Geometric Modeling with Conical Meshes and Developable Surfaces

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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

SIGGRAPH 2006

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

problem
• mesh suitable to architecture, especially for layered glass structure
• 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

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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
• 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
• 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
• discrete counterpart of developable surface

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### PQ mesh

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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)
• 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’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’d)
• example #3: principle curvature lines
• example #4: isoparameter lines of a translational surface

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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’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’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 mesh
• discrete analogue of conjugate curves network (example #2)

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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

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

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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)
• 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’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’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

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

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