Art-Inspired Modeling and Visualization

Galina Pasko, Alexander PaskoEuropean University of Lefke, Northern CyprusBournemouth University, UKEmail: gip@pasko.orgThe mathematical investigations of Roger Pen-rose inspired Dutch artist M.C. Escher to illustratein his drawings the constructive elements of thisworld. We introduce unusual higher-dimensionalcomputer reconstructions of some drawings for pre-senting our research results in solving geometricmodeling problems arising from analysis ofEscher’s artworks. The following identified prob-lems had no general solutions in geometric model-ing: the metamorphosis between 2D arbitrary poly-gons similar to shape transformations in “Day &Night” (1938), the simulation of 3D shape recon-struction from its 2D projection as it happens in“Reptiles” (1943), and the surface trimming de-picted in the “Rind” (1955) and other Escher’sdrawings. We applied the Function Representation(FRep) of geometric shapes by continuous func-tions of point coordinates [1, 4] to solve these prob-lems.A visually smooth transformation (or metamor-phosis) between two arbitrary shapes can find a gen-eral solution if both shapes are represented by realfunctions and the transformation is described as aninterpolation between these functions. First, werepresent a 2D polygon by a real function of pointcoordinates taking zero value at polygon edges.An arbitrary polygon can be represented by a set-theoretic expression involving union and intersec-tion operations on the half-planes passing throughthe polygon edges. The formula for the functiondefining a polygon can be obtained from the set-theoretic expression by replacing each half-plane byits defining function while set-theoretic operationsare replaced by the corresponding R-functions, seedetails in [2].We implemented the above polygon-to-functionconversion approach and applied it to several ge-ometric modeling problems including shape meta-morphosis and reconstruction from projections.The original Escher’s drawing and our reconstruc-a bFigure 1: Shape metamorphosis:a) the original transformation (fragment of “Day &Night”,