Computer Aided Engineering DesignAnupam Saxena

Associate ProfessorIndian Institute of Technology KANPUR 208016

Implementation and Coding

Parameterization and Knot Vector generation

Geometric/PARAMETRIC Modeling

Solid Modeling

Perception of Solids

Topology and Sol ids

Solid Modeling 1-2

Transformations and Projections 1-2

Modeling of Curves

Representation, Differential Geometry

Ferguson Segments

Bezier Segments 1-2

B-spl ine curves 1-5

NURBS

Modeling of Surfaces (Patches)

Differential Geom etry

Tensor Product

Boundary Interpolating

Composite

NURBS

Examples

Geometric/PARAMETRIC Modeling

Solid Modeling

Perception of Solids

Topology and Solids

Solid Modeling 1-2

Transformations and Projections 1-2

Modeling of Curves

Representation, Differential Geometry

Ferguson Segments

Bezier Segments 1-2

B-spline curves 1-5

NURBS

Modeling of Surfaces (Patches)

Differential Geometry

Tensor Product

Boundary Interpolating

Composite

NURBS

Geometric/PARAMETRIC Modeling

Solid Modeling

Perception of Solids

Topology and Sol ids

Solid Modeling 1-2

Transformations and Projections 1-2

Modeling of Curves

Representation, Differential Geometry

Ferguson Segments

Bezier Segments 1-2

B-spl ine curves 1-5

NURBS

Modeling of Surfaces (Patches)

Differential Geom etry

Tensor Product

Boundary Interpolating

Composite

NURBS

Lecture #32

Interpolation with B-spline curves

NURBS

Geometric/PARAMETRIC Modeling

Solid Modeling

Perception of Solids

Topology and Sol ids

Solid Modeling 1-2

Transformations and Projections 1-2

Modeling of Curves

Representation, Differential Geometry

Ferguson Segments

Bezier Segments 1-2

B-spl ine curves 1-5

NURBS

Modeling of Surfaces (Patches)

Differential Geom etry

Tensor Product

Boundary Interpolating

Composite

NURBS

Interpolation with B-spline curves

Given n+1 data points p0, p1, ..., pn

fit them with a B-spline curve of given order p n

a set of parameters u0, u1, ..., un may be generated

the number of knots m + 1 may be computed

knot vector [t0, t1, …, tm] may then be computed

Basis functions known

n + 1 conditions

Interpolation with B-spline curves

n

iiipp tNt

0, )()( bb

REQUIRED TO FIND THE INTERPOLATING B-SPLINE CURVE

Control points bi’s are (n+1) unknowns

Consider pk = b(uk) =

n

iikipp uN

0, )( b

p0

p1

p2

…pn

Np,p(u0) Np,p+1(u0) Np,p+2(u0) … Np,n+p(u0)Np,p(u1) Np,p+1(u1) Np,p+2(u1) … Np,n+p(u1)Np,p(u2) Np,p+1(u2) Np,p+2(u2) … Np,n+p(u2)… … … … …Np,p(un) Np,p+1(un) Np,p+2(un) … Np,n+p(un)

P = =

b0

b1

b2

…bn

= NB

k = 0, …, n

Examples

NURBS

Short for Non-Uniform Rational B-Splines

Recall from Rational Bézier curves that

n

ii

ni

n

iii

ni

wtB

wtBt

0

0

)(

)()(

Pb

Likewise, NURBS can be computed as

n

iippi

n

iiippi

tNw

tNwt

0,

0,

)(

)()(

bb

weights wi specified by the user to gain additional design freedom

non-uniform: knots are not placed at regular intervals wi = 0: location of bi does not affect the curve’s shapeFor larger values of wi, the curve gets pushed towards bi

Offer great flexibility in designPossess local shape control & all otherProperties of B-spline curves

Widely used in freeform curve designCan also model analytical curves

Examples

B-spline and Bernstein polynomialsFor an order p curve …

Repeat the first knot ‘0’ p times

Repeat the last knot ‘1’ p times

Consider n + 1 = p B-spline basis functions/ Control points

number of knots (m + 1); m = n + p = p + p = 2p

THE B-SPLINE BASIS FUNCTIONS DEGENERATE TO BERNSTEIN POLYNOMIALS

Examples