An analysis of symmetry breaking in the taxicab ellipse

Abstract

This article analyzes the behavior of the ellipse in Taxicab Geometry (or Manhattan Geometry), a non-Euclidean geometry in which the metric in an  space is defined by the sum of the absolute values of the differences between coordinates. As a direct consequence of this alteration in the distance function, conics such as the circle, ellipse, parabola, and hyperbola are no longer curves, and are instead represented by line segments. This work has the objective of presenting the construction of the ellipse as an octagon and investigating the effect of reflection and rotation transformations on the figure. It is proved that such operations, isometric in Euclidean Geometry, do not preserve distances in the taxicab metric, deforming the geometric structure of the conic. Applying the matrices of these transformations, we demonstrate that oblique angles maintain the octagonal figure, while orthogonal applications change the figure into a hexagon. Finally, we present the deduction that proves the reduction of the figure from eight to six sides.