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id: 2026-05-31T10:31:14-0400
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title: 2026-05-31 10:31:14
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tags: []
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daily: "[[2026-05-31]]"
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---
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# 2026-05-31 10:31:14
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## Taxicab Geometry Proofs
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In [[2026-03-31_14-53-42]]
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I mentioned proofs that I had written on [[taxicab-geometry]]
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for use in [[statistical-modeling-for-construction-estimating]],
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but that I had not yet added to [[this-notebook]].
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I was hoping to add these with their history,
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unfortunately they predate my understanding of [[git]].
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Last modified dates suggest I started work prior to 2024-08.
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## Prior Work
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In attempting to find more detailed sources for [^2],
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I've realized that there may be more to read
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before I'm confident in my math.
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For this reason (but mostly because I've become bored)
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I've stopped early in moving over the notes.
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See [[#Converting the Old Notes]] for work before I gave up.
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### Multiple Integral Approach
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I'm most interested in the double/quadruple integral approaches
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given in some of the articles I've seen
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$$
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\int_{0}^{1}\int_{-1}^{0}
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\int_{-1/2}^{1/2}\int_{-1/2}^{1/2}
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\sqrt{ (x_{1}-x_{2})^{2} + (y_{1}-y_{2})^{2} }\
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dx_1\ dx_2\ dy_1\ dy_2
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$$
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which gives the average distance between points in two adjacent unit squares.
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The use of multiple integrals is not intuitive.
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The following explanation is adapted from a deleted user's comment
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on [this r/askmath post](https://www.reddit.com/r/askmath/comments/dyxb4i/average_distance_between_two_random_points_in_a/):
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> Basically, the usual formula for the average of finitely many values
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> is $(x_1+\dots+x_n)/n$, right?
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> 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 $i$,
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> where the interval we're taking the area over is $[0,n]$
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> (try and draw this if it's confusing).
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> Then to get the average we take that area
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> and divide it by the length of the interval we're taking the area over.
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>
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> This generalizes to finding averages of general functions.
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> Specifically, the average of $f$ over the interval $[a,b]$
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> is the integral of $f$ on $[a,b]$ divided by $b-a$.
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> Similarly, if we have a function of 2 variables,
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> the average value of $f$ over some domain $A$
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> is the integral of $f$ over $A$ divided by the area of $A$.
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> In the case of your video, $A$ is the unit square which has area 1.
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> Hence the average of the distance function in the unit square
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> is just the integral of $f$ over the unit square.
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## Converting the Old Notes
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To preserve the files as they were would be a nightmare.
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They can still be found at `ZaneMeyers/old-notes`
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in `./risk-oriented-estimating/03_geometric-estimating`.
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* convert the original [[asciidoc]] to [[markdown]]
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* re-wrap the prose using [[semantic-line-breaks]]
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* fix glaring typos
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* remove references to
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[spaces](https://en.wikipedia.org/wiki/Space_(mathematics)).
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Change references to "measurement in spaces of _X_ geometry"
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to "measurement under _X_ geometry**.[^1]
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[^1]: I was trying not to get involved to this degree,
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but I couldn't let this point go.
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Removing spaces makes this subject much more approachable
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The universe may be flat,
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but on earth we treat it as spherical when it makes sense.
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Despite the fact they intersect,
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lines of longitude are considered straight and parallel,
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because we view them under spherical geometry.
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Buildings aren't $L^{1}$ space,
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but it is some times useful to measure length as if they are.
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Fair warning for the faint of heart:
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I have been told that my writing is overformal to the point of being opaque.
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That used to be worse, and so did the rest of it.
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## Geometric Estimating
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### Introduction
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This chapter details heuristics
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that can be used for project measurements,
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primarily those of lengths for feeders and branch wiring,
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in lieu of the manual alternative.
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The purpose of this chapter
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is primarily to prove the legitimacy of such methods
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for use in formulas and analysis.
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### Introduction to Taxicab Geometry
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In electrical installations,
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shortest-path distance is an uncommon measurement.
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Conduit and cables are installed parallel to building lines,
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Luminaires are placed in grids,
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and devices are installed along walls.
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Were an estimator to measure the distance
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between an outlet and the panel that feeds it
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as the crow flies (the Euclidean distance)
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they could be up to almost 30% short.
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Estimators compensate for this by measuring distance
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as components will be installed,
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however when calculating averages,
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common formulas used in other fields are incompatible.
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**Euclidean geometry** is the kind taught in elementary school,
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and the kind used in the vast majority of cases,
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since it appears to be accurate to our universe.
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In Euclidean geometry,
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the minimum distance between two points
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is defined by the Pythagorean theorem,
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$a^2+b^2=c^2$.
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**Taxicab geometry**, also called Manhattan geometry,
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is geometry where the minimum distance between two points
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is defined to be the sum of the absolute differences
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of their respective Cartesian (x and y) coordinates.
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In taxicab geometry, the equivalent of the Pythagorean theorem is $a+b=c$.
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Taxicab distance can be thought of as the route a car (or taxicab)
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would have to take to travel through a grid-planned city (like Manhattan),
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or the way a rook must move on a chess board.
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Where the Euclidean distance between two points
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can be measured given only their positions,
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under taxicab geometry we need the additional information
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of the absolute axes to measure along (the **grid**).
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For the purposes of estimation,
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we can define the grid by the lines of the building structure,
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the column lines.
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This description will be familiar to estimators
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as it is how most length measurements are taken.
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#### Average Distance Between Two Random Points In A Region
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##### Euclidean Distance
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To illustrate the ease of use of taxicab geometry,
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the equivalent formula for Euclidean space
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is given by
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$$
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\frac1{15} \left( \frac{L_w^3}{L_h^2} + \frac{L_h^3}{L_w^2} +d\left( 3 - \frac{L_w^2}{L_h^2} - \frac{L_h^2}{L_w^2} \right) + \frac{5}{2} \left( \frac{L_h^2}{L_w}\log\frac{L_w+d}{L_h} + \frac{L_w^2}{L_h}\log\frac{L_h+d}{L_w} \right) \right),
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$$
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where $d=\sqrt{L_w^2+L_h^2}$.[^2]
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[^2]: [[burgstaller_2009_average-distance]], [[mathai_1999_random-points]]
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