CAD Tools for Creating 3D Escher Tiles

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CAD Tools for Creating 3D Escher Tiles
Mark Howison and Carlo H. Séquin University of
California, Berkeley Computer-Aided Design and
Applications June 11, 2009
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Overview

• 2½D Tilings
• Constrained Delaunay Triangulation in Java
• Mesh-Cutting Algorithm
• Visual Debugging
• 3D Tilings
• User Interface Issues

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Introduction

• M.C. Escher popularized intricately decorated
  isohedral tilings.
• Planar tilings can be designed with available
  tools on the web.
• Specialized CAD tools help address the
  challenges of tiling other 2-manifolds.
• What are interesting tilings of 3D-space?

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Tiling on 2-manifolds
In the plane
On the sphere
On a genus-3 Tetrus surface
In the Poincaré disk
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2½D Tilings

• Warm-up exercise before tackling full 3D.
• Extruded 2D tilings form layers in 3-space.
• Trivial case extrude vertically, edit height
  field.
• Fancier case choose an offset between adjacent
  layers.

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Constrained Delaunay Triangulation in Java

• Meshes provide boundary representations of tiles.
• Delaunay triangulation produces aesthetically
  pleasing meshes with minimal sliver triangles.
• Tile decorations can be specified as
  constraints.
• Could not find existing CDT library for Java.
• Many available for C, such as Jonathan
  Shewchuks Triangle.
• Want interactive triangulation, not batch-mode
  processing.
• Users are adding and moving vertices and
  constrained edges.
• Developed our own library in Java!
• Open-source (BSD license), available from Google
  codehttp//code.google.com/p/jmescher

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jmEscher CDT Library for Java

• Delaunay triangulation uses Lawsons (1977)
  incremental insertion algorithm, backed by a
  half-edge data structure.
• Typically O(n log n) Performance is limited by
  how well you can locate which triangle contains
  the insertion site.
• Uses edge flips to turn a non-Delaunay
  triangulation into a Delaunay one.
• Constrained edge insertion uses algorithms by
  Anglada (1997).
• Supports non-convex boundaries and interactive
  relocation of boundary vertices.
• Robustness is provided by floating-point filters
  and arbitrary precision arithmetic (apfloat Java
  package).

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Locating Insertion Sites

• Heuristic use last inserted site as the search
  origin, since designer will often add vertices
  in localized groups.
• Easy if you have a convex boundary Walk along
  the triangles!
• For non-convexboundaries, we loadcopies of
  theneighboring tiles tofill concavities
  asnecessary.

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Mesh-Cutting Algorithm

• Need to form the bottom face of an offset 2½D
  tile.
• Use a cookie cutter to truncate the geometry
  in the underlying landscape.

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Mesh-Cutting Algorithm (Contd)

• User specifies lateral offset.
• Construct cookie cutter as a boundary shell
  filled with temporary edges.
• Walk along the boundary to identify intersections
  with the underlying landscape.
• Extend the landscape as needed by loading
  additional mesh copies.

1.
2.
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Mesh-Cutting Algorithm (Contd)

• Maintain a list of edge fragments that lie inside
  the cookie cutter.
• Test for intersections among fragments.
• After the boundary walk, add all fragments to the
  cookie cutter as constrained edges.
• Perform a flood search to find the remaining
  geometry inside the cookie cutter.

3.
4.
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Visual Debugging

• Animation and visualization help identify bugs
  and difficult or unexpected cases in geometric
  algorithms.
• We implement visual breakpoints in Java2D by
  overriding the repaintmechanism.
• Can insert breakpointsin mid-algorithm.
• Can specify whichgeometric featuresto highlight.

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Results 2½D Bird Tile
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3D Tilings

• Exploring two fundamental domains
• 1 Truncated octahedron, derived from
  the body-centered cubic lattice.

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3D Tilings

• Exploring two fundamental domains
• 2 Rhombic dodecahedron, based on the
  densest sphere packing.

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Overview of 3D Editing Interface

• Phase I
• Individual panes of the 3D tile are Delaunay
  triangulated.
• Vertices can be moved in 2D within pane interior.
• Boundary vertices cannot be moved yet, since this
  would create non-planar panes.
• Phase II
• All vertices can be moved in 3D
• Last selected point defines an extrusion vector.
• Can move points along extrusion vector or
  parallel to the edit pane.
• Local edits available by trisecting faces, but no
  Delaunay guarantee.
• Limited roll-back to Phase I.

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User Interface Issues

• Occlusion is an obstacle to free-form editing of
  3D tiles.
• Because of symmetry, edits in the current view
  will also change opposite faces.
• Creating a convex feature (e.g. a fish fin)
  creates a corresponding concave feature (e.g.
  eye socket) on the opposite side.
• Dual cameras show pairingsconvex/concave.

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User Interface Issues (Contd)

• 3D domains can be scaled/skewed and remain
  space-filling.
• What is the easiest way to manipulate this affine
  transform?
• Created a widget with 9 control points, each
  restricted to one degree of freedom.
• Widget maintains same orientation as camera.

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User Interface Issues (Contd)

• 3D tilings can have complicated interlocking
  features.
• Nearest neighbors can be scaled/translated to
  reveal the interface between adjoining tiles.

Scaled to 85 translated 0.6 tile lengths away
Scaled to 85
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Results 3D Fish Tile
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Conclusion

• Specialized CAD tools make it possible to design
  and fabricate space-filling Escher tiles.
• In 2½D, we can draw on an existing vocabulary
  of 2D tilings from Eschers sketchbook.
• 3D cubic lattice tiles are more difficult to
  design.
• The entire editable surface is constrained to fit
  seamlessly with adjacent tiles.
• In the 2D case, only the 1D border is subject to
  symmetry constraints, while the interior can be
  decorated freely
• There is no Escher sketchbook for 3D.
• Artists needed!