Computer Aided Engineering DesignAnupam Saxena

Associate ProfessorIndian Institute of Technology KANPUR 208016

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

Lecture # 36Design of Surface Patches

Design of Surface patches

Surface patches can be modeled mathematically in parametric form as

A closed, connected composite surface represents the shape of a solid.

This surface, in turn, is composed of surface patches,

aesthetics, aerodynamics, fluid flow etc. may influence surface design

Surfaces of aircraft wings and fuselage, car body and its doors, seats, and windshields are all designed by combining surface patches at their boundaries.

]1,0[],1,0[),(),(),(),( vuvuzvuyvuxvuf

),(),,(),,( vuzvuyvux scalar polynomials in parameters (u, v)

Types of Surface patches

Tensor product surface patches

Boundary interpolating patches

Sweep surfaces

Quadric (Analytic) surface patches

Tensor Product Surface patchLet and be univariate functions such that

n

jjm

ii vu00 )(,)(

uU and vV

n

j

m

ijiij vuvu

0 0

)()(),( Cr

is called a tensor product surface with domain UV

Cij 3

The surface is bi-quadratic for m = n = 2 and bi-cubic for m = n = 3e.g.

22

210

10

)1(2)1()()()()(

,)1()()()(

vvvvvvvv

uuuuu

2

2

121110

020100 )1(2

)1(

)3,1,1()4,2,1()0,2,1(

)3,1,0()4,2,0()0,0,0()1(][),(

v

vv

v

uuzyxvuCCC

CCCr

Tensor Product Surface patch…

v = constant

u = constant

Tensor Product Surface patch…Generalization

1

.

.

..

.....

.....

..

..

1..),(

)1(

00)1(00

0)1()1)(1()1(

0)1(

1

0 0

n

n

nn

mnmnm

mnmmn

mmn

j

m

i

jiij

v

v

uuvuvu

DDD

DDD

DDD

Dr

m and n are user-chosen degrees in parameters u and v

For a bi-cubic surface patch, one needs to specify 16 sets of data as control points and/or slopes One for each Dij

patches with degrees in u and v greater than 3 can be modeled

one can as well choose the degrees unequal in parameters

for most applications, use of bi-cubic surface patches seems adequate

Ferguson’s Bicubic Patch

3

0

3

0

)()(),(j i

jiij vuvu Cr

)()(

),2()(

),32()(

),132()(

233

232

231

230

uuu

uuuu

uuu

uuu

Hermite functions

)(

)(

)(

)(

)()()()(

3

2

1

0

33323130

23222120

13121110

03020100

3210

v

v

v

v

uuuu

CCCC

CCCC

CCCC

CCCC

In matrix form

Ferguson’s Bicubic Patch …

(v =0)

(u =0)(v =1)

(u =1)

00r

01r

10r

11r

u (0,0)rv (0,0)r

u (0,1)rv (0,1)r

u (1,1)r

v (1,1)r

u (1,0)r

v (1,0)r

Ferguson’s Bicubic Patch …

33231303

32221202

31211101

30201000

)1,1(;)1,0(;)1,1(;)1,0(

)0,1(;)0,0(;)0,1(;)0,0(

)1,1(;)1,0(;)1,1(;)1,0(

)0,1(;)0,0(;)0,1(;)0,0(

CrCrCrCr

CrCrCrCr

CrCrCrCr

CrCrCrCr

uvuvvv

uvuvvv

uu

uu

)1,1(),(

)0,1(),(

)1,0(),(

)0,0(),(

1,1

2

0,1

2

1,0

2

0,0

2

uv

vu

uv

vu

uv

vu

uv

vu

udv

vu

udv

vu

udv

vu

udv

vu

rr

rr

rr

rr

)(

)(

)(

)(

)1,1()0,1()1,1()0,1(

)1,0()0,0()1,0()0,0(

)1,1()0,1()1,1()0,1(

)1,0()0,0()1,0()0,0(

)()()()(),(

3

2

1

0

3210

v

v

v

v

uuuuvu

uvuvuu

uvuvuu

vv

vv

rrrr

rrrr

rrrr

rrrr

r

Ferguson’s patch

= UMGMT VT Geometric matrix

Ferguson coefficient matrix

Example A simple Ferguson Bicubic Patch

Specifying twist vectors is not easy; we assign them 0 values

r(u, v) = UMGMT VT =

]6,587,6137[ 2323 vuuuu

00)0,0,5()0,0,5(

00)0,5,0()0,5,0(

)6,0,0()6,0,0()6,6,0()0,6,0(

)6,0,0()6,0,0()6,0,6()0,0,6(

G

02

46

02

46

0

2

4

6

Example A simple Ferguson Bicubic Patch

)100,0,100()100,0,100()0,0,5()0,0,5(

)100,0,100()100,0,100()0,5,0()0,5,0(

)6,0,0()6,0,0()6,6,0()0,6,0(

)6,0,0()6,0,0()6,0,6()0,0,6(

G

05

1015

0

2

4

6

-5

0

5

10