Curve Surface
Transcript of Curve Surface
-
8/13/2019 Curve Surface
1/43
Representation of Curves &Surfaces
Prof. Lizhuang Ma
Shanghai Jiao Tong University
-
8/13/2019 Curve Surface
2/43
Contents
Specialized Modeling Techniques
Polygon Meshes
Parametric Cubic Curves Parametric Bi-Cubic Surfaces
Quadric Surfaces
Specialized Modeling Techniques
-
8/13/2019 Curve Surface
3/43
The Teapot
-
8/13/2019 Curve Surface
4/43
Representing Polygon Meshes
Explicit representation
By a list of vertex coordinates
Pointers to a vertex list
Pointers to an edge list
)),,(),...,,,(),,,(( 222111 nnn zyxzyxzyxP
-
8/13/2019 Curve Surface
5/43
Pointers to A Vertex List
),,,( 4321 VVVVV
)),,(),...,,,(( 444111 zyxzyx
)3,2,4(
)4,2,1(
2
1
P
P
-
8/13/2019 Curve Surface
6/43
Pointers to An Edge List
),,,( 4321 VVVVV
)),,(),...,,,(( 444111 zyxzyx
),,(
),,(
),,,(
),,,(
),,,(
),,,(),,,(
4322
5411
1145
21244
2433
2322
1211
EEEP
EEEP
PVVE
PPVVE
PVVE
PVVEPVVE
-
8/13/2019 Curve Surface
7/43
Parametric Cubic Curves
The cubic polynomials that define a curve
segment are of the form:
10)(
)(
)(
,
23
23
23
tdtctbtatz
dtctbtaty
dtctbtatx
zzzz
yyyy
xxxx
TtztytxtQ )]()()([)(
-
8/13/2019 Curve Surface
8/43
Parametric Cubic Curves
The curve segment can be rewrite as
Where
zzzz
yyyy
xxxx
T
dcba
dcba
dcba
C
tttT 123
TCtztytxtQ T )]()()([)(
-
8/13/2019 Curve Surface
9/43
Parametric Cubic Curves
-
8/13/2019 Curve Surface
10/43
Tangent Vector
Ttzdt
dty
dt
dtx
dt
dtQtQ
dt
d)]()()([)()( '
T
ttCTCdt
d
]123[ 2
T
zzzyyyxxx ctbtactbtactbta ]232323[ 222
-
8/13/2019 Curve Surface
11/43
Continuity
Between Curve Segments G0 geometric continuity
Two curve segments join together
G1 geometric continuity
The directions (but not necessarily the magnitudes)
of the two segments tangent vectors are equal at a
join point
-
8/13/2019 Curve Surface
12/43
Continuity
Between Curve Segments C1 continuous
The tangent vectors of the two cubic curve segments
are equal (both directions and magnitudes) at thesegments joint point
Cncontinuous
The direction and magnitude of through
the nth derivative are equal at the joint point
)]([/ tQdtd nn
-
8/13/2019 Curve Surface
13/43
Continuity
Between Curve Segments
-
8/13/2019 Curve Surface
14/43
Continuity
Between Curve Segments
-
8/13/2019 Curve Surface
15/43
Three Types of
Parametric Cubic Curves
Hermite Curves Defined by two endpoints and two endpoint tangent
vectors Bzier Curves
Defined by two endpoints and two control pointswhich control the endpoint tangent vectors
Splines Defined by four control points
-
8/13/2019 Curve Surface
16/43
Parametric Cubic Curves
Representation:
Rewrite the coefficient matrix as
where Mis a 4x4 basis matrix, Gis called the
geometry matrix So
1)(
)(
)(
)(2
3
44342414
43332313
42322212
41312111
4321t
t
t
mmmm
mmmm
mmmm
mmmm
GGGG
tz
ty
tx
tQ
TCtQ )(
MGC
4 endpoints or tangent vectors
-
8/13/2019 Curve Surface
17/43
Parametric Cubic Curves
Where is called the blending functions
BGTMGtQ )(
TMB
-
8/13/2019 Curve Surface
18/43
Hermite Curves
Given the endpointsP1andP4and tangent vectors atthemR1andR4
What is Hermite basis matrix MH Hermite geometry vector GH Hermite blending functions BH
By definition
GH=[P1 P4 R1 R4]
-
8/13/2019 Curve Surface
19/43
Hermite Curves
Since
THH
T
HH
T
HH
T
HH
MGPQMGRQ
MGPQ
MGPQ
0123)1(0100)0(
1111)1(
1000)0(
4
'
1
'
4
1
0011
1110
2010
3010
4141 HHH MGRRPPG
-
8/13/2019 Curve Surface
20/43
Hermite Curves
So
And
0011
0121
0032
1032
0011
1110
2010
3010 1
HM
TH tttttttttB 23232323 232132
HHHH BGTMGtQ )(
-
8/13/2019 Curve Surface
21/43
Given the endpointsP1andP4and two control
pointsP2andP3which determine the endpoints
tangent vectors, such that
What is Bzier basis matrixMB Bzier geometry vector GB Bzier blending functionsBB
Bzier Curves
)(3)1(
)(3)0(
34
'
4
12'
1
PPQR
PPQR
-
8/13/2019 Curve Surface
22/43
Bzier Curves
By definition
Then
So
HBB MGPPPP
3011
3000
0300
0301
4321
4321 PPPPGB
4141 RRPPGH
TMMGTMGtQHHBBHH
)()(
TMGTMMG BBHHBB )(
-
8/13/2019 Curve Surface
23/43
Bzier Curves
And
HBBHHBB MGMMM
0001
0033
0363
1331
4
3
3
2
2
2
1
3 )1(3)1(3)1()( PtPttPttPttQ
T
ttttttBB 3223
)1(3)1(3)1( (Bernstein polynomials)
-
8/13/2019 Curve Surface
24/43
Convex Hull
-
8/13/2019 Curve Surface
25/43
The polynomial coefficients for natural cubic
splines are dependent on all n control points
Has one more degree of continuity than is inherent inthe Hermite and Bzier forms
Moving any one control point affects the entire curve
The computation time needed to invert the matrix can
interfere with rapid interactive reshaping of a curve
Spline
-
8/13/2019 Curve Surface
26/43
B-Spline
-
8/13/2019 Curve Surface
27/43
Cubic B-Spline
Has m+1control points
Has m-2cubic polynomial curve segments
Uniform The knots are spaced at equal intervals of the
parameter t Non-rational Not rational cubic polynomial curves
Uniform NonRational B-Splines
3,,..., 10 mPPP m
mQQQ ,...,, 43
-
8/13/2019 Curve Surface
28/43
Curve segment Qi is defined by points
B-Spline geometry matrix
If Then
Uniform NonRational B-Splines
iiii PPPP ,,, 223
miPPPPG iiiiBSi 3,123
T
iiii ttttttT 1)()()(
23
1,)( iiiBsBsii tttTMGtQ
-
8/13/2019 Curve Surface
29/43
So B-Spline basis matrix
B-Spline blending functions
Uniform NonRational B-Splines
TBs tttttttB 323233 1333463)1(6
1
0001
1333
4063
1331
6
1BsM
-
8/13/2019 Curve Surface
30/43
The knot-value sequence is a nondecreasing
sequence
Allow multiple knot and the number of identical
parameter is the multiplicity Ex. (0,0,0,0,1,1,2,3,4,4,5,5,5,5)
So
Uniform NonRational B-Splines
)()()()()( 4,4,114,224,33 tBPtBPtBPtBPtQ iiiiiiiii
-
8/13/2019 Curve Surface
31/43
Uniform NonRational B-Splines
Where isj th-order blending function for
weighting control pointPi
)()()(
)()()(
)()()(
,0
,1)(
3,1
14
43,
3
4,
2,1
13
32,
2
3,
1,1
12
21,
1
2,
1
1,
tBtt
tttB
tt
tttB
tB
tt
tttB
tt
tttB
tBtt
tttB
tt
tttB
otherwise
ttttB
i
ii
ii
ii
ii
i
ii
ii
ii
ii
i
ii
ii
ii
ii
ii
i
)(, tB ji
-
8/13/2019 Curve Surface
32/43
Knot Multiplicity & Continuity
Since is within the convex hull of and
If is within the convex hull of ,
and and the convex hull of and ,so it
will lie on If will lie on
If will lie on both and , and the
curve becomes broken
)( itQ 23 , ii PP 1iP
)(,1 iii tQtt 23, ii PP
1iP 12 , ii PP 1iP
12 ii PP
)(,21 iiii tQttt 1iP
)(,321 iiiii tQtttt 1iP iP
-
8/13/2019 Curve Surface
33/43
Multiplicity 1 : C2 continuity
Multiplicity 2 : C1 continuity
Multiplicity 3 : C0 continuity Multiplicity 4 : no continuity
Knot Multiplicity & Continuity
-
8/13/2019 Curve Surface
34/43
NURBS:
NonUniform Rational B-Splines
Rational
and are defined as the ratio of two
cubic polynomials
Rational cubic polynomial curve segments areratios of polynomials
Can be Bzier, Hermite, or B-Splines
),(),( tytx
)(
)()(
tW
tXtx
)(tz
)(
)()(
tW
tYty
)(
)()(
tW
tZtz
-
8/13/2019 Curve Surface
35/43
Parametric Bi-Cubic Surfaces
Parametric cubic curves are
So, parametric bi-cubic surfaces are
If we allow the points in Gto vary in 3D along
some path, then
Since are cubics:
, where
SMtGtGtGtGtsQ )()()()(,( 4321
TMGtQ )(
SMGtQ )(
)(tGi
TMGtGi )( 4321 iiiii ggggG
-
8/13/2019 Curve Surface
36/43
Parametric Bi-Cubic Surfaces
So
SM
gggg
gggg
gggg
gggg
MTtsQ TT
44342414
43332313
42322212
41313211
),(
1,0, tsSMGMT TT
-
8/13/2019 Curve Surface
37/43
Hermite Surfaces
SMGMTtsQ HHT
H
T ),( TMGtRtRtPtP HH ()()( 4141
-
8/13/2019 Curve Surface
38/43
Bzier Surfaces
SMGMTtsQ BsBsT
Bs
T ),(
-
8/13/2019 Curve Surface
39/43
Normals to Surfaces
Ss
MGMTtsQs
TT
),(
TTT ssMGMT 0123 2
SMGMTt
tsQs
TT
)(),(
SMGMss TT 0123 2
),(),( tsQt
tsQs
normal vector
-
8/13/2019 Curve Surface
40/43
Quadric Surfaces
Implicit surface equation
An alternative representation
with
0 PQPT
0222222),,( 222 kjzhygxfxzeyzdxyczbyaxzyxf
1
z
yx
P
kjhg
jcef
hebdgfda
Q
-
8/13/2019 Curve Surface
41/43
Advantages:
Computing the surface normal
Testing whether a point is on the surface Computingzgivenxandy
Calculating intersections of one surface with another
Quadric Surfaces
-
8/13/2019 Curve Surface
42/43
Fractal Models
-
8/13/2019 Curve Surface
43/43
Grammar-Based Models
L-grammars
A -> AA
B -> A[B]AA[B]