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
Shape Space Exploration of Constrained Meshes
Yas Island Marina Hotel
Abu Dhabi
Architect: Asymptote Architecture
Steel/glass construction: Waagner Biro
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

Deformation field d applied to the current mesh x
yields a new mesh x + d

Distance measure
point
Shape Space Exploration of Constrained Meshes
Constrained Mesh Manifold

Constrained mesh manifold M:


represents all meshes satisfying the given
constraints
Individual constraint

defines a hypersurface in
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
Example: PQ Mesh Manifold

PQ mesh manifold M: represents all PQ meshes

Constraints (planarity per face)

each face
(signed diagonal distance)

deviation from planarity

10mm allowance for 2m x 2m panels
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 path
consider 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
nd
2
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
E : 1  3  
c
i
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
Shape Space Exploration of Constrained Meshes