[Paper Review] Taxicab Angles and Trigonometry

Kevin P. Thompson, Tevian Dray|arXiv (Cornell University)|Jan 14, 2011

Mathematics and Applications3 references22 citations

TL;DR

This paper introduces a natural definition of angles and trigonometric functions in taxicab geometry using the taxicab metric, defining a t-radian as the angle subtending an arc length of 1 on the unit taxicab circle. It derives formulas for taxicab angle measures, establishes that congruent triangles require strict conditions (three sides and two angles), and demonstrates an exact parallax method for computing Euclidean distance using taxicab geometry, offering a precise alternative to the standard Euclidean approximation.

ABSTRACT

A natural analogue to angles and trigonometry is developed in taxicab geometry. This structure is then analyzed to see which, if any, congruent triangle relations hold. A nice application involving the use of parallax to determine the exact (taxicab) distance to an object is also discussed.

Motivation & Objective

• To develop a consistent definition of angles and trigonometric functions in taxicab geometry that mirrors Euclidean trigonometry.
• To analyze which triangle congruence relations hold under the taxicab metric, given the non-Euclidean nature of the geometry.
• To demonstrate an exact method for determining the distance to a distant object using parallax in taxicab geometry, contrasting with the standard Euclidean approximation.
• To establish a link between taxicab and Euclidean parallax formulas, showing the Euclidean version as an approximation.

Proposed method

• Defining a t-radian as the angle subtending an arc length of 1 on the unit taxicab circle, which has a total circumference of 8 t-radians.
• Deriving a formula for the taxicab measure of a Euclidean angle in standard position: $ \theta = \frac{2\sin_e\phi_e}{\sin_e\phi_e + \cos_e\phi_e} $.
• Extending the formula to non-standard position angles using reference angles and trigonometric identities.
• Using geometric transformations (translations and reflections) to relate angles and distances in parallax measurements.
• Applying the taxicab arc length formula $ \ell = r \cdot \theta $ to derive the exact distance formula $ d = \frac{s}{\beta - \alpha} $, where $ s $ is the baseline and $ \beta - \alpha $ is the taxicab parallax angle.
• Linking the taxicab and Euclidean parallax formulas by substituting Euclidean distances and angles, showing the Euclidean formula as an approximation valid only for small angles.

Experimental results

Research questions

• RQ1 How can a consistent definition of angles and trigonometric functions be established in taxicab geometry, analogous to Euclidean trigonometry?
• RQ2 Which triangle congruence relations remain valid in taxicab geometry, and what strict conditions are required?
• RQ3 Can the parallax method for measuring distance be made exact in taxicab geometry, and how does it compare to the standard Euclidean approach?
• RQ4 What is the relationship between the exact taxicab parallax formula and the commonly used Euclidean approximation?

Key findings

• The taxicab unit circle has a circumference of 8 t-radians, so a full circle is 8 t-radians, and a right angle measures exactly 2 t-radians.
• The taxicab measure of a Euclidean angle depends on its orientation, not just its magnitude, meaning angles are not rotation-invariant in this geometry.
• Congruent triangles in taxicab geometry require all three sides and two angles to be congruent, indicating that standard congruence theorems like SAS or ASA do not hold.
• An exact formula for distance using parallax is derived: $ d = \frac{s}{\beta - \alpha} $, where $ s $ is the baseline and $ \beta - \alpha $ is the taxicab parallax angle, providing an exact result rather than an approximation.
• The standard Euclidean parallax formula $ d_e \approx \frac{s_e}{\beta_e - \alpha_e} $ is shown to be an approximation valid only for small angles, while the taxicab version is exact under the same geometric setup.
• The derivation confirms that moving along lines $ y = x $ or $ y = -x $ preserves distance to a target, enabling exact parallax measurements in taxicab geometry.