diff --git a/.obsidian/plugins/obsidian-latex-suite/data.json b/.obsidian/plugins/obsidian-latex-suite/data.json index f06433e..7588b9b 100644 --- a/.obsidian/plugins/obsidian-latex-suite/data.json +++ b/.obsidian/plugins/obsidian-latex-suite/data.json @@ -13,7 +13,7 @@ "loadSnippetVariablesFromFile": false, "snippetsFileLocation": "", "snippetVariablesFileLocation": "", - "concealEnabled": false, + "concealEnabled": true, "concealRevealTimeout": 300, "colorPairedBracketsEnabled": true, "highlightCursorBracketsEnabled": true, diff --git a/burgstaller_2009_average-distance.md b/burgstaller_2009_average-distance.md new file mode 100644 index 0000000..b46260a --- /dev/null +++ b/burgstaller_2009_average-distance.md @@ -0,0 +1,13 @@ +--- +title: The Average Distance Between Two Points +tags: [] +author: B. Burgstaller & F. Pillichshammer +doi: 10.1017/S0004972709000707 +journal: Bulletin of the Australian Mathematical Society +month: 10 +number: 3 +pages: 353--359 +volume: 80 +year: 2009 +--- +# The Average Distance Between Two Points diff --git a/mathai_1999_random-points.md b/mathai_1999_random-points.md new file mode 100644 index 0000000..46796a2 --- /dev/null +++ b/mathai_1999_random-points.md @@ -0,0 +1,12 @@ +--- +title: Random Points Associated with Rectangles +tags: [] +author: A. M. Mathai & P. Moschopoulos & G. Pederzoli +doi: 10.1007/BF02844387 +journal: Rendiconti Del Circolo Matematico Di Palermo +number: 1 +pages: 163--90 +volume: 48 +year: 1999 +--- +# Random Points Associated with Rectangles diff --git a/taxicab-geometry.md b/taxicab-geometry.md new file mode 100644 index 0000000..d01af7c --- /dev/null +++ b/taxicab-geometry.md @@ -0,0 +1,7 @@ +--- +title: Taxicab Geometry +tags: [] +--- +# Taxicab Geometry + +[Wikipedia](https://en.wikipedia.org/wiki/Taxicab_geometry) diff --git a/timestamped/2026-05-31_10-31-14.md b/timestamped/2026-05-31_10-31-14.md new file mode 100644 index 0000000..ecb42bd --- /dev/null +++ b/timestamped/2026-05-31_10-31-14.md @@ -0,0 +1,169 @@ +--- +id: 2026-05-31T10:31:14-0400 +title: 2026-05-31 10:31:14 +tags: [] +daily: "[[2026-05-31]]" +--- +# 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 [^2], +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](https://www.reddit.com/r/askmath/comments/dyxb4i/average_distance_between_two_random_points_in_a/): + +> 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](https://en.wikipedia.org/wiki/Space_(mathematics)). + Change references to "measurement in spaces of _X_ geometry" + to "measurement under _X_ geometry**.[^1] + +[^1]: 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. + +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 $d=\sqrt{L_w^2+L_h^2}$.[^2] + +[^2]: [[burgstaller_2009_average-distance]], [[mathai_1999_random-points]]