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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

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geometric modeling with conical meshes and developable surfaces

Geometric Modeling with Conical Meshes and Developable Surfaces

SIGGRAPH 2006

Yang Liu, Helmut Pottmann, Johannes Wallner, Yong-Liang Yang and Wenping Wang

problem
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

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conical meshes in action
conical meshes in action

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pq planar quad strip

PQ (Planar Quad) strip

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ruled surface
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

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developable 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 surfacewith K=0 everywhere
  • examples: http://www.rhino3.de/design/modeling/developable

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tangent surface examples
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

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pq strip
PQ strip
  • discrete counterpart of developable surface

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pq mesh

PQ mesh

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conjugate curves
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

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conjugate curves cont d
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

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conjugate curves cont d12
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

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conjugate curves cont d13
conjugate curves (cont’d)
  • example #3: principle curvature lines
  • example #4: isoparameter lines of a translational surface

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conjugate curves cont d14
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

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conjugate curves cont d15
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

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conjugate curves cont d16
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

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pq mesh17
PQ mesh
  • discrete analogue of conjugate curves network (example #2)

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pq mesh cont d
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

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conical mesh

conical mesh

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

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conical mesh21
conical mesh
  • all the vertices of valence 4 are conical vertices of which adjacent faces are tangent to a common sphere

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conical mesh cont d
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

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conical mesh cont d23
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”

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conical mesh cont d24
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

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getting pq conical meshes

getting PQ/conical meshes

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getting pq mesh
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

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getting conical 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

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

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