Two-Dimensional Image Rotation

We study the problem of approximating a rotation of the plane, α : R ² → R ² α(x,y) = (x cos θ + y sin θ, y cos θ − x sin θ), by a bijection β: Z ² → Z ². We show by an explicit construction that one may choose β so that \( sup_{z \in Z^2 } \left| {\alpha \left( z \right) - \beta \left( z \right)} \right| \leqslant \frac{1} {{\sqrt 2 }}\frac{{1 + r}} {{\sqrt {1 + r^2 } }}, \). Where r = tan(θ/2). The scheme is based on those invented and patented by the second author in 1994.

Authors and Affiliations

1. Department of Mathematics, University of Toledo, Toledo, Ohio, USA

   Ivan Sterling

2. 1715 Chandler, Ann Arbor, Michigan, USA

   Thomas Sterling

Editors and Affiliations

1. Wissenschaftliche Visualisierung, Konrad-Zuse-Zentrum für Informationstechnik Berlin, Takustraße 7, D-14195, Berlin, Germany

   Hans-Christian Hege

2. Fachbereich 3, Mathematik, Technische Universität Berlin, Straße des 17. Juni 136, D-10623, Berlin, Germany

   Konrad Polthier

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