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Kevin's Corner
The first dimensional measure we will examine is the simplest of the measures: the measurement of length. This is also, for line segments at least, the most well understood measure in taxicab geometry.
One-dimensional Length
The simplest measurement of length is the length of a line segment in one dimension. On this point, Euclidean and taxicab geometry are in complete agreement. The length of a line segment from a point A=a to another point B=b is simply the number of unit lengths covered by the line segment,
Two-dimensional Linear Length
In two dimensions, however, the Euclidean and taxicab metrics are not always in agreement on the length of a line segment. For line segments parallel to a coordinate axis, such as the line segment with endpoints A = (x1, y) and B = (x2, y), there is agreement since both metrics reduce to one-dimensional measurement: de(A, B) = dt(A, B) = |x2 - x1|. Only when the line segment is not parallel to one of the coordinate axes do we finally see disagreement between the Euclidean and taxicab metrics. The taxicab length of such a line segment can be viewed as the sum of the Euclidean lengths of the projections of the line segment onto the coordinate axes (Figure 1),
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FIGURE 1: Taxicab length as a sum of Euclidean projections onto the coordinate axes.
The Pythagorean Theorem tells us the Euclidean and taxicab lengths will generally not agree for line segments that are not parallel to one of the coordinate axes. Line segments of the same Euclidean length will have various taxicab lengths as their position relative to the coordinate axes changes. If one were to place a scale on a diagonal line, the Euclidean and taxicab markings would differ with the largest discrepancy being at a 45 degree angle to the coordinate axes (Figure 2).
FIGURE 2: The scale of measurement differs along a diagonal line for Euclidean
geometry (below the line) and taxicab geometry (above the line).
Three-dimensional Linear Length
The taxicab length of a line segment in three dimensions is a natural extension of the formula in two dimensions. For a line segment with endpoints (x1, y1, z1) and (x2, y2, z2),
A nice discussion of the three-dimensional metric and other properties is given in the references. There are no surprises here, and the taxicab length can be viewed as the sum of the Euclidean lengths of the projections of the line segment onto the three coordinate axes.
References
[1] Thompson, Kevin P. The Nature of Length, Area, and Volume in Taxicab Geometry, International Electronic Journal of Geometry, Vol. 4, No. 2 (2011), pp. 193-207.
[2] Ziya Akça and Rüstem Kaya. On the Distance Formulae in Three Dimensional Taxicab Space, Hadronic Journal, Vol. 27, No. 5 (2004), pp. 521-532.
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