Add: Useful Formulae, with MathJax formatting

docs/assets/javascript/mathjax.js

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+window.MathJax = {
+  tex: {
+    inlineMath: [["\\(", "\\)"]],
+    displayMath: [["\\[", "\\]"]],
+    processEscapes: true,
+    processEnvironments: true,
+  },
+  options: {
+    ignoreHtmlClass: ".*|",
+    processHtmlClass: "arithmatex",
+  },
+};
+
+document$.subscribe(() => {
+  MathJax.startup.output.clearCache();
+  MathJax.typesetClear();
+  MathJax.texReset();
+  MathJax.typesetPromise();
+});

docs/guides/useful-formulae.md

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+# Useful Formulae
+
+## Ordinary Objects
+An ordinary object is any object that does not fall withing any of the other categories described below.
+
+Let the 3D vectors $p_d$ and $p_s$ represent the object’s position and the warp’s origin, respectively; and $\vec{v}$ the directional vector from $p_s$ to $p_d$. Let $r$ be the object’s radius.
+
+The object’s warp-in point is the vector $p_s + \vec{v} - r\hat{v}$.
+
+## Large Objects
+A large object is any celestial body whose radius exceeds 90 kilometres (180 kilometres in diameter), except planets.
+
+Let $x$, $y$, and $z$ represent the object’s coordinates. Let $r$ be the object’s radius.
+
+The object’s warp-in point is the vector $\left(x + (r + 5000000)\cos{r} \\, y + 1.3r - 7500 \\, z - (r + 5000000)\sin{r} \\ \right).$
+
+## Planets
+
+The warp-in point of a planet is determined by the planet's ID, its location, and radius.
+
+Let $x$, $y$, and $z$ represent the planet's coordinates. Let $r$ be the planet's radius.
+
+The planet's warp-in point is the vector $\left(x + d \sin{\theta}, y + \frac{1}{2} r \sin{j}, z - d \cos{\theta}\right)$
+where:
+
+$$
+d = r(s + 1) + 1000000
+$$
+
+$$
+\theta = \sin^{-1}\left(\frac{x}{|x|} \cdot \frac{z}{\sqrt{x^2 + z^2}}\right) + j
+$$
+
+$$
+s|_{0.5 \leq s \leq 10.5} = 20\left(\frac{1}{40}\left(10\log_{10}\left(\frac{r}{10^6}\right) - 39\right)\right)^{20} + \frac{1}{2}
+$$
+
+Now, $j$ is a special snowflake. Its value is the Python equivalent of `(random.Random(planetID).random() - 1.0) / 3.0`.
+
+<h3>Example</h3>
+
+--8<-- "snippets/formulae/warp-in.md"
+
+## Skillpoints needed per level
+
+The skillpoints needed for a level depend on the skill rank.
+
+$y_{skillpoints} = 2^{2.5(x_{skilllevel}-1)} \cdot 250 \cdot r_{skillrank}$
+
+<h3>Skillpoints for common ranks</h3>
+
+| Rank   | Level 1 | Level 2 | Level 3 | Level 4 | Level 5   |
+|--------|---------|---------|---------|---------|-----------|
+| **1**  | 250     | 1.414   | 8000    | 45,254  | 256,000   |
+| **2**  | 500     | 2,828   | 16,000  | 90,509  | 512,000   |
+| **3**  | 750     | 4,242   | 24,000  | 135,764 | 768,000   |
+| **4**  | 1000    | 5,656   | 32,000  | 181,019 | 1,024,000 |
+| **5**  | 1250    | 7,071   | 40,000  | 226,274 | 1,280,000 |
+| **6**  | 1500    | 8,485   | 48,000  | 271,529 | 1,536,000 |
+| **7**  | 1750    | 9,899   | 56,000  | 316,784 | 1,792,000 |
+| **8**  | 2000    | 11,313  | 64,000  | 362,039 | 2,048,000 |
+| **9**  | 2250    | 12,727  | 72,000  | 407,293 | 2,304,000 |
+| **10** | 2500    | 14,142  | 80,000  | 452,548 | 2,560,000 |
+| **11** | 2750    | 15,556  | 88,000  | 497,803 | 2,816,000 |
+| **12** | 3000    | 16,970  | 96,000  | 543,058 | 3,072,000 |
+| **13** | 3250    | 18,384  | 104,000 | 588,312 | 3,328,000 |
+| **14** | 3500    | 19,798  | 112,000 | 633,567 | 3,584,000 |
+| **15** | 3750    | 21,213  | 120,000 | 678,822 | 3,840,000 |
+| **16** | 4000    | 22,627  | 128,000 | 724,077 | 4,096,000 |
+
+## Skillpoints per minute
+
+
+The skillpoints generated each minute depend on the primary $(a_{primary})$ and secondary attribute $(a_{secondary})$ of the skill.
+
+$$
+y_{skillpointsPerMinute} = a_{primary} + {a_{secondary} \over 2}
+$$
+
+## Target lock time
+
+The target lock time ($t_{targetlock}$) in seconds depends on the ship's scan resolution ($s$) and the target's signature radius ($r$)
+
+$$
+t_{targetlock} = {40000 \over s \cdot asinh(r)^2}
+$$
+
+## Alignment time
+
+The ship alignment time ($t_{align}$) depends on the ship's inertia modifier ($i$) and the ships mass ($m$)
+
+$$
+t_{align} = { ln(2) \cdot i \cdot m \over 500000 }
+$$

mkdocs.yml

@@ -9,7 +9,7 @@ hooks:
   - scripts/generate-snippets.py
 
 nav:
-  - "Home": 
+  - Home: 
     - Welcome: index.md
     - Resources: resources/index.md
     - Support: support.md
@@ -19,7 +19,8 @@ nav:
     - "Image Server": services/image-server/index.md
     - "Static Data Export": services/sde/index.md
     - "Single-Sign-On": services/sso/index.md
-  - "Guides and Examples": guides/index.md
+  - Guides and Examples:
+    - Useful Formulae: guides/useful-formulae.md
   - Community: community/community-services.md
 not_in_nav: |
   community/*/index.md
@@ -59,6 +60,8 @@ extra_css:
 
 extra_javascript:
   - assets/javascript/theme.js
+  - assets/javascript/mathjax.js
+  - https://cdnjs.cloudflare.com/ajax/libs/mathjax/3.2.2/es5/tex-mml-chtml.min.js
 
 extra:
   homepage: !ENV [SITE_HOME, 'https://developers.eveonline.com/']
@@ -142,7 +145,8 @@ plugins:
   - search
   - social
   - tags
-  - privacy
+  - privacy:
+      assets: false
   - macros:
       on_undefined: keep
 
@@ -151,6 +155,8 @@ markdown_extensions:
   - admonition
   - attr_list
   - md_in_html
+  - pymdownx.arithmatex:
+      generic: true
   - pymdownx.emoji:
       emoji_index: !!python/name:material.extensions.emoji.twemoji
       emoji_generator: !!python/name:material.extensions.emoji.to_svg

snippets/formulae/warp-in.py

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+import math
+import random
+
+
+def warpin(id, x, y, z, r):
+    j = (random.Random(id).random() - 1.0) / 3.0
+    t = math.asin(x / abs(x) * (z / math.sqrt(x**2 + z**2))) + j
+    s = 20.0 * (1.0 / 40.0 * (10 * math.log10(r / 10**6) - 39)) ** 20.0 + 1.0 / 2.0
+    s = max(0.5, min(s, 10.5))
+    d = r * (s + 1) + 1000000
+
+    return (x + d * math.sin(t), y + 1.0 / 2.0 * r * math.sin(j), z - d * math.cos(t))