surface modeling through geodesic reporter: hongyan zhao date: apr. 18th email:...
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Surface modelingthrough geodesic
Guo-jin Wang, Kai Tang, Chiew-Lan Tai. Parametric representation of a surface pencil with a common spatial geodesic. Computer Aided Design 36 (2004) 447-459.
Marco Paluszny. Cubic Polynomial Patches though Geodesics.
***, Wenping Wang, ***. Geodesic-Controlled Developable Surfaces for Modeling Paper Bending.
Background
Geodesic A geodesic is a locally length-minimizing
curve. In the plane, the geodesics are straight lines. On the sphere, the geodesics are great circles.For a parametric representation surface, the
geodesic can be found ……http://mathworld.wolfram.com/Geodesic.html
Background
Applications of geodesics(1)Geodesic Dome
tent manufacturing
Background
Applications of geodesics(2)Shoe-making industry
Garment industry
Surface modelingthrough geodesic
Guo-jin Wang, Kai Tang, Chiew-Lan Tai. Parametric representation of a surface pencil with a common spatial geodesic. Computer Aided Design 36 (2004) 447-459.
Marco Paluszny. Cubic Polynomial Patches though Geodesics.
***, Wenping Wang, ***. Geodesic-Controlled Developable Surfaces for Modeling Paper Bending.
Parametric representation of a surface pencil with a common spatial geodesicGuo-jin Wang, Kai Tang, Chiew-Lan Tai
Computer Aided Design 36 (2004) 447-459
Author Introduction
Kai Tang http://ihome.ust.hk/~mektang/
Department of Mechanical Engineering,
Hong Kong University of Science & Technology.
Chiew-Lan Taihttp://www.cs.ust.hk/~taicl/
Department of Computer Science & Engineering,
Hong Kong University of Science & Technology.
Parametric representation of a surface pencil with a common spatial geodesicBasic idea
Representation of a surface pencil through the given curve
Isoparametric and geodesic requirements
Representation of a surface pencil through Representation of a surface pencil through the given curvethe given curve
Parametric representation of a surface pencil with a common spatial geodesicBasic idea
Representation of a surface pencil through the given curve
Isoparametric and geodesic requirements
Representation of a surface pencil through the given curve
( )
( , ) ( ) ( ( , ), ( , ), ( , )) ( )
( )
r
s t r u r t v r t w r t r
r
T
P R N
B
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Isoparametric and geodesic requirements
Isoparametric requirements
Geodesic requirementsAt any point on the curve, the principal normal to the
curve and the normal to the surface are parallel to each other.
0( , ) ( )s t rP R 0 0 0( , ) ( , ) ( , ) 0u r t v r t w r t
0 0 0 0
0 0 0 0
0 0 0 0
, , , ,0,
, , , ,1 0,
, , , ,1 0
v r t w r t w r t v r t
r t r t
u r t w r t w r t u r t
r t r t
u r t v r t v r t u r t
r t r t
0
0
,0,
,0
v r t
t
w r t
t
Isoparametric and geodesic requirements
( , ) ( ) ( , ) ( ) ( , ) ( ) ( , ) ( )s t r u r t r v r t r w r t r P R T N B
( )( ) ,
| ( ) |
( ) ( )( ) ,
| ( ) ( ) |
( ) ( ) ( )
rr
r
r rr
r r
r r r
RT
R
R RB
R R
N B T
The representation with isoparametric and geodesic requirements
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Cubic Polynomial Patches though Geodesics
Marco Paluszny
Author introduction
Marco Paluszny
Professor
Universidad Central de Venezuela
Cubic Polynomial Patches though Geodesics
GoalExhibit a simple method to create low degree
and in particular cubic polynomial surface patches that contain given curves as geodesics.
Cubic Polynomial Patches though Geodesics
OutlinePatch through one geodesic
Representation– Ribbon (ruled patch)– Non ruled patch
Developable patchesPatch through pairs of geodesics
Using Hermite polynomialsJoining two cubic ribbons
G1 joining of geodesic curves
Patch through one geodesicPatch through one geodesic
Cubic Polynomial Patches though Geodesics
Patch through one geodesicRepresentation
Ribbon (ruled patch)Non ruled patch
Developable patchesPatch through pairs of geodesics
Using Hermite polynomialsJoining two cubic ribbons
G1 joining of geodesic curves
Patch through one geodesic
RepresentationRibbon (ruled surface)
Non ruled surface
ˆ( , ) ( ) { ( ) ( ) ( )}s t t s t t t X X X X X
2( , ) ( , ) ( , )s t s t s s t Y X P
Patch through one geodesic
Developable patches
Then the surface patch
is developable.
2
2
( ) || ( ) || ( ( ) ( )) ( )
ˆ ( ) || ( ) ( ) ||
t t t t t
t t t
X X X X
X X
ˆ( , ) ( ) { ( ) ( ) ( )}s t t s t t t X X X X X
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Patch through pairs of geodesics
Patch through pairs of geodesics
Using Hermite polynomials3 2
0
21
22
2 33
( ) (1 ) 3(1 ) ,
( ) (1 ) ,
( ) (1 ) ,
( ) 3(1 ) .
H s s s s
H s s s
H s s s
H s s s s
0 1 1 1 1 1 1 1
2 2 2 2 2 2 3 2
ˆ( , ) ( ) ( ) ( ){ ( ) ( ) ( )}
ˆ ( ){ ( ) ( ) ( ) ( )} ( )
s t H s t H s t t t
H s t t t t H t
X X X X X
X X X X
Patch through pairs of geodesics
Joining two cubic ribbons
X00
X02X01
X03
X11
X10
X12
X13
Y00
Y02Y01
Y03
Y11
Y10
Y12
Y13
X00
X02X01
X03
X11
X10
X12
X13
Y10
Y12Y11
Y13
Y01
Y00
Y02
Y03 Return
G1 joining of geodesic curves
G1 connection of two ribbons containing G1 abutting geodesics(1)
G1 joining of geodesic curves
G1 connection of two ribbons containing G1
abutting geodesics(2)The tangent vectors and are parallel.The ribbons share a common ruling segment at
.The tangent planes at each point of the com-
mon segment are equal for both patches.
(0)X (1)Y
(0) (1)X Y
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Geodesic-Controlled Developable Surfaces for Modeling Paper Bending
***, Wenping Wang, ***
Author introduction
Wenping Wang
Associate Professor B.Sc. and M.Eng, Shandong University, 1983, 1986;
Ph.D., University of Alberta, 1992. Department of Computer Science,The University of Hong Kong.
Email: [email protected]
Geodesic-Controlled Developable Surfaces
Goal: modeling paper bending
Geodesic-Controlled Developable Surfaces
OutlinePropose a representation of
developable surface Rectifying developable (geodesic-
controlled developable)
Composite developableModify the surface by modifying the
geodesicMove control pointsMove control handles Preserve the curve length
Propose a representation of developable surface
Geodesic-Controlled Developable Surfaces
OutlinePropose a representation of developabl
e surface Rectifying developable (a geodesic-
controlled developable) Composite developable
Modify the surface by modifying the geodesic Move control points Move control handles Preserve the curve length
Rectifying developable
Definition Rectifying plane: The plane
spanned by the tangent vector and binormal vector
Given a 3D curve with non-vanishing curvature, the envelope of its rectifying planes is a developable surface, called rectifying developable.
Rectifying developable
Representation
or
where is arc length.
The surface possesses as a directrix as well
as a geodesic!
( , ) ( ) ( )s t s t X p T B
( , ) ( ) ( ( ) ( ))s t s t s s X p p p
s
( )sp
Rectifying developable
Curve of regressionWhy?
A general developable surface is singular along the curve of regression.
GoalKeep singularities out of
region of interest
Definition:
limit intersection of rulings
Rectifying developable
Compute Paper boundaryGoal
Keep singularities out of region of interest
Keep the paper shape when bending
MethodCompute the ruling length of each
curve point
Rectifying developable
Keep singularities out of region of interest
Composite developable
Why?A piece of paper consists of several parts
which cannot be represented by a one-parameter family of rulings from a single developable.
Composite developable
DefinitionA composite developable surface is made of
a union of curved developables joined together by transition planar regions.
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Interactive modifying
Move control points
Interactive modifying
Move control handles(1)Why?
Users usually bend a piece of paper by holding to two positions on it.
Give:positions and orientation vectors
at the two ends.Want:
a control geodesic meeting those conditions
Interactive modifying
Move control handles(2)
When the constraints are not enough, minimize
20 10 1
0 1
interpolate normalinterpolate normal
( ) | | | | | || | | |
f X N N
N N XN N
Interactive modifying
Preserve curve length
Composite developable
Boundary planar region
Composite developable
Control a composite developable
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Application
Texture mappingThe algorithm computing paper boundary.
Surface approximation
VIDEOVIDEOVIDEOVIDEO
Future work
Investigate the representation of the control geodesic curve with length preserving property.3D PH curve
The endThe endThe endThe end
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