Geometric Modeling with Conical Meshes and Developable
SurfacesSIGGRAPH 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: http://math.arizona.edu/~models/Ruled_Surfaces
ruled surface
)()(),( uvuvu δbx
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:
http://www.rhino3.de/design/modeling/developable
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
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 transl
ational 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 norm
als 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 connectivi
ty– mesh normals of adjacent vertices intersect th
us resulting in natural support structure
getting PQ/conical meshes
getting PQ mesh
• optimization!• a quad is planar iff the sum of four inner a
ngles 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