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

3. Introduction 3  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 4 In the plane In the Poincaré disk On the sphere On a genus-3 “Tetrus” surface

5. 2½D Tilings 5  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 6  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 7  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 8  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-convex boundaries, we load copies of the neighboring tiles to fill concavities as necessary.

9. Mesh-Cutting Algorithm 9  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) 10 1. User specifies lateral offset. 2. Construct cookie cutter as a boundary “shell” filled with temporary edges. o Walk along the boundary to identify intersections with the underlying landscape. o Extend the landscape as needed by loading additional mesh copies. 1. 2.

11. Mesh-Cutting Algorithm (Cont’d) 11 3. Maintain a list of edge fragments that lie inside the cookie cutter. o Test for intersections among fragments. o After the boundary walk, add all fragments to the cookie cutter as constrained edges. 4. Perform a flood search to find the remaining geometry inside the cookie cutter. 3. 4.

12. Visual Debugging 12  Animation and visualization help identify bugs and difficult or unexpected cases in geometric algorithms.  We implement visual “breakpoints” in Java2D by overriding the repaint mechanism.  Can insert breakpoints in mid-algorithm.  Can specify which geometric features to highlight.

13. Results: 2½D Bird Tile 13

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

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

16. Overview of 3D Editing Interface 16  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.  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 pairings convex/concave. User Interface Issues 17

18. User Interface Issues (Cont’d) 18  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) 19  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

20. Results: 3D Fish Tile 20

21. Conclusion 21  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!