lotta text added

.vscode/ltex.dictionary.en-US.txt

@@ -0,0 +1 @@
+DebriSat

.vscode/ltex.hiddenFalsePositives.en-US.txt

@@ -1 +1,3 @@
 {"rule":"UPPERCASE_SENTENCE_START","sentence":"^\\Qmodels that are in the \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q format.\\E$"}
+{"rule":"MORFOLOGIK_RULE_EN_US","sentence":"^\\QWikipedia gives a different definition^[Wikipedia: Characteristic Length] that defines characteristic length as the volume divided by the surface area.\\E$"}
+{"rule":"MORFOLOGIK_RULE_EN_US","sentence":"^\\QCalculation of Characteristic Length, @hillMeasurementSatelliteImpact slide 9\\E$"}

citations.bib

@@ -24,3 +24,19 @@
   url     = {https://grabcad.com/library/6u-cubesat-model-1},
   urldate = {2022-02-15}
 }
+
+@article{hillMeasurementSatelliteImpact,
+  title   = {Measurement of Satellite Impact Test Measurement of Satellite Impact Test Fragments for Modeling Orbital Debris},
+  author  = {Hill, Nicole M.},
+  url     = {https://ntrs.nasa.gov/api/citations/20090019123/downloads/20090019123.pdf},
+  urldate = {2009}
+}
+
+@misc{plane_line,
+  title        = {determine where a vector will intersect a plane},
+  author       = {David Mitra (https://math.stackexchange.com/users/18986/david-mitra)},
+  howpublished = {Mathematics Stack Exchange},
+  note         = {URL:https://math.stackexchange.com/q/100447 (version: 2012-01-19)},
+  eprint       = {https://math.stackexchange.com/q/100447},
+  url          = {https://math.stackexchange.com/q/100447}
+}

report.qmd

@@ -1,8 +1,13 @@
 ---
 title: "Characterization of Space Debris using Machine Learning Methods"
 subtitle: "Advanced processing of 3D meshes using Julia, and data science in Matlab."
-author: Anson Biggs
+author:
+  - name: Anson Biggs
+    url: https://ansonbiggs.com
+
 date: "4/30/2022"
+thanks: Mehran Andalibi
+
 latex-auto-install: true
 format:
   html:
@@ -19,6 +24,7 @@ execute:
 
 filters:
   - filter.lua
+# theme: litera
 ---
 
 ## Introduction
@@ -98,17 +104,66 @@ The algorithm is finding the area of a triangle made up by 3 points in 3D space
 3 vector math. The area of a triangle is $S=\dfrac{|\mathbf{AB}\times\mathbf{AC}|}2$, then the sum
 of all the triangles that make up the mesh should produce the surface area. This can be inaccurate
 if a mesh is missing triangles, has triangles that overlap, or is a mesh that has internal geometry.
+None of which should be an issue for 3D scans.
 
 ### Characteristic Length
 
+This is a property that is heavily used by the DebriSat^[@DebriSat2019] program, so it is definitely
+worth having specifically since DebriSat is a very interesting partner for this project. It is
+unclear if the current implementation is the same value that DebriSat uses. Wikipedia gives a
+different
+definition^[[Wikipedia: Characteristic Length](https://en.wikipedia.org/wiki/Characteristic_length)]
+that defines characteristic length as the volume divided by the surface area. DebriSat mentions it
+in various papers with slightly different definitions.
+
+The current implementation is quite complicated and is the most involved algorithm used. The key
+fact that makes this algorithm work is that the eigenvectors are orthonormal and oriented so that
+they are pointing in the direction of most mass. Assuming the mesh has uniform density, this means
+that the eigenvectors are pointing in the direction of the points furthest from the center of
+gravity.
+
+The algorithm begins by taking the eigenvectors $\lambda$, and making vectors from the center of
+gravity to every point that makes up the mesh, $\bar{P_i}$. Then the angle, $\theta$, between the
+eigenvectors and all the point vectors needs to be calculated as follows:
+
+$$\theta_i = arccos\left(\frac{\lambda \cdot P_i}{\| \lambda \| \| P_i \| }\right)$$
+
+For almost every mesh there won't be a point that has a direct intersection with the eigenvector,
+i.e. $\theta \approx 0$. To deal with this the 3 points with the smallest angle can be turned into a
+plane, then the intersection of the eigenvector and the plane can be calculated^[I won't go into
+depth of this math, but you can find a great explanation here [@plane_line]]. This process then
+needs to be repeated for the largest 3 angles to be able to get two points that a distance can be
+calculated between to be used in the characteristic length. The characteristic length, $L_C$, can
+then be calculated simply by taking the average of the distances found for each eigenvector.
+
+$$L_c = \frac{\mathbf{A} + \mathbf{B} + \mathbf{C}}{3}$$
+
 ### Solid Body Values
 
+The current algorithm just takes all the points that make up the mesh, then finds the `minmax` for
+each axis then averages them like the figure above. No consideration is taken for the orientation of
+the model, but I believe that shouldn't matter. Definitely an algorithm that deserves a revisit
+especially if DebriSat could be contacted for a rigorous definition of the quantity.
+
+![Calculation of Solid Body Values, @hillMeasurementSatelliteImpact [slide 9]](Figures/solid_body.png)
+
 ### `find_scale` and `fast_volume` Functions
 
-```{julia}
+These functions are used together to scale the models for machine learning. When performing machine
-#| echo: false
+learning on a dataset with properties of varing magnitudes it is important to do some sort of
-#| output: false
+scaling. It was determined that scaling by volume was optimal for many reasons, but mainly because
+the algorithm to calculate the volume is one of the fastest. The `fast_volume` function is just a
+modified version of the main volume function that has all the extra calculations required for center
+of gravity and inertia removed. The `find_scale` function uses the `fast_volume` function and a
+derivative root finding algorithm to find a scaling factor that can be used to get a volume of 1
+from a model. This process while iterative works incredibly fast and with the current dataset only
+adds about 20% to the computation time to calculate all the properties for a mesh.
 
+## stl-process Workflow
+
+### Imports
+
+```{julia}
 using FileIO
 using MeshIO
 
@@ -116,15 +171,14 @@ using stlProcess
 
 using CSV
 using DataFrames
-using Plots
-theme(:ggplot2)
 
 using LinearAlgebra
 using Statistics
 ```
 
+### `loop` Setup
+
 ```{julia}
-#| code-fold: true
 #| output: false
 
 # local path to https://gitlab.com/orbital-debris-research/fake-satellite-dataset
@@ -145,6 +199,8 @@ df = DataFrame(;
 )
 ```
 
+### The `loop`
+
 ```{julia}
 #| output: false
 
@@ -167,6 +223,8 @@ for path in dataset_path * "\\" .* folders
 end
 ```
 
+### Output
+
 :::{.column-body-outset}
 
 ```{julia}
@@ -177,23 +235,6 @@ describe(df)
 
 :::
 
-```{julia}
-S = cov(Matrix(df))
-
-eig_vals = eigvals(S);
-
-# sorting eigenvalues from largest to smallest
-sort_index = sortperm(eig_vals; rev=true)
-
-lambda = eig_vals[sort_index]
-names_sorted = names(df)[sort_index]
-
-lambda_ratio = cumsum(lambda) ./ sum(lambda)
-
-plot(lambda_ratio, marker=:x)
-xticks!(sort_index,names(df), xrotation = 15)
-```
-
 ## Gathering Data
 
 To get started on the project before any scans of the actual debris are made available, I opted to