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