Advertisement

Homeomorphisms Between the Circular Disc and the Square

  • Chamberlain FongEmail author
Living reference work entry

Abstract

The circle and the square are among the most common shapes used by mankind. Consequently, it is worthwhile to study the mathematical correspondence between the two. This chapter discusses three different ways of mapping a circular region to a square region and vice versa. Each of these mappings has nice closed-form invertible equations and different interesting properties. In addition, this chapter will present artistic applications of these mappings such as converting the Poincaré disk to a square as well as molding rectangular artworks into oval-shaped ones.

Keywords

Conformal square Escheresque artworks Invertible mappings Non-Euclidean geometry Poincaré disk squircles 

References

  1. Abramowitz M, Stegun I (1972) Handbook of mathematical functions. Dover Publications Inc., New YorkzbMATHGoogle Scholar
  2. Carlson B (1977) Special functions of applied mathematics. Academic, New YorkzbMATHGoogle Scholar
  3. Dunham D (2009) Hamiltonian paths and hyperbolic patterns. Contemp Math 479:51–65MathSciNetCrossRefGoogle Scholar
  4. Fernandez-Guasti M (1992) Analytic geometry of some rectilinear figures. Int J Math Educ Sci Technol 23:895–901CrossRefGoogle Scholar
  5. Floater M, Hormann K (2005) Surface parameterization: a tutorial and survey. In: Dodgson N, Floater M, Sabin M (eds) Advances in multiresolution for geometric modelling. Springer, New York, pp 157–186CrossRefGoogle Scholar
  6. Fong C (2014) Analytical methods for squaring the disc. In: Seoul ICM 2014Google Scholar
  7. Fong C (2019) Elliptification of rectangular imagery. In: Joint mathematics meeting SIGMAA-ARTSGoogle Scholar
  8. Frederick C, Schwarz E (1990) Conformal image warping. IEEE Comput Graph Appl 10(2):54–61CrossRefGoogle Scholar
  9. Hancock H (1958) Elliptic integrals. Dover Publications Inc., New YorkzbMATHGoogle Scholar
  10. Langer J, Singer D (2011) The lemniscatic chessboard. Forum Geometricorum 11:183–199MathSciNetzbMATHGoogle Scholar
  11. Nowell P (2005) Mapping a square to a circle (blog). http://mathproofs.blogspot.com/2005/07/mapping-square-to-circle.html
  12. Press W, Flannery B, Teukolsky S, Vetterling W (1992) Numerical recipes in C. The art of scientific computing, 2nd edn. Cambridge University Press, CambridgezbMATHGoogle Scholar
  13. Rice A, Brown E (2012) Why ellipses are not elliptic curves. Math Mag 85(3):163–176MathSciNetCrossRefGoogle Scholar
  14. Shirley P, Chiu K (1997) A low distortion map between disk and square. J Graph Tools 2:45–52CrossRefGoogle Scholar
  15. Taimina D (2009) Crocheting adventures with hyperbolic planes. AK Peters, WellesleyCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.exile.orgSan FranciscoUSA

Section editors and affiliations

  • Bharath Sriraman
    • 1
  • Kyeong-Hwa Lee
    • 2
  1. 1.Department of Mathematical SciencesThe University of MontanaMissoulaUSA
  2. 2.Department of Mathematics Education, College of EducationSeoul National UniversitySeoulSouth Korea

Personalised recommendations