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