vault backup: 2026-05-31 15:24:48
This commit is contained in:
@@ -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]]
|
||||
Reference in New Issue
Block a user