degree elevation - vracpublic.vrac.iastate.edu/~oliver/courses/me625/week10b.pdfua inserted 2 times...

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Degree Elevation

Let be an n-th degree

polynomial curve. Clearly, we can set an+1 = 0and write Cn(u) as an (n + 1)-th degree curve

Cn u( ) aiui

i 0=

n

∑=

Cn u( ) Cn 1+ u( ) aiui

i 0=

n 1+

∑= =

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From a vector space point of view, Cn+1(u) issimply Cn(u) embedded in a higherdimensional space.

Now let be a p-th

degree NURBS curve on the knot vector U.

Cpw u( ) Ni p, u( ) Pi

w

i 0=

n

∑=

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Since it is a piecewise polynomial curve, itshould be possible to elevate its degree top + 1. Thus there must be a set of controlpoints Qw

i and knot vector such that

where Cwp(u) and Cw

p + 1(u) are the samecurve, geometrically and parametrically.Degree elevation refers to the process offinding the unknown Qw

i and

U

Cpw u( ) Cp 1+

w Ni p 1+, u( ) Qiw

i 0=

n

∑= =

U

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Applications of elevation include the following:

• The construction of certain types of surfacesfrom a set of curves. Using tensor productsurfaces requires that these curves have acommon degree.

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• Let C1, ..., Cn, (n ≥ 2) be a sequence ofNURBS curves with the property that the endpoint of Ci is coincident with the start point ofCi+1. Then the curve may be composed intoone NURBS curve. One step in the processis to elevate to a common degree.

As usual, degree elevation algorithms areapplied to Cw

p(u) in 4D space, and hence, wedrop the w for the remainder of this discussion.

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There are three unknowns in the aboveequation, , and the {Qw

i}. and can bedetermined by the following reasoning.Suppose U has the form

n U n U

U u0 … um, ,{ }=

a … a u1 … u1 … us … us b … b, , , , ,, , , , , , ,{ }=

p + 1 p + 1m1 ms

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where m1, ..., ms denote the multiplicities ofthe interior knots. Now, Cp(u) is a polynomialcurve on each nondegenerate knot span;hence, its degree can be elevated to p + 1 oneach such knot span.

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At a knot of multiplicity mi, Cp(u)is -continuous. Since the degreeelevated curve, Cp+1(u), must have the samecontinuity, it follows that the knot must havemultiplicity mi + 1 for Cp+1(u). This yields:

and

Cp mi–

n n s 1+ +=

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where

U u0 … um, ,{ }=

a … a u1 … u1 … us … us b … b, , , , ,, , , , , , ,{ }=

p + 2 p + 2m1 + 1 ms + 1

m m s 2+ +=

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The only remaining problem is to compute the{Qi}. An obvious (but very inefficient) methodto do this is to solve a system of linearequations. Setting,

and evaluating the Ni,p(u) and Ni,p+1(u) atappropriate + 1 values, yields a bandedsystem of + 1 linear equations in theunknowns Qi.

Ni p 1+, u( ) Qii 0=

n

∑ Ni p, u( ) Pii 0=

n

∑=

nn

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Most algorithms in the literature are moreefficient but are mathematically morecomplicated, and are designed to raise thedegree by 1. The one presented here isefficient, mathematically simple, and designedto raise the degree by more than one (in onestep).

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The degree elevation algorithm applies thefollowing steps:

• Using knot insertion, extract the i-th Beziersegment from the curve.

• Degree elevate the i-the Bezier segment.

• Remove unnecessary knots separating the (i- 1)-th and i-th segments.

The following example illustrates its operation.

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original curveU={0 0 0 0 ua ub 1 1 1 1}

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ua inserted 2 times(mua = 3)

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Bezier segment on [0, ua]degree elevated (4pts -> 5pts)

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ub inserted 2 times

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Bezier segment on [ua, ub]degree elevated

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knot ua removed twice

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last segment on [ub, 1]degree elevated

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knot ub removed twice

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Step one is a simple application of the knotinsertion algorithm until the i-th knot hasmultiplicity p.

The second step is more interesting. Letsderive the formula to elevate a Bezier curvefrom p to p + 1. Let,

Cp u( ) Bi p, u( ) Pii 0=

p

∑=

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be a p-th degree Bezier curve. Itsrepresentation as a (p + 1)-th degree curve is

Cp 1+ u( ) Bi p 1+, u( ) Qii 0=

p 1+

∑=

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Setting these equal and multiplying Cp(u) by(u + (1 - u)) (= 1) yields:

Bi p 1+, u( ) Qii 0=

p 1+

∑ =

u 1 u–( )+( ) Bi p, u( ) Pii 0=

p

∑=

1 u–( ) Bi p, uBi p,+( ) Pi

i 0=

p

∑=

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Using the notation,

and applying the definition of the Bezierblending functions Bi,p(u), we obtain,

p!i! p i–( ) !------------------------ p

i =

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p 1+i

ui 1 u–( ) p 1 i–+ Qii 0=

p 1+

∑ =

pi

i 0=

p

∑ ui 1 u–( ) p 1 i–+(=

ui 1+ 1 u–( ) p i– ) Pi+

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p 1+i

ui 1 u–( ) p 1 i–+ Qii 0=

p 1+

∑ =

pi

ui 1 u–( ) p 1 i–+ Pii 0=

p

∑=

pi 1–

ui 1 u–( ) p 1 i–+ Pi 1–i 1=

p 1+

∑+

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Equating coefficients ofyields,

ui 1 u–( ) p 1 i–+

p 1+i

Qipi

Pip

i 1– Pi 1–+=

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And, from

pi

p 1+i

÷ p!i! p 1 i–+( ) !i! p i–( ) ! p 1+( ) !----------------------------------------------=

p 1 i–+p 1+

-------------------- 1 ip 1+------------–==

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and

it follows that:

pi 1–

p 1+i

÷ =

p!i! p 1 i–+( ) !i 1–( ) ! p 1 i–+( ) ! p 1+( ) !

-----------------------------------------------------------------------=

ip 1+------------=

Qi 1 αi–( ) Pi αiPi 1–+=

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where,

Note that this equation represents a cornercutting process analogous to the deCastlejaualgorithm.

There are two important points regarding thisformula with respect to algorithmicimplementation,

αii

p 1+------------= i, 0 … p 1+, ,=

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• It can be applied recursively to elevate thedegree t times.

• The α’s depend only on the degree, not onthe particular Bezier segment to which theyare applied. Hence, they can be computedone time and stored in a local array beforeprocessing of the segments begins.

The final step of the degree elevationalgorithm, knot removal, is simpler than thegeneral case for the following two reasons.

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• We know how many knots are removable,i.e., the number that was inserted. Thus thereis no need to check point coincidence andthe associated intermediate steps can beavoided.

• The knot vector has a specific structure in theneighborhood of the knot being removed(again, because we know the original)

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Let

be a NURBS surface. Degree elevation isaccomplished for surfaces by applying it to therows/columns of control points.

Sw u v,( ) Ni p, u( ) Nj q, v( ) wijPijw

j 0=

m

∑i 0=

n

∑=

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In particular, we elevate the degree p(u-direction) by applying algorithm A5.9 toeach of the m + 1 columns of control points.The v-direction degree q is elevated byapplying the algorithm to each of the n + 1rows of control points.

An efficient implementation of the surfacedegree elevation algorithm requires storage ofBezier strips.