Surface modeling through geodesic Reporter: Hongyan Zhao Date: Apr. 18th

Surface modeling through 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)  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 ……

Background  Applications of geodesics(1) Geodesic Dome  tent manufacturing

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

Surface modeling through 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) Parametric representation of a surface pencil with a common spatial geodesic  Marco Paluszny. Cubic Polynomial Patches though Geodesics.Cubic Polynomial Patches though Geodesics  ***, Wenping Wang, ***. Geodesic-Controlled Developable Surfaces for Modeling Paper Bending.Geodesic-Controlled Developable Surfaces for Modeling Paper Bending

Parametric representation of a surface pencil with a common spatial geodesic Guo-jin Wang, Kai Tang, Chiew-Lan Tai Computer Aided Design 36 (2004)

Author Introduction  Kai Tang Department of Mechanical Engineering, Hong Kong University of Science & Technology.  Chiew-Lan Tai Department of Computer Science & Engineering, Hong Kong University of Science & Technology.

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 Representation of a surface pencil through the given curve

Parametric representation of a surface pencil with a common spatial geodesic  Basic idea Representation of a surface pencil through the given curve Representation of a surface pencil through the given curve Isoparametric and geodesic requirements

Representation of a surface pencil through the given curve

Isoparametric and geodesic requirements IIsoparametric requirements G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.

 The representation with isoparametric and geodesic requirements

Cubic Polynomial Patches though Geodesics Marco Paluszny

Author introduction Marco Paluszny Professor Universidad Central de Venezuela

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.

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 G 1 joining of geodesic curves Patch through one geodesic Patch through one geodesic

Cubic Polynomial Patches though Geodesics  Patch through one geodesic Patch through one geodesic Representation  Ribbon (ruled patch)  Non ruled patch Developable patches  Patch through pairs of geodesics Patch through pairs of geodesics Using Hermite polynomials Joining two cubic ribbons  G 1 joining of geodesic curves G 1 joining of geodesic curves

Patch through one geodesic  Representation Ribbon (ruled surface) Non ruled surface

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

Patch through pairs of geodesics

 Using Hermite polynomials

Patch through pairs of geodesics  Joining two cubic ribbons X 00 X 02 X 01 X 03 X 11 X 10 X 12 X 13 Y 00 Y 02 Y 01 Y 03 Y 11 Y 10 Y 12 Y 13 X 00 X 02 X 01 X 03 X 11 X 10 X 12 X 13 Y 10 Y 12 Y 11 Y 13 Y 01 Y 00 Y 02 Y 03

G 1 joining of geodesic curves  G 1 connection of two ribbons containing G 1 abutting geodesics(1)

G 1 joining of geodesic curves  G 1 connection of two ribbons containing G 1 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.

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, Department of Computer Science, The University of Hong Kong.

Geodesic-Controlled Developable Surfaces  Goal: modeling paper bending

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 Propose a representation of developable surface

Geodesic-Controlled Developable Surfaces  Outline Propose a representation of developable surface Propose a representation of developable surface  Rectifying developable (a geodesic- controlled developable)  Composite developable Modify the surface by modifying the geodesic 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 !

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

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

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

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

Interactive modifying  Preserve curve length

Composite developable  Boundary planar region

Composite developable  Control a composite developable

Application  Texture mapping The algorithm computing paper boundary.  Surface approximation

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

The end Thank you!