A new method of coding geodesics on surfaces whose fundamental regions consist of the union of elementary triangles derived from the Farey series

Abstract

In this paper, a new method of coding geodesics is presented. This method generalizes the method proposed by Series and it is based on the set of hyperbolic tessellations whose corresponding fundamental regions are made up of elementary triangles derived from the Farey series, and it makes use of the concepts of continued fraction expansion and cutting sequences. These tessellations have interesting properties with applications in (lossless) source coding, topological quantum codes, design of digital signal constellations, study of second-order ordinary differential equations of the Fuchsian type besides coding of geodesics related to dynamical systems. This method promises to open up new perspectives of applications of the Farey series in digital signal processing. The union of these triangles provides a systematic procedure to obtain families of tessellations and one of the main characteristics of the Farey series is the easiness in obtaining and characterizing the triangles of the new tessellations. In applications related to hyperbolic tessellations, knowing the genus of the surface is quite important due to the fact that it is related to the characteristic of the transmission channel as well as to the algebraic–geometric code to be used in the channel encoder. We consider the analyses of the surfaces, with genus $$g=1$$ , showing that the results may be extended to any value of $$n$$ ( $$n>0$$ ), as well as to any $$g\ge 2$$ . Finally, the proposed method for coding geodesics in the hyperbolic plane is presented and the new codes are tabulated.