Cassini sets in taxicab geometry

TL;DR

This work studies Cassini ovals under the taxicab metric by developing a region-based geometric framework with coordinate and guide lines. It obtains explicit piecewise descriptions in standard position, showing that $K(p,q;r)$ consists of segments on guide lines in each quadrant, a central rectangle segment, and hyperbola-like arcs in half-strips, with a critical transition at $r^* = \frac{1}{2}d(p,q)$ from two closed curves to one. The paper proves that $K(p,q;r)$ is invariant under isometries and scales with dilations, and gives two dual representations of the filled Cassini set $L(p,q;r)$ as unions of intersections and intersections of unions of guide Cassini sets, highlighting a deeper duality in taxicab geometry. It also discusses the special case where $q=-p$ (standard position), and suggests that the observed union/intersection dualities may extend to broader families of focal-distance sets, indicating a path toward a unified theory for taxicab conic-like loci.

Abstract

Given two points $p$ and $q$ in the plane and a nonnegative number $r$, the Cassini oval is the set of points $x$ that satisfy $d(x, p) d(x, q) = r^2$. In this paper, we study this set using the taxicab metric. We find that these sets have characteristics that are qualitatively similar to their Euclidean counterparts while also reflecting the underlying taxicab structure. We provide a geometric description of these sets and provide a characterization in terms of intersections and unions of a restricted family of such sets analogous to that found recently for taxicab Apollonian sets.

Figures (7)

• Figure 1: Cassini ovals in Euclidean space with lighter shading corresponding to increasing $r$. The second-from-outermost curve corresponds to $r = \sqrt{2}\,r_E^*$.
• Figure 2: Cassini sets when the foci are in general position (a) and (b), and when the foci share a coordinate line (c). As in the Euclidean case, a transition from a pair of simple closed curves to a single simple closed curve occurs at $r = r^* = \frac{1}{2} d(p, q)$.
• Figure 3: (a) Coordinate lines, dotted; guide lines, dashed; coordinate complements and guide complements. (b) The regions defined by $p$ and $q$. (c) The values of $\sigma_p$ and $\sigma_q$ in each region.
• Figure 4: The sets $E(p, q)$ when $p$ and $q$ do not share a guide line (a) and when $p$ and $q$ do share a guide line (b), and the sets $H(p, q)$ when $p$ and $q$ do not share a guide line (c) and when $p$ and $q$ do share a guide line (d).
• Figure 5: The Cassini set $K(p, q; r)$ intersects the quadrants and central rectangle along guide lines (top row) and intersects half-strips in hyperbolas (bottom row). Dashed curves are centered at $g^-$ and dotted curves are centered at $g^+$. On the left, $p = (8, 3)$, $q = -p$, and $r = 16$. On the right, $p = (12, 1)$, $q = -p$, and $r = 12$.

Theorems & Definitions (11)

• Lemma 2.1
• Theorem A
• proof
• Lemma 4.1
• proof
• Theorem B
• proof
• Theorem C
• proof
• Corollary 4.1.1