CAD Tools for Creating 3D Escher Tiles

1. 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

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

3. 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?

4. Tiling on 2-manifolds In the plane On the sphere On a genus-3 “Tetrus” surface In the Poincaré disk

5. 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.

6. 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 Shewchuk’s 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 code:http://code.google.com/p/jmescher

7. jmEscher: CDT Library for Java • Delaunay triangulation uses Lawson’s (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).

8. 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.

9. 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.

10. Mesh-Cutting Algorithm (Cont’d) • 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.

11. Mesh-Cutting Algorithm (Cont’d) • 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.

12. 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.

13. Results: 2½D Bird Tile

14. 3D Tilings • Exploring two fundamental domains: • #1: Truncated octahedron, derived from the body-centered cubic lattice.

15. 3D Tilings • Exploring two fundamental domains: • #2: Rhombic dodecahedron, based on the densest sphere packing.

16. 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.

17. 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.

18. User Interface Issues (Cont’d) • 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.

19. User Interface Issues (Cont’d) • 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%

20. Results: 3D Fish Tile

21. 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 Escher’s 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!