Shape Space Exploration of Constrained Meshes

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Shape Space Exploration of Constrained Meshes

• Yongliang Yang, Yijun Yang, Helmut Pottmann,
  Niloy J. Mitra

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Meshes and Constraints

• Meshes as discrete geometry representations
• Constrained meshes for various applications

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Yas Island Marina Hotel Abu Dhabi Architect
Asymptote Architecture Steel/glass construction
Waagner Biro
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Constrained Mesh Example (1)

• Planar quad (PQ) meshes Liu et al. 2006

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Constrained Mesh Example (2)

• Circular/conical meshes Liu et al. 2006

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Problem Statement

• Givensingle 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

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Example
meshes found via exploration
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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

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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

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Constrained Mesh Manifold

• Constrained mesh manifold M
• represents all meshes satisfying the given
  constraints
• Individual constraint
• defines a hypersurface in

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Constrained Mesh Manifold

• Involving m constraints in
• M is the intersection of m hypersurfaces
• dimension D-m (tangent space)
• codimension m (normal space)

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Example PQ Mesh Manifold

• PQ mesh manifold M
• Constraints (planarity per face)
• each face (signed diagonal
  distance)
• deviation from planarity
• 10mm allowance for 2m x 2m panels

represents all PQ meshes
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Tangent Space

• starting mesh
• Geometrically, intersection of the tangent
  hyperplanes of the constraint hypersurfaces

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Walking on the Tangent Space
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Better Approximation ?

• Better approximation - 2nd order approximant

curved path consider the curvature of the manifold
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a simple idea

• m hypersurfaces Ei 0 (i1, 2, ..., m)
• osculating paraboloid Si
• the intersection of all osculating paraboloids
• hard to compute
• not easy to use for exploration

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Compute Osculant

• Generalization of the osculating paraboloid of a
  hypersurface osculant
• Has the following form
• Second order contact with each of the constraint
  hypersurfaces

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2nd order contact
amounts to solving linear systems
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Walking on the Osculant
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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

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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

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Commonly used Energies

• Fairness energies
• smoothness of the poly-lines
• Orthogonality energy
• generate large visible shape changes

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Applications
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Spectral Analysis

• Good (desirable) subspaces to explore
• 2D-slice of design space

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2D Subspace Exploration
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Handle Driven Exploration
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stiffness analysis
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Circular Mesh Manifolds

• Circular Meshes (discrete principal curvature
  param.)
• Each face has a circumcircle

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moving out into space
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Combined Constraint Manifolds
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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
  doesnt really work (because of low degrees of
  freedom) form-finding

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Acknowledgements

• Bailin Deng, Michael Eigensatz, Mathias Höbinger,
  Alexander
• Schiftner, Heinz Schmiedhofer, Johannes
  Wallner
• Funding agencies FWF, FFG
• Asymptote Architecture

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What Do We Gain?