10. Bezier Curves, B-Splines & NURBS

8/11/2019 10. Bezier Curves, B-Splines & NURBS

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Bezier Curves, B-Splines, NURBS



Example Application:

Font Design an Displa!● Curved objects are

everywhere●

There is always need for: – mathematical fidelity

– high precision

– artistic freedom andflexibility

– physical realism



Example Application:

"rap#ic Design an Arts



Example Application:$ool %at# "eneration an

&otion %lanning



Functional Representations

● Explicit Functions: – representing one variable with another

–

fine if only one x value for each y value – Problem: what if I have a sphere?

● Multiple values ! "not used in graphics#

z = r 2− x

2− y

2

∃




● Implicit Functions: – curves$surfaces represented as %the &eros'

–

good for rep! of n-1() objects in n() space – *phere example:

– +hat class of function?● polynomial : linear combo of integer powers of x,y,z ●

algebraic curves & surfaces: rep,d by implicit polynomialfunctions

● polynomial degree: total sum of powers-i!e! polynomial of degree .:

02222 =−++ r z y x

02222

=−++ r z y x






● +hich is best?? – It depends on the application –

Implicit is good for ● computing ray$surface intersection● point inclusion "inside$outside test#● mass 3 volume properties

– Parametric is good for ● subdivision- faceting for rendering● *urface 3 area properties● popular in graphics



'ssues in Speci(!ing)Designing

Curves)Sur(aces● 2ote: the internal mathematical representation

can be very complex –

high degree polynomials – hard to see how parameters relate to shape

● 4ow do we deal with this complexity? – 5se curve control points and either

● Interpolate● 6pproximate



%oints to Curves

● The Lagrangian interpolating polynomial

– n+1 points- the uni7ue polynomial of degree n

– curve wiggles thru each control point

– Issue: not good if you want smooth or flat curves

● Approximation of control points – points are weights that tug on the curve or surface



%arametric Curves

● 8eneral rep:

●

Properties: – individual functions are single(valued – approximations are done with

piecewise poly curves – 1ach segment is given by three cubic

polynomials "x-y-&# in parameter t – Concise representation



Cu*ic %arametric Curves

● 9alance between – Complexity

–

Control – +iggles

– 6mount of computation

– 2on(planar



%arametric Curves

● Cubic Polynomialsthat define aparametric curve

segment

are of the form

●

2otice we restrict theparameter t to be



%arametric Curves

● If coefficients arerepresented as a matrix

and

then:



• Q(t) can be defined with four constraints – ewrite the coefficient matrix C as

where M is a ;x; basis matrix - and G is a four(element constraintmatrix "geometry matrix #

● 1xpanding gives:

Q(t) is a weighted sum of the columns of thegeometry matrix- each of which represents a pointor vector in 0(space

%arametric Curves



%arametric Curves

● Multiplying out gives

"i!e! just weighted sums of the elements#● The weights are cubic polynomials in t "called

the blending functions- B=MT #

• M and G matrices vary by curve – 4ermite- 9<&ier- spline- etc!



+arning, +arning, +arning:

%ening Notation A*use● t and u are used interchangeably as a

parameteri&ation variable for functions● +hy?

– t historically is %time'- certain parametric functions candescribe %change over time' "e!g! motion of a camera-physics models#

– u comes from the 0) world- i!e! where two variablesdescribe a 9(spline surface

● u and v are the variables for defining a surface

● Choice of t or u depends on the text$reference



Continuit!

Two types:● 8eometric Continuity- Gi:

– endpoints meet

– tangent vectors, directions are e7ual

● Parametric Continuity- C i: – endpoints meet

– tangent vectors, directions are e7ual

– tangent vectors, magnitudes are e7ual

● In general: C implies G but not vice versa



%arametric Continuit!

● Continuity "recall from the calculus#: – Two curves are C i continuous at a point p iff the

i-th derivatives of the curves are e7ual at p



Continuit!

● The derivative of is the parametric tangentvector of the curve:



Continuit!

● +hat are the conditions for C 0 and C 1 continuity atthe joint of curves xl and xr ? – tangent vectors at end points e7ual

– end points e7ual

xl xr



Continuit!

● In 0)- compute this for each component of theparametric function – =or the x component:

● *imilar for the y and z components!

xl xr



Convex ulls

● The smallest convexcontainer of a set ofpoints

● 9oth practically andtheoretically useful ina number ofapplications



Some $!pes o( Curves

● 4ermite – def,d by two end points

and two tangent

vectors● Bzier

– two end points plustwo control points forthe tangent vectors

● Splines – 9asis *plines

– def,d w$ ; control points

– 5niform- nonrational B-splines

– 2onuniform- nonrational 9(splines

– 2onuniform- rational B-splines NURBS !



Bzier Curves

● Pierre 9<&ier ><nault @A.B

● 9asic idea – four points

– *tart point P

– 1nd point P !

– Tangent at P - P P "

– Tangent at P !- P ! P #



Bzier Curves

6n 1xample:● $eometry matrix is

where P i are controlpoints for the curve

● Basis %atrix is

convex hull



Bzier Curves

● The general representation of a9<&ier curve is

whereG B ( 9<&ier 8eometry Matrix

M B ( 9<&ier 9asis Matrix

which is "multiplying out#:

convex hull



Bernstein %ol!nomials

● The general form for the i-th 9ernstein polynomialfor a degree " 9<&ier curve is

● *ome properties of 9Ps – Invariant under transformations

– =orm a partition of unity - i!e! summing to @

–

ow degree 9Ps can be written as high degree 9Ps – 9P derivatives are linear combo of 9Ps

– =orm a basis for space of polynomials w$ degD"



"eneral Bezier Curve

∑=

=n

i

ini t B pt s0

, )()(

ini

in t t i

nt B −−

= )1()(,9ernsteinbasis

The Euadratic and Cubic Curves of Fava /)are 9e&ier Curves with nG/ and nG0

The pi are the control points



Bernstein %ol!nomials

● =or those that forget combinatorics

iik ik uu

ik ik ub −−−= )1(

)!(!!)(



.oining Bzier Segments:

$#e Bernstein %ol!nomials● Hbserve

The =our Bernstein polynomials – also defined by

●

These represent the blending proportionsamong the control points




$#e Bernstein %ol!nomials● The four cubic Bernstein

polynomials

● Hbserve: – at tGB- only BB" is B

● curve interpolates P@

– at tG@- only BB& is B● curve interpolates P;




$#e Bernstein %ol!nomials● Cubic 9ernstein

blending functions

● Hbserve: thecoefficients are justrows in Pascal,striangle







Bezier Curves, B-Splines,NURBS



Some $!pes o( Curves

● 4ermite – def,d by two end points

and two tangent

vectors● Bzier

– two end points plustwo control points forthe tangent vectors

● Splines – 9asis *plines

– def,d w$ ; control points

– 5niform- nonrational B-splines

– 2onuniform- nonrational 9(splines

– 2onuniform- rational

B-splines NURBS !



Bzier Curves

● Pierre 9<&ier ><nault @A.B

● 9asic idea

– four points – *tart point P

– 1nd point P !

– Tangent at P - P P "

– Tangent at P !- P ! P #







B-Spline Curve

∑=

=n

i

ik i t N pt p0

, )()(

)()()(

otherwise,0),[,1)(

1,1

11

1,1,

1,0

t N t t

t t t N

t t

t t t N

t t t t N

ik

ik i

k iik

ik i

iik

iii

+−

+++

++−

+

+

−

−+

−

−=

∈=2ormali&ed 9(

spline blendingfunctions

nJ@ control points and nJKJ/ parameters Known as Knots

)efined only on Lt0- t

nJK(/#





B-splines: Basic 'eas

● *imilar to 9<&ier curves – *mooth blending function times control points

●

9ut: – 9lending functions are non(&ero over only a smallpart of the parameter range"giving us local support #

– +hen non&ero- they are the %concatenation' ofsmooth polynomials



B-spline Blening Functions

● is a step function that is @ in theinterval

● spans two intervals and is apiecewise linear function that goesfrom B to @ "and bacK#

● spans three intervals and is apiecewise 7uadratic that grows from Bto @$;- then up to 0$; in the middle ofthe second interval- bacK to @$;- andbacK to B

● is a cubic that spans four intervals

growing from B to @$. to /$0- thenbacK to @$. and to B

9(spline blending functions

)(0, t Bk

Bk ,1

( t )

)(2, t Bk

)(3, t Bk

B li Bl i F ti



B-spline Blening Functions:Example (or 0n 1rer Splines

● 2ote: can,t define apolynomial with theseproperties "both B and non(

&ero for ranges#● Idea: subdivide theparameter space intointervals and build a piecewise polynomial – 1ach interval gets different

polynomial function

B li Bl i F ti



B-spline Blening Functions:Example (or 2 1rer Splines

● Hbserve: – at tGB and tG@ just

three of the functionsare non(&ero

– all are GB and sum to@- hence the convexhull property holds foreach curve segment ofa9(spline

@AA; =oley$an)am$=iner$4uges$Phillips IC8



B-splines: Setting t#e 1ptions

● *pecified by –

– m1 control points- ' ( ) ' m

– m-* cubic polynomial curve segments- +,)+m

– m-1 'not points- t t m1

– segments +i of the 9(spline curve are● defined over a Knot interval

●

defined by ; of the control points- ' i-, ) ' i

– segments +i of the 9(spline curve are blended together into

smooth transitions via"the new 3 improved# blending functions

],[ 1+ii t t

3≥m

E l C ti B li



Example: Creating a B-splineCurve Segment

ii

t t 1−

P i

Qi



B-splines: 3not Selection

● Instead of worKing with the parameterspace - use

● The 'not points – joint points between

curve segments- +i

– 1ach has a"not value

– m-1 Knots form1 points

10 ≤≤ t max1210min ... t t t t t t m ≤≤≤≤≤ −

@AA; =oley$an)am$=iner$4uges$Phillips IC8



B-spline: 3not Se4uences

● 1ven distribution of Knots – uniform 9(splines

– Curve does not interpolate end points● first blending function not e7ual to @ at tGB

● 5neven distribution of Knots – non-uniform 9(splines

– 6llows us to tie down the endpoints by repeating Knot values"in Cox(de9oor- B$BG@#

– If a Knot value is repeated- it increases the effect "weight# of the

blending function at that point – If Knot is repeated d times- blending function converges to @ and the

curve interpolates the control point

Creating a Non Uni(orm



Creating a Non-Uni(ormB-spline: 3not Selection

● 8iven curve of degree d.,- with m1 controlpoints – first- create m-1*d-1! Knot points

–

use Knot values (/(/(/1/*/)/ m-*/ m-1/m-1/m-1! "adding two extra ( ,s and m-1,s#

– 2ote● Causes Cox(de9oor to give

added weight in blending to the

first and last points when t isnear t min and t max

Pics$Math courtesy of 8! =arin > 6*5

+atc#ing E((ects



+atc#ing E((ectso( 3not Selection

● N Knot points "initially# – 2ote: Knots are distributed

parametrically based on t -hence why they %move'

● @B control points● Curves have as many

segments as they have non(&ero intervals in u

degree ofcurve



B-splines: 5ocal Control %ropert!

● ocal Control – polynomial coefficients

depend on a few points

–

moving control point "' #affects only local curve

– +hy: 9ased on curve def,n-affected region extends atmost @ Knot point away





B-splines: Setting t#e 1ptions

● 4ow to space the "not points0

O Uni(orm● e7ual spacing of Knots along the curve

O Non-Uni(orm

● +hich type of parametric function?

O Rational●

x(t)* y(t)* +(t) defined as ratio of cubic polynomials O Non-Rational

Documents

10. Bezier Curves, B-Splines & NURBS