computer aided engineering design

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 #34

Differential Geometry of Surfaces


Solid Modeling


Topology and Sol ids

Solid Modeling 1-2


Modeling of Curves


Ferguson Segments

Bezier Segments 1-2

B-spl ine curves 1-5

NURBS


Differential Geom etry

Tensor Product


Composite

NURBS

Curves on a surface

dt

dvdt

du

v

z

u

zv

y

u

yv

x

u

x

dt

dvdt

du

dt

dvdt

du

vu Arr

r(u, v)

c(t)= r(u(t), v(t))

tangent to the curve

dt

dv

v

vu

dt

du

u

vu

dt

vud

),(),(),( rrcT


Solid Modeling



Solid Modeling 1-2


Modeling of Curves


Ferguson Segments

Bezier Segments 1-2


NURBS



Tensor Product


Composite

NURBS

Curves on a surface

dtdt

vudds

),(c dt

dt

dv

dt

duvu rr

r(u, v)

c(t) = r(u(t), v(t))

differential arc ds length of the curve

dtdt

dv

dt

du

dt

dv

dt

duvuvu

rr.rr

dt

dt

dvdt

du

dt

dv

dt

duvu

v

u

rr

r

rdt

dt

dvdt

du

dt

dv

dt

du

vvuv

vuuu

.rr.rr

.rr.rrdt

dt

dvdt

du

dt

dv

dt

du

G AA

.rr.rr

.rr.rrG T

vvuv

vuuu

where

Symmetric G is called the first fundamental matrix of the surface


Solid Modeling



Solid Modeling 1-2


Modeling of Curves


Ferguson Segments

Bezier Segments 1-2


NURBS



Tensor Product


Composite

NURBS

Curves on a surface …unit tangent t to the curve

dtdv

vvu

dtdu

uvu

dtdv

vvu

dtdu

uvu

),(),(

),(),(

rr

rr

t

dtdvdtdu

dtdv

dtdu

dtdv

vvu

dtdu

uvu

G

rr ),(),(

for t to exist

G should be always be positive definite

011 uu .rrG

G11G22 – G12G21 2)())(( vuvvuu .rr.rr.rr )()( vuvu rrrr > 0

implies that G is always positive definite


Solid Modeling



Solid Modeling 1-2


Modeling of Curves


Ferguson Segments

Bezier Segments 1-2


NURBS



Tensor Product


Composite

NURBS

Curves on a surface …

length of the curve segment in t0 t t1

11 t

t

t

t oo

dt

dt

dvdt

du

dt

dv

dt

dudss G

c(t1) and c(t2) as two curves on the surface r(u, v) that intersect the angle of intersection is given by

cos

),(),(),(),(

2

2

22

22

1

1

11

1121

dt

dvdt

du

dt

dv

dt

du

dt

dv

v

vu

dt

du

u

vu

dt

dvdt

du

dt

dv

dt

du

dt

dv

v

vu

dt

du

u

vu

G

rr

.

G

rr

.tt


Solid Modeling



Solid Modeling 1-2


Modeling of Curves


Ferguson Segments

Bezier Segments 1-2


NURBS



Tensor Product


Composite

NURBS

Curves on a surface …

two curves are orthogonal to each other if

0or

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

2122

122112

2111

2211

dt

dv

dt

dv

dt

dv

dt

du

dt

dv

dt

du

dt

du

dt

du

dt

dv

v

vu

dt

du

u

vu

dt

dv

v

vu

dt

du

u

vu

GGG

rr.

rr

If u t1 and v t2

vvuu

vu

.rr.rr

.rr.tt cos21

cos

),(),(),(),(

2

2

22

22

1

1

11

1121

dt

dvdt

du

dt

dv

dt

du

dt

dv

v

vu

dt

du

u

vu

dt

dvdt

du

dt

dv

dt

du

dt

dv

v

vu

dt

du

u

vu

G

rr

.

G

rr

.ttGeometric/PARAMETRIC Modeling

Solid Modeling



Solid Modeling 1-2


Modeling of Curves


Ferguson Segments

Bezier Segments 1-2


NURBS



Tensor Product


Composite

NURBS

Area of the surface patch

u = u0

u = u0 + du

v = v0

v = v0 + dv

r(u0, v0)

r(u0 + du, v0)r(u0, v0 + dv)

rudurvdv

dudvdvdudA vuvu rrrr

dudvdudv ||2122211 GGGG

dudvADomain ||G


Solid Modeling



Solid Modeling 1-2


Modeling of Curves


Ferguson Segments

Bezier Segments 1-2


NURBS



Tensor Product


Composite

NURBS

Surface from the tangent plane: Derivation

n

P R

d

nrr ),(),( 0000 vudvvduud

nrr

rrr

2

2

22

2

2

2

)(2

1)(

2

1dv

vdu

u

dudvvu

dvv

duud

22

2

22

22

u

)(2

1

)(2

1)(

dvv

duu

dudvvu

dvdu v

nr

nr

nr

nrnr


Solid Modeling



Solid Modeling 1-2


Modeling of Curves


Ferguson Segments

Bezier Segments 1-2


NURBS



Tensor Product


Composite

NURBS

Surface from the tangent plane: Derivation

n

P R

dn is perpendicular to the tangent plane, ru.n = rv.n = 0

dv

dudvdu

dvv

duu

dudvvu

d

vvuv

uvuu

nrnr

nrnr

nr

nr

nr

2

1

)(2

1)(

2

1)( 2

2

22

2

22

second fundamental matrix D


Solid Modeling



Solid Modeling 1-2


Modeling of Curves


Ferguson Segments

Bezier Segments 1-2


NURBS



Tensor Product


Composite

NURBS

Second fundamental matrix

22 )(2)(2, dvNMdudvduLdor

nrnrnr vvuvuu NML ,,

22 )(2)(2 dvdudvdudv

dudvdud vvuvuu

vvuv

uvuu nrnrnrnrnr

nrnr

L, M and N are called the second fundamental form coefficients

NM

MLD

nrnr

nrnr

vvuv

uvuu

)()(

)()(12

122211 vuvvvuuv

vuuvvuuu

rrrrrr

rrrrrr

GGGD

2122211 GGG

rr

rr

rrn

vu

vu

vu

use

Second fundamental matrix …

,,,where uvvuuvvuuvvu yxyxCxzxzBzyzyA

ruu = xuui + yuuj + zuuk

ruv = xuvi + yuvj + zuvk

rvv = xvvi + yvvj + zvvkvvv

uuuvu

zyx

zyx

kji

rr

2221

1211

222

1

DD

DD

CBAD

vvv

uuu

vvvvvv

vvv

uuu

uvuvuv

vvv

uuu

uuuuuu

zyx

zyx

zyx

D

zyx

zyx

zyx

DD

zyx

zyx

zyx

D 22211211 ,,

Classification of pointson the surface

)2(2

1 22 NdvMdudvLdud

tangent plane intersects the surface at all points where d = 0

dvL

LNMMduNdvMdudvLdu

222 02

02 LNMCase 1: No real value of du

P is the only common point between the tangent plane and the surface

No other point of intersection

P ELLIPTICAL POINT

Classification of pointson the surface

Case 2: 02 LNM L2+M2+N2 > 0 du = (M/L)dv

u – u0 = (M/L)(v – v0)

tangent plane intersects the surface along this straight line

P PARABOLIC POINT

Case 3: 02 LNM two real roots for du

tangent plane at P intersects the surface along two lines passing through P P HYPERBOLIC POINT

Case 4: L = M = N = 0 P FLAT POINT

computer aided engineering design

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