,2 u 5,2 u 6,2 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)

Optical illusion notwithstanding, this picture shows that oneadditional basis function needs to be evaluated at each lower levelusing the recursive relation except the last level Ni,1 which isevaluated using the unit step function part of basis function

equations.


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Find the complete parametric equation of the quadratic B-Splinecurve with the above given control points.


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General Matrix form for any k value

[ ]

[ ]1][

2:][]][[)(

21=

+=

uuuU

kniiPMUuP

kk

ksi

L

[ ]2:1][][ += kiijPPw ere

jk


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PERIODIC OR CLOSEDB-SPLINE CURVES

P i di Cl d B S li C


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Periodic or Closed B-Spline Curves

( )

=

nAifnAofremainder ,/

The mod function enables the periodic (cyclic) translation of the canonicalbasis function N0,k, which is the same as that for non-periodic curve.

zero.becomersdenominatoif10

0Choose

function.stepunitaisThis,0

,1

)(

)()(

)(

)()()(

1

1,

1

1,1

1

1,

,

=

=

+

=

+

++

+

+

+

otherwise

uuuN

where

uu

uNuu

uu

uNuuuN

ii

i

iki

ki

ki

iki

ki

iki

A=4, n=4, (A) mod (n) = 0


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Characteristics of Periodic B-Splines

The difference from non-periodic curves is

the choice of knots and basis functions Periodic B-spline basis functions with

These basis functions are cyclic translatesof a single canonical function with a period

(interval) of kfor support.


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Characteristics of Periodic B-Splines

They also have the properties of partitionof unity, positivity, local support andcontinuity.

They do not pass through the end controlpoints

Therefore they are not tangent to the firstand last control segments


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k=2 (linear); uj = [0 1 2]

N0,2

0 1 2

1

u

N0,3 k=3 (quadratic); uj= [0 1 2 3]

03

3/4

N0,4

k=4 (cubic); uj = [0 1 2 3 4]

0 4

2/3

PARAMETRIC FORM OF PERIODIC CURVE


34/53

))1(mod)1(()(equations.followingthefromcalculatedbetoare

instead,curves,periodicforfunctionsbasisspline-BtheHowever,

0)()(

ispointscontrol1)(ndefined1)-(kdegreeofcurveperiodicThe

,0,

0

max,

=

+++=

=

+

=

nniuNuN

uuuNu

kki

n

i

kiiPPPPPPPP

36mod90;8mod83.5;6mod3.5example,For

,ofreminder

,0

,

mod

asdefinedisItfunction.modulotheis1)mod(nThe

,

===

>

=

7 curvesbspline rev 2

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