2026-05-31 10:31:14

Taxicab Geometry Proofs

In 2026-03-31_14-53-42 I mentioned proofs that I had written on taxicab-geometry for use in statistical-modeling-for-construction-estimating, but that I had not yet added to this-notebook. I was hoping to add these with their history, unfortunately they predate my understanding of git. Last modified dates suggest I started work prior to 2024-08.

Prior Work

In attempting to find more detailed sources for ¹, I've realized that there may be more to read before I'm confident in my math.

For this reason (but mostly because I've become bored) I've stopped early in moving over the notes. See #Converting the Old Notes for work before I gave up.

Multiple Integral Approach

I'm most interested in the double/quadruple integral approaches given in some of the articles I've seen


\int_{0}^{1}\int_{-1}^{0}
\int_{-1/2}^{1/2}\int_{-1/2}^{1/2}
\sqrt{ (x_{1}-x_{2})^{2} + (y_{1}-y_{2})^{2} }\
dx_1\ dx_2\ dy_1\ dy_2

which gives the average distance between points in two adjacent unit squares.

The use of multiple integrals is not intuitive. The following explanation is adapted from a deleted user's comment on this r/askmath post:

Basically, the usual formula for the average of finitely many values is (x_1+\dots+x_n)/n, right? If you think about it, x_1+\dots+x_n is the area under the graph of the function given by f(x)=x_i if x is between i-1 and i, where the interval we're taking the area over is [0,n] (try and draw this if it's confusing). Then to get the average we take that area and divide it by the length of the interval we're taking the area over.

This generalizes to finding averages of general functions. Specifically, the average of f over the interval [a,b] is the integral of f on [a,b] divided by b-a. Similarly, if we have a function of 2 variables, the average value of f over some domain A is the integral of f over A divided by the area of A. In the case of your video, A is the unit square which has area 1. Hence the average of the distance function in the unit square is just the integral of f over the unit square.

Converting the Old Notes

To preserve the files as they were would be a nightmare. They can still be found at ZaneMeyers/old-notes in ./risk-oriented-estimating/03_geometric-estimating.

convert the original asciidoc to markdown
re-wrap the prose using semantic-line-breaks
fix glaring typos
remove references to spaces. Change references to "measurement in spaces of X geometry" to "measurement under X geometry**.²

Fair warning for the faint of heart: I have been told that my writing is overformal to the point of being opaque. That used to be worse, and so did the rest of it.

Geometric Estimating

Introduction

This chapter details heuristics that can be used for project measurements, primarily those of lengths for feeders and branch wiring, in lieu of the manual alternative.

The purpose of this chapter is primarily to prove the legitimacy of such methods for use in formulas and analysis.

Introduction to Taxicab Geometry

In electrical installations, shortest-path distance is an uncommon measurement. Conduit and cables are installed parallel to building lines, Luminaires are placed in grids, and devices are installed along walls. Were an estimator to measure the distance between an outlet and the panel that feeds it as the crow flies (the Euclidean distance) they could be up to almost 30% short.

Estimators compensate for this by measuring distance as components will be installed, however when calculating averages, common formulas used in other fields are incompatible.

Euclidean geometry is the kind taught in elementary school, and the kind used in the vast majority of cases, since it appears to be accurate to our universe.

In Euclidean geometry, the minimum distance between two points is defined by the Pythagorean theorem, a^2+b^2=c^2.

Taxicab geometry, also called Manhattan geometry, is geometry where the minimum distance between two points is defined to be the sum of the absolute differences of their respective Cartesian (x and y) coordinates.

In taxicab geometry, the equivalent of the Pythagorean theorem is a+b=c.

Taxicab distance can be thought of as the route a car (or taxicab) would have to take to travel through a grid-planned city (like Manhattan), or the way a rook must move on a chess board.

Where the Euclidean distance between two points can be measured given only their positions, under taxicab geometry we need the additional information of the absolute axes to measure along (the grid).

For the purposes of estimation, we can define the grid by the lines of the building structure, the column lines. This description will be familiar to estimators as it is how most length measurements are taken.

Average Distance Between Two Random Points In A Region

Euclidean Distance

To illustrate the ease of use of taxicab geometry, the equivalent formula for Euclidean space is given by


\frac1{15} \left( \frac{L_w^3}{L_h^2} + \frac{L_h^3}{L_w^2} +d\left( 3 - \frac{L_w^2}{L_h^2} - \frac{L_h^2}{L_w^2} \right) + \frac{5}{2} \left( \frac{L_h^2}{L_w}\log\frac{L_w+d}{L_h} + \frac{L_w^2}{L_h}\log\frac{L_h+d}{L_w} \right) \right),

where d=\sqrt{L_w^2+L_h^2}.¹

burgstaller_2009_average-distance, mathai_1999_random-points ↩︎
I was trying not to get involved to this degree, but I couldn't let this point go. Removing spaces makes this subject much more approachable The universe may be flat, but on earth we treat it as spherical when it makes sense. Despite the fact they intersect, lines of longitude are considered straight and parallel, because we view them under spherical geometry. Buildings aren't L^{1} space, but it is some times useful to measure length as if they are. ↩︎

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