PARABOLAS IN GENERALIZED TAXICAB PLANE
Abstract
This study investigates parabolas in the generalized taxicab plane, a non-Euclidean geometry where distance is measured using weighted coordinate axes with positive parameters (a,b). Using the focus-directrix definition, it examines the structures of generalized taxicab parabolas (briefly, GTPs) with respect to the positions of their directrices. It is determined that generalized taxicab parabolas are simple rectilinear figures. It further provides a detailed analysis of GTPs, including their axes, vertices, latus rectum, and focal lengths. It reveals that the latus rectum length of a GTP is four times its focal length regardless of the directrix type. Also, the algorithm is presented to visualize GTPs for all types of directrices. Additionally, the study identifies degenerate cases in which the focus is on the directrix, and it is demonstrated that the obtained geometric structures reduce to single lines or unions of planar regions defined by vertical and horizontal lines through the focus.
Keywords
Generalized taxicab distance, Generalized taxicab parabola, Focus-directrix conics, Degenerate conics, non-Euclidean geometry
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