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Surface modeling through geodesic

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Guo-jin Wang, Kai Tang, Chiew-Lan Tai. Parametric representation of a surface ... Shoe-making industry. Garment industry. Surface modeling. through geodesic ... – PowerPoint PPT presentation

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Title: Surface modeling through geodesic


1
Surface modelingthrough geodesic
  • Reporter Hongyan Zhao
  • Date Apr. 18th
  • Email Hongyanzhao_cn_at_yahoo.com.cn

2
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.

3
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
4
Background
  • Applications of geodesics(1)
  • Geodesic Dome
  • tent manufacturing

5
Background
  • Applications of geodesics(2)
  • Shoe-making industry
  • Garment industry

6
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.

7
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
8
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.

9
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
10
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

11
Representation of a surface pencil through the
given curve
Return
12
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.

13
Isoparametric and geodesic requirements
  • The representation with isoparametric and
    geodesic requirements

Return
14
Cubic Polynomial Patches though Geodesics
  • Marco Paluszny

15
Author introduction
  • Marco Paluszny
  • Professor
  • Universidad Central de Venezuela

16
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.

17
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
18
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

19
Patch through one geodesic
  • Representation
  • Ribbon (ruled surface)
  • Non ruled surface

20
Patch through one geodesic
  • Developable patches
  • Then the surface patch
  • is developable.

Return
21
Patch through pairs of geodesics
22
Patch through pairs of geodesics
  • Using Hermite polynomials

23
Patch through pairs of geodesics
  • Joining two cubic ribbons

Return
24
G1 joining of geodesic curves
  • G1 connection of two ribbons containing G1
    abutting geodesics(1)

25
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
26
Geodesic-Controlled Developable Surfaces for
Modeling Paper Bending
  • , Wenping Wang,

27
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

28
Geodesic-Controlled Developable Surfaces
  • Goal modeling paper bending

29
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
30
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

31
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.

32
Rectifying developable
  • Representation
  • or
  • where is arc length.
  • The surface possesses as a directrix as
    well
  • as a geodesic!

33
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

34
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

35
Rectifying developable
  • Keep singularities out of region of interest

36
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.

37
Composite developable
  • Definition
  • A composite developable surface is made of a
    union of curved developables joined together by
    transition planar regions.

Return
38
Interactive modifying
  • Move control points

39
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

40
Interactive modifying
  • Move control handles(2)
  • When the constraints are not enough, minimize

41
Interactive modifying
  • Preserve curve length

42
Composite developable
  • Boundary planar region

43
Composite developable
  • Control a composite developable

Return
44
Application
  • Texture mapping
  • The algorithm computing paper boundary.
  • Surface approximation

VIDEO
45
Future work
  • Investigate the representation of the control
    geodesic curve with length preserving property.
  • 3D PH curve

46
The end
Thank you!
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