Shape Space Exploration of Constrained Meshes
Shape Space Exploration of Constrained Meshes
Yongliang Yang, Yijun Yang, Helmut Pottmann, Niloy J. Mitra
Shape Space Exploration of Constrained Meshes
Meshes and Constraints Meshes as discrete geometry representations
Constrained meshes for various applications
Yas Island Marina HotelAbu DhabiArchitect: Asymptote ArchitectureSteel/glass construction: Waagner Biro
Shape Space Exploration of Constrained Meshes
Constrained Mesh Example (1) Planar quad (PQ) meshes [Liu et al. 2006]
Shape Space Exploration of Constrained Meshes
Constrained Mesh Example (2) Circular/conical meshes [Liu et al. 2006]
Shape Space Exploration of Constrained Meshes
Shape Space Exploration of Constrained Meshes
Problem Statement Given:
single input mesh with a set of non-linear constraints in terms of mesh vertices
Goal:
explore neighboring meshes respecting the prescribed constraints
based on different application requirements, navigate only the desirable meshes according to given quality measures
Shape Space Exploration of Constrained Meshes
Example
input
meshes found via exploration
Shape Space Exploration of Constrained Meshes
Basic Idea Exploration of a high dimensional manifold
Meshes with same connectivity are mapped to points
Constrained meshes are mapped to points in a manifold M
Extract and explore the desirable parts of the manifold M
Shape Space Exploration of Constrained Meshes
Map Mesh to Point
The family of meshes with same combinatorics Mesh point
Deformation field d applied to the current mesh x yields a new mesh x + d
Distance measure
Shape Space Exploration of Constrained Meshes
Constrained mesh manifold M:
represents all meshes satisfying the given constraints
Individual constraint
defines a hypersurface in
Constrained Mesh Manifold
Shape Space Exploration of Constrained Meshes
Constrained Mesh Manifold Involving m constraints in
M is the intersection of m hypersurfaces
dimension D-m (tangent space)
codimension m (normal space)
Shape Space Exploration of Constrained Meshes
PQ mesh manifold M:
Constraints (planarity per face)
each face (signed diagonal distance)
deviation from planarity
10mm allowance for 2m x 2m panels
Example: PQ Mesh Manifoldrepresents all PQ meshes
Shape Space Exploration of Constrained Meshes
Tangent Space starting mesh
Geometrically, intersection of the tangent hyperplanes of the constraint hypersurfaces
Shape Space Exploration of Constrained Meshes
Walking on the Tangent Space
Shape Space Exploration of Constrained Meshes
Better Approximation ? Better approximation - 2nd order approximant
curved pathconsider the curvature of the manifold
Shape Space Exploration of Constrained Meshes
a simple idea
m hypersurfaces: Ei = 0 (i=1, 2, ..., m)
osculating paraboloid Si
the intersection of all osculating paraboloids:
hard to compute
not easy to use for exploration
Shape Space Exploration of Constrained Meshes
Compute Osculant Generalization of the osculating paraboloid of a
hypersurface: osculant
Has the following form:
Second order contact with each of the constraint hypersurfaces
Shape Space Exploration of Constrained Meshes
2nd order contact
amounts to solving linear systems
Shape Space Exploration of Constrained Meshes
Walking on the Osculant
Shape Space Exploration of Constrained Meshes
Mesh Quality? Osculant respects only the constraints
Quality measures based on application
Mesh fairness: important for applications like architecture
Extract the useful part of the manifold
Shape Space Exploration of Constrained Meshes
Extract the Good Regions
Abstract aesthetics and other properties via functions F(x) defined on
Restricting F(x) to the osculant S(u) yields an intrinsic Hessian of the function F
Shape Space Exploration of Constrained Meshes
Commonly used Energies Fairness energies
smoothness of the poly-lines
Orthogonality energy
generate large visible shape changes
Shape Space Exploration of Constrained Meshes
Applications
Shape Space Exploration of Constrained Meshes
Spectral Analysis
Good (desirable) subspaces to explore
2D-slice of design space
Shape Space Exploration of Constrained Meshes
2D Subspace Exploration
Shape Space Exploration of Constrained Meshes
Handle Driven Exploration
Shape Space Exploration of Constrained Meshes
stiffness analysis
Shape Space Exploration of Constrained Meshes
Circular Mesh Manifolds
Circular Meshes (discrete principal curvature param.)
Each face has a circumcircle
1 3:ciE
Shape Space Exploration of Constrained Meshes
moving out into space
Shape Space Exploration of Constrained Meshes
Shape Space Exploration of Constrained Meshes
Combined Constraint Manifolds
Shape Space Exploration of Constrained Meshes
Future Work multi-resolution framework
osculant surfaces
update instead of recompute (quasi-Newton)
other ways of exploration
interesting curves and 2-surfaces in M, ….
applications where handle-driven deformation doesn’t really work (because of low degrees of freedom): form-finding