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

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 

conical meshes in action 

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

surface that can be swept by moving a line in space Gaussian curvature on a ruled regular surface is everywhere non-positive (MathWorld) examples: ruled surface 

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 surface with K=0 everywhere examples: 

tangent surface examples of helix (animation): of twisted cubic: surface0.html surface0.html 

PQ strip discrete counterpart of developable surface 

PQ mesh 

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 

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 

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 

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

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 

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 

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 

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

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 

conical mesh 

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

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

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 

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” 

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 

getting PQ/conical meshes 

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 

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 

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