Surface modeling through geodesic

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Surface modelingthrough geodesic

• Reporter Hongyan Zhao
• Date Apr. 18th
• Email Hongyanzhao_cn_at_yahoo.com.cn

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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.

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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
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Background

• Applications of geodesics(1)
• Geodesic Dome
• tent manufacturing

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Background

• Applications of geodesics(2)
• Shoe-making industry
• Garment industry

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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.

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Parametric representation of a surface pencil
with a common spatial geodesic

• Guo-jin Wang, Kai Tang, Chiew-Lan Tai

Computer Aided Design 36 (2004) 447-459
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Author Introduction

• Kai Tang
• http//ihome.ust.hk/mektang/
• Department of Mechanical Engineering,
• Hong Kong University of Science Technology.

• Chiew-Lan Tai
• http//www.cs.ust.hk/taicl/
• Department of Computer Science Engineering,
• Hong Kong University of Science Technology.

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Parametric representation of a surface pencil
with a common spatial geodesic

• Basic idea
• Representation of a surface pencil through the
  given curve
• Isoparametric and geodesic requirements

Representation of a surface pencil through the
given curve
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Parametric representation of a surface pencil
with a common spatial geodesic

• Basic idea
• Representation of a surface pencil through the
  given curve
• Isoparametric and geodesic requirements

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Representation of a surface pencil through the
given curve
Return
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Isoparametric and geodesic requirements

• Isoparametric requirements
• Geodesic requirements
• At any point on the curve, the principal normal
  to the curve and the normal to the surface are
  parallel to each other.

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Isoparametric and geodesic requirements

• The representation with isoparametric and
  geodesic requirements

Return
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Cubic Polynomial Patches though Geodesics

• Marco Paluszny

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Author introduction

• Marco Paluszny
• Professor
• Universidad Central de Venezuela

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Cubic Polynomial Patches though Geodesics

• Goal
• Exhibit a simple method to create low degree and
  in particular cubic polynomial surface patches
  that contain given curves as geodesics.

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Cubic Polynomial Patches though Geodesics

• Outline
• Patch through one geodesic
• Representation
• Ribbon (ruled patch)
• Non ruled patch
• Developable patches
• Patch through pairs of geodesics
• Using Hermite polynomials
• Joining two cubic ribbons
• G1 joining of geodesic curves

Patch through one geodesic
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Cubic Polynomial Patches though Geodesics

• Patch through one geodesic
• Representation
• Ribbon (ruled patch)
• Non ruled patch
• Developable patches
• Patch through pairs of geodesics
• Using Hermite polynomials
• Joining two cubic ribbons
• G1 joining of geodesic curves

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Patch through one geodesic

• Representation
• Ribbon (ruled surface)
• Non ruled surface

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Patch through one geodesic

• Developable patches
• Then the surface patch
• is developable.

Return
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Patch through pairs of geodesics
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Patch through pairs of geodesics

• Using Hermite polynomials

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Patch through pairs of geodesics

• Joining two cubic ribbons

Return
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G1 joining of geodesic curves

• G1 connection of two ribbons containing G1
  abutting geodesics(1)

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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.

Return
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Geodesic-Controlled Developable Surfaces for
Modeling Paper Bending

• , Wenping Wang,

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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 wenping_at_cs.hku.hk

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Geodesic-Controlled Developable Surfaces

• Goal modeling paper bending

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Geodesic-Controlled Developable Surfaces

• Outline
• Propose a representation of developable surface
• Rectifying developable (geodesic-controlled
  developable)
• Composite developable
• Modify the surface by modifying the geodesic
• Move control points
• Move control handles
• Preserve the curve length

Propose a representation of developable surface
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Geodesic-Controlled Developable Surfaces

• Outline
• Propose a representation of developable 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

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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.

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Rectifying developable

• Representation
• or
• where is arc length.
• The surface possesses as a directrix as
  well
• as a geodesic!

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Rectifying developable

• Curve of regression
• Why?
• A general developable surface is singular along
  the curve of regression.
• Goal
• Keep singularities out of region of interest
• Definition
• limit intersection of rulings

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Rectifying developable

• Compute Paper boundary
• Goal
• Keep singularities out of region of
  interest
• Keep the paper shape when bending
• Method
• Compute the ruling length of each curve point

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Rectifying developable

• Keep singularities out of region of interest

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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.

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Composite developable

• Definition
• A composite developable surface is made of a
  union of curved developables joined together by
  transition planar regions.

Return
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Interactive modifying

• Move control points

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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

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Interactive modifying

• Move control handles(2)
• When the constraints are not enough, minimize

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Interactive modifying

• Preserve curve length

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Composite developable

• Boundary planar region

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Composite developable

• Control a composite developable

Return
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Application

• Texture mapping
• The algorithm computing paper boundary.
• Surface approximation

VIDEO
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Future work

• Investigate the representation of the control
  geodesic curve with length preserving property.
• 3D PH curve

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The end
Thank you!