Surface modeling through geodesic Reporter: Hongyan Zhao Date: Apr. 18th Email: [email protected].

Surface modeling

through geodesic

Reporter: Hongyan ZhaoDate: Apr. 18thEmail: [email protected]

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

ReturnReturn

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

ReturnReturn

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

Return

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

ReturnReturn

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.

Return

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

Return

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

Thank you!Thank you!Thank you!Thank you!