Text of Escherization and Ornamental Subdivisions. M.C. Escher
Escherization and OrnamentalSubdivisions
• ``Escherization,'' by Craig S. Kaplan and David H. Salesin. SIGGRAPH 2000, the 27th International Conference on Computer Graphics and Interactive Techniques. New Orleans, Louisiana, USA, 25-27 July 2000.
• Computer Graphics and Geometric Ornamental Design
Craig Kaplan, University of Washington 2002
Escherization• Problem statement
Given a closed plane figure S (the “goal shape”), find a new closed figure T such that:
– 1. T is as close as possible to S; and
– 2. copies of T fit together to form a tiling of the plane.
• Geometric pattern, which is able to fill an infinite plane without any overlaps or gaps
• Individual tiles can undergo rigid body transformations
• N-hedral• Monohedral
• Trivial dihedral case
• Symmetry groups
Measure of Closeness
• How to compare two shapes?
• Metric insensitive to scaling, rotation, and translation
• Polygon Turning NumbersArkin, E.M., Chew, L.P., Huttenlocher, D.P., Kedem, K., and Mitchell, J.S.B. An Efficiently Computable Metric for Comparing Polygonal Shapes. PAMI(13), No. 3, March 1991, pp. 209-216.
Polygon Turning Numbers
Optimizing over Tiling Space
• function FINDOPTIMALTILING(GOALSHAPE ; FAMILIES ):INSTANCES CREATEINSTANCES (FAMILIES )while || INSTANCES || > 1 do
for each i in INSTANCES do
– ANNEAL(i; GOALSHAPE )end for
return CONTENTS (INSTANCES)
Results of System
• Performs well on convex and “nearly convex” shapes
Results of System
• System can fail on an already
• System tends to fail on shapes with
long, complicated tiling edges
• Vertices can be converted into control points to form curves
• User manipulation can improve results
• ``Voronoi Diagrams and Ornamental Design,'' by Craig S. Kaplan. ISAMA '99, The first annual symposium of the International Society for the Arts, Mathematics, and Architecture. San Sebastián, Spain, 7-11 June 1999, pp. 277-283.
• Division of a plane based upon the proximity to a set of point or line generators