On the group of isometries of the generalized taxicab plane

2019

Turk J Math

In this paper, we first determine the generalized taxicab distance formulae between a point and a line and two parallel lines in the real plane, then we determine the generalized taxicab distance formulae between a point and a plane, two parallel planes, a point and a line, two parallel lines and two skew lines in three dimensional space, giving also the relations between these formulae and their well-known Euclidean analogs. Finally, we give the generalized taxicab distance formulae between a point and a plane, a point and a line and two skew lines in n-dimensional space, by generalizing the concepts used for three dimensional space to n-dimensional space.

Section: Introductionmentioning

confidence: 99%

On the distance formulae in the generalized taxicab geometry

2019

Turk J Math

“…During the recent years, metric geometries based on these metrics have been studied and developed in many directions. See [1], [2], [4], [5], [6], [7], [9], [10], [11], [12], [16], and [17] for some of studies.…”

Section: Introductionmentioning

confidence: 99%

On generalized taxicab metric in three dimensional space

Communications Faculty of Science University of Ankara Series A1Mathematics and Statistics

2019

In this paper, we de…ne the generalized taxicab distance function in three dimensional space, which includes the taxicab distance function as a special case, and we show that three dimensional generalized taxicab distance function determines a metric. Then we give some properties of three dimensional generalized taxicab metric, and determine Euclidean isometries of the space preserving the generalized taxicab metric.

“…During the recent years, metric geometries based on these metrics have been studied and developed in many directions. See [1], [2], [3], [4], [5], [6], [7], [12], [13], and [14] for some of studies. Let P 1 = (x 1 , y 1 ) and P 2 = (x 2 , y 2 ) be two points in R 2 .…”

Section: Introductionmentioning

confidence: 99%

The Generalized Taxicab Group

International Electronic Journal of Geometry

2018

In this study, we determine the generalized taxicab group consisting all isometries of the real plane endowed with the generalized taxicab metric. First we develop natural analogues of Euclidean reflection and rotation notions, and then determine all isometries in the generalized taxicab plane. Finally, we show that the generalized taxicab group is semidirect product of the translation group and the generalized taxicab symmetry group of the unit generalized taxicab circle, as Euclidean group. We also see that there are transformations of the real plane onto itself which preserve the generalized taxicab distance, but not preserve the Euclidean distance.