On the Ratio of Directed Lengths in the Taxicab Plane and Related Properties

On The Chinese Checkers Spherical Inversions in Three Dimensional Chinese Checkers Space

“…It is known that the cross ratio is positive if both C and D are between A and B or if neither C nor D is between A and B, whereas the cross ratio is negative if the pairs fA; Bg and fC; Dg separate each other. Also, the cross ratio is an invariant under inversion in a sphere whose center is not any of the four points A; B; C and D in the taxicab plane; [11]. Similarly, this property is valid in Chinese Checkers sphere.…”

“…The Chinese Checkers directed distance from the point A to the point B along a line l in R [11]. The Chinese Checkers cross ratio is preserved by the inversion in the Chinese Checkers circle as in the taxicab plane in [17].…”

On Directed Length Ratios in the Lorentz-Minkowski Plane

“…ii) If 𝑚 = 0 or 𝑚 → ∞, it is clear that 𝑑 𝐸 (𝑃, 𝑄) = 𝑑 𝐿 (𝑃, 𝑄). That is, the ratios of the Euclidean and Lorentz directed lengths are the same [9].…”

On the ratio of directed lengths on the plane with generalized absolute value metric and related properties

“…Since the plane with the generalized absolute value metric has distance function different from that in the Euclidean plane, it is interesting to study on the plane with the generalized absolute value metric of topics that include the distance concept in the Euclidean plane. ( [1], [2], [4], [5], [6], [7], [9], [12], [10], [11], [13], [14], [16], [23], [24], [25], [19], [26], [27], [28], [29]) These topics are division point, directed lengths, ratio of directed lengths, Menelaus'es Theorem, Ceva's Theorem and the theorem of directed lines. In this paper, GAM is the abbreviation for the plane geometry with the generalized absolute value metric.…”