Geometric Modeling with Conical Meshes and Developable Surfaces

1. Geometric Modeling with Conical Meshes and Developable Surfaces SIGGRAPH 2006 Yang Liu, Helmut Pottmann, Johannes Wallner, Yong-Liang Yang and Wenping Wang

2. problem • mesh suitable to architecture, especially for layered glass structure • planar quad faces • nice offset property – offsetting mesh with constant results in the same connectivity • natural support structure orthogonal to the mesh 

3. conical meshes in action 

4. PQ (Planar Quad) strip 

5. ruled surface • surface that can be swept by moving a line in space • Gaussian curvature on a ruled regular surface is everywhere non-positive (MathWorld) • examples: http://math.arizona.edu/~models/Ruled_Surfaces 

6. developable surface • surface which can be flattened onto a plane without distortion • cylinder, cone and tangent surface • part of the tangent surface of a space curve, called singular curve • a ruled surfacewith K=0 everywhere • examples: http://www.rhino3.de/design/modeling/developable 

7. tangent surface examples • of helix (animation): http://www.ag.jku.at/helixtang_en.htm • of twisted cubic: http://math.umn.edu/~roberts/java.dir/JGV/tangent_surface0.html 

8. PQ strip • discrete counterpart of developable surface 

9. PQ mesh 

10. conjugate curves • two one parameter families A,B of curves which cover a given surface such that for each point p on the surface, there is a unique curve of A and a unique curve of B which pass through p 

11. conjugate curves (cont’d) • example #1: (conjugate surface tangent) rays from a (light) source tangent to a surface and the tangent line of the shadow contour generated by the light source 

12. conjugate curves (cont’d) • example #2: (general version of previous example) for a developable surface enveloped by the tangent planes along a curve on the surface, at each point, one family curve is the ruling and the other is tangent to the curve at the point- they are symmetric 

13. conjugate curves (cont’d) • example #3: principle curvature lines • example #4: isoparameter lines of a translational surface 

14. conjugate curves (cont’d) • example #5: (another generalization of example #1?) contour generators on a surface produced by a movement of a viewpoint along some curve in space and the epipolar curves which can be found by integrating the (light) rays tangent to the surface 

15. conjugate curves (cont’d) • example #6: intersection curves of a surface with the planes containing a line and the contour generators for viewpoints on the line asymptotic lines: self-conjugate 

16. conjugate curves (cont’d) • example #7: isophotic curves (points where surface normals form constant angle with a given direction) and the curves of steepest descents w.r.t. the direction 

17. PQ mesh • discrete analogue of conjugate curves network (example #2) 

18. PQ mesh (cont’d) • if a subdivision process, which preserves the PQ property, refines a PQ mesh and produces a curve network in the limit, then the limit is a conjugate curve network on a surface 

19. conical mesh 

20. circular mesh • PQ mesh where each of the quad has a circumcircle • discrete analogue of principle curvature lines 

21. conical mesh • all the vertices of valence 4 are conical vertices of which adjacent faces are tangent to a common sphere 

22. conical mesh (cont’d) • three types of conical vertices: hyperbolic, elliptic and parabolic • conical vertex  1+3=2+4 • the spherical image of a conical mesh is a circular mesh 

23. conical mesh (cont’d) • discrete analogue of principle curvatures • “in differential geometry, the surface normals of a smooth surface along a curve constitute a developable surface iff that curve is a principle curvature line” 

24. conical mesh (cont’d) • nice properties • all quads are planar, of course • offsetting a conical mesh keeps the connectivity • mesh normals of adjacent vertices intersect thus resulting in natural support structure 

25. getting PQ/conical meshes 

26. getting PQ mesh • optimization! • a quad is planar iff the sum of four inner angles is 2 • minimizes bending energy • minimizes distance from input quad mesh 

27. getting conical mesh • optimization with different constraint • to get a conical mesh of an arbitrary mesh, first compute the quad mesh extracted from its principle curvature lines and uses it as the input mesh 

28. refinement • alternates subdivision (Catmull-Clark or Doo-Sabin) and perturbation • for PQ strip, uses curve subdivision algorithm, e.g, Chaikin’s 