Topic

# Taxicab geometry

About: Taxicab geometry is a research topic. Over the lifetime, 169 publications have been published within this topic receiving 4055 citations. The topic is also known as: Manhattan distance & Manhattan length.

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01 Sep 1984-Graphical Models \/graphical Models and Image Processing \/computer Vision, Graphics, and Image Processing

TL;DR: The purpose of this paper is to generalize these distance transformation families to higher dimensions and to compare the computed distances with the Euclidean distance.

Abstract: In many applications of digital picture processing, distances from certain feature elements to the nonfeature elements must be computed. In two dimensions at least four different families of distance transformations have been suggested, the most popular one being the city block/chessboard distance family. The purpose of this paper is twofold: To generalize these transformations to higher dimensions and to compare the computed distances with the Euclidean distance. All of the four distance transformation families are presented in three dimensions, and the two fastest ones are presented in four and arbitrary dimensions. The comparison with Euclidean distance is given as upper limits for the difference between the Euclidean distance and the computed distances.

870 citations

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TL;DR: In this article, the authors consider Steiner minimal trees in the plane with rectilinear distance and show that the rectilINear distance between two points is at most 2.3.

Abstract: We consider Steiner minimal trees in the plane with rectilinear distance. The rectilinear distance $d(p_1 ,p_2 )$ between two points $p_1 $, $p_2 $ is $| {x_1 - x_2 } | + | {y_1 - y_2 } |$, where the $(x_i ,y_i )$ are the Cartesian coordinates of the $p_i $. For a given finite set P of points, let $l_s $ denote the length of a Steiner minimal tree and $l_m $ the length of a minimal spanning tree. The main result of the memorandum is that ${l_s / l_m } \geqq \frac{2}{3}$.

413 citations

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360 citations

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TL;DR: An efficient algorithm for finding minimal distance feasible paths between the points, assuming that all travel occurs according to the rectilinear distance metric is developed.

Abstract: Given a set of origin-destination points in the plane and a set of polygonal barriers to travel, this paper develops an efficient algorithm for finding minimal distance feasible paths between the points, assuming that all travel occurs according to the rectilinear distance metric. By geometrical arguments the problem is reduced to a finite network problem. The nodes are the origin-destination points and the barrier vertices. The links designate those node pairs that ”communicate” in a simple way, where communication implies the existence of a node-to-node rectilinear path that is not made longer by the barriers. The weight of each link is the rectilinear distance between its two corresponding nodes. Solution of the minimal distance path problem on the network procedes in two steps. First, for a given origin or root node, a tree is generated containing a minimal distance path to each node that communicates with the root node. Second, a modified Diikstra-type iteration is utilized, starting with the nodes of the tree, sequentially adding nodes according to minimum “penalty distance,” where the penalty is the extra travel distance caused by the barriers. The paper concludes with a discussion of the computational complexity of the procedure, followed by a numerical example.

132 citations