Escherization and OrnamentalSubdivisions

M.C. Escher

Escherization

• ``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.

Escherization

Tessellations

• Geometric pattern, which is able to fill an infinite plane without any overlaps or gaps

• Individual tiles can undergo rigid body transformations

N-hedral Property

• N-hedral• Monohedral

• Trivial dihedral case

N-hedral Property

Symmetry

• 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

INSTANCES PRUNE(INSTANCES)

end while

return CONTENTS (INSTANCES)

end function

Results of System

• Performs well on convex and “nearly convex” shapes

Results of System

• System can fail on an already

repeatable tile

• 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

• ``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.

Voronoi Diagrams

• Division of a plane based upon the proximity to a set of point or line generators

• Generators with weights

Voronoi Diagrams

Parquet Deformation

Applications to Other Works

• Parquet Deformation• Circle and Square Limits