Post on 11-Jan-2016
description
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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