Upload wing_generation_explained_katex.md

wing_generation_explained_katex.md

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+# How the 500 Wing Samples Are Generated
+
+This document explains how the Python pipeline creates the **500 aircraft‑style wings** in your dataset—what varies, what’s fixed, and the exact math behind the values stored.
+
+---
+
+## 1) Inputs & Reproducibility
+
+- **Airfoil geometry**: Each airfoil file contains two columns \((x, y)\) forming a closed perimeter ordered **TE → upper → LE → lower → TE**.
+  - On load, \(x\) is normalized to \([0,1]\); the perimeter is rotated so the first point is closest to the **trailing edge** \((1,0)\) and the **upper surface** comes first.
+- **Polars (optional)**: QBlade/XFOIL exports with \(\alpha, C_l, C_d, C_m\).
+  - Files are matched to airfoils by flexible filename heuristics.
+  - Polars are parsed (3–4 numeric columns), **deduplicated in \(\alpha\)**, and **sorted by \(\alpha\)**.
+- **Random seed**: We use `np.random.default_rng(42)` so all 500 wings are reproducible bit‑for‑bit.
+
+---
+
+## 2) Per‑Wing Planform Sampling (What Varies)
+
+We generate **500** wings. Airfoils are cycled round‑robin (≈50 wings per foil). For each wing we sample:
+
+- **Half‑span** \(s\) (inches):  
+  \(s \sim \mathcal{U}[60,\;120]\).
+- **Root chord** \(c_{\text{root}}\) (inches):  
+  \(c_{\text{root}} \sim \mathcal{U}[18,\;36]\).
+- **Taper ratio** \(\lambda\):  
+  \(\lambda \sim \mathcal{U}[0.25,\;0.50]\), so \(c_{\text{tip}}=\lambda\,c_{\text{root}}\).
+- **Twist endpoints (washout)** (degrees):  
+  \(i_{\text{root}} \sim \mathcal{U}[0,\;2]\), \(i_{\text{tip}} \sim \mathcal{U}[-6,\;-2]\).
+  - The final twist distribution is **linear** from root to tip; then we **pin** the very first station to **0°** so the wing “hinges” at the root plane (legacy convention).
+
+We use **20 stations** along the half‑span:
+
+$$
+\text{Dis}[j] = y_j = \frac{j-1}{19}\; s,\quad j=1..20.
+$$
+
+---
+
+## 3) Chord Distribution: Schrenk’s Approximation
+
+We blend a trapezoid with an ellipse to approximate an elliptical lift distribution:
+
+- Linear (trapezoid) chord:
+
+$$
+c_{\text{trap}}(y) = c_{\text{root}} + (c_{\text{tip}} - c_{\text{root}})\,\frac{y}{s}.
+$$
+
+- Elliptic surrogate:
+
+$$
+c_{\text{ell}}(y) = c_{\text{root}}\sqrt{1 - \left(\frac{y}{s}\right)^2}.
+$$
+
+- **Schrenk chord** at each station:
+
+$$
+c(y) = \tfrac{1}{2}\left[c_{\text{trap}}(y) + c_{\text{ell}}(y)\right].
+$$
+
+A small clamp prevents pathological tips for extreme tapers:
+
+$$
+c(y) \leftarrow \max\!\big(c(y),\;0.25\cdot\min\nolimits_y c_{\text{trap}}(y)\big).
+$$
+
+This yields the **Cho** vector (inches) over the 20 stations.
+
+---
+
+## 4) Twist (Washout) Distribution
+
+With sampled endpoints \(i_{\text{root}}\) and \(i_{\text{tip}}\), define a **linear** twist:
+
+$$
+\text{Twi}[j] = i_{\text{root}} + \big(i_{\text{tip}} - i_{\text{root}}\big)\frac{y_j}{s},\quad j=1..20.
+$$
+
+Then set \(\text{Twi}[1]=0^\circ\) (root plane hinge).
+
+---
+
+## 5) Lofting the 3D Wing for Previews
+
+Given the normalized perimeter \((\bar{x},\bar{y})\) (TE→upper→LE→lower→TE), we scale and twist each section about the **quarter‑chord** \(x_{\text{pivot}}=0.25\):
+
+1. Shift section to pivot origin: \(x_c = \bar{x} - 0.25\).
+2. Scale to local chord \(c_j\) (inches): \(X_s = x_c\,c_j,\; Y_s = \bar{y}\,c_j\).
+3. Rotate by \(\theta_j=\text{Twi}[j]\cdot\pi/180\):
+
+$$
+\begin{bmatrix} Y \\ Z \end{bmatrix}
+= \begin{bmatrix} \cos\theta_j & -\sin\theta_j \\ \sin\theta_j & \cos\theta_j \end{bmatrix}
+\begin{bmatrix} X_s \\ Y_s \end{bmatrix}.
+$$
+
+4. Spanwise coordinate for the whole perimeter at station \(j\): \(S=y_j\) (inches).
+
+The arrays \(S, Y, Z\) generate a fast **wireframe 3D PNG** used as the dataset’s `render_png` field.
+
+---
+
+## 6) Planform Integrals & Derived Metrics
+
+Treat `Dis`/`Cho` as samples over the **half‑span** \([0,s]\) in **inches**. We integrate with the trapezoidal rule, then convert to SI for storage.
+
+- **Wing area** (full wing):
+
+$$
+S_{\tfrac{1}{2}} = \int_0^s c(y)\,dy \;\approx\; \operatorname{trapz}(\text{Dis},\text{Cho})\quad [\text{in}^2],
+\qquad
+S_{\text{full}} = 2\,S_{\tfrac{1}{2}}.
+$$
+
+Convert: \(S_{\text{full (m}^2)} = S_{\text{full}}\cdot(0.0254)^2\).
+
+- **Mean Aerodynamic Chord** (full wing):
+
+$$
+\text{MAC} = \frac{2}{S_{\text{full}}} \int_{-s}^{s} c(y)^2\,dy
+= \frac{4}{S_{\text{full}}} \int_0^s c(y)^2\,dy.
+$$
+
+We integrate on the half‑span (inches), then convert MAC to meters.
+
+- **Aspect Ratio** (full span \(b=2s\cdot 0.0254\) meters):
+
+$$
+\text{AR} = \frac{b^2}{S_{\text{full (m}^2)}}.
+$$
+
+- **Polar‑derived metrics** (if a polar is found):
+  - \(C_{l,\max}=\max C_l\) at \(\alpha_{C_{l,\max}}\).
+  - \(C_{d,\min}=\min C_d\) at \(\alpha_{C_{d,\min}}\).
+  - \((L/D)_{\max} = \max(C_l/C_d)\) at \(\alpha_{(L/D)_{\max}}\).
+  - Small‑angle lift slope and zero‑lift angle (linear fit on \(\alpha\in[-5^\circ,5^\circ]\)):
+
+$$
+C_l \approx m\,\alpha_{\deg} + b
+\;\Rightarrow\;
+C_{l_\alpha}\;[\text{per rad}] = m\cdot \frac{180}{\pi},
+\quad
+\alpha_{0L}\;[^\circ] = -\frac{b}{m}.
+$$
+
+  - Also stored as objective‑style **scores**:
+    - `score_min_cd = min(C_d)`
+    - `score_max_cl = max(C_l)`
+    - `score_max_ld = max(C_l/C_d)`
+
+If a polar is not found, these fields are **NaN**; geometry is still fully populated.
+
+---
+
+## 7) Station Count & Units
+
+- **Stations**: always **20** over the half‑span (keeps compatibility with legacy builders).
+- **Units**: dataset stores **SI** (`dis_m`, `chord_m`, `span_m`, `area_m2`, `mac_m`). Inches are used internally during generation for readability and then converted.
+
+---
+
+## 8) Image Preview Column
+
+Each wing includes a 3D wireframe PNG (`render_png`) created from the lofted \(S,Y,Z\) arrays:
+- Section loops at every station,
+- ~14 spanwise polylines to suggest the surface,
+- Isometric view (elev \(20^\circ\), azim \(35^\circ\)),
+- Title string with span, root/tip chords, and taper.
+
+This renders directly in the Hugging Face Dataset viewer.
+
+---
+
+## 9) Why These Look Like Transport Wings
+
+- **Aspect ratio** typically in the 7–11 range (after area settles) due to sampled spans/cords.
+- **Taper** \(0.25\to 0.50\) and **washout** \(-6^\circ\to -2^\circ\) are characteristic of transport wings aimed at cruise efficiency and benign stall.
+- **Quarter‑chord pivot** is the standard torsion axis.
+- **Schrenk chord** smooths the planform compared to pure linear taper, approximating more elliptical loading.
+
+---
+
+## 10) Worked Example (Representative Draw)
+
+Let \(s=100\) in, \(c_{\text{root}}=30\) in, \(\lambda=0.35 \Rightarrow c_{\text{tip}}=10.5\) in, stations=20.  
+Twist endpoints: \(i_{\text{root}}=1.0^\circ\), \(i_{\text{tip}}=-4.0^\circ\), then \(\text{Twi}[1]=0^\circ\).
+
+Mid‑span chord \(y=50\) in:
+
+$$
+c_{\text{trap}}(50) = 30 + (10.5-30)\cdot 0.5 = 20.25\text{ in},\quad
+c_{\text{ell}}(50) = 30\sqrt{1-0.5^2} \approx 25.98\text{ in},
+$$
+
+$$
+c(50) \approx \tfrac{1}{2}(20.25+25.98) \approx 23.11\text{ in}.
+$$
+
+If \(S_{\text{full}} \approx 5000\;\text{in}^2 \Rightarrow S_{\text{full}} \approx 3.226\;\text{m}^2\) and full span \(b=200\) in \(=5.08\) m, then:
+
+$$
+\text{AR} = \frac{b^2}{S} \approx \frac{(5.08)^2}{3.226} \approx 8.0.
+$$
+
+MAC comes from the \(c(y)^2\) integral and is often in the \(0.3\text{–}0.5\) m range here.
+
+---
+
+## 11) What Makes Each of the 500 Unique?
+
+- **Airfoil choice** (≈50 samples per foil) → geometry & polar behavior differ.
+- **Planform**: each wing draws a new \((s,\;c_{\text{root}},\;\lambda)\) → different area, AR, MAC.
+- **Twist**: each wing draws \((i_{\text{root}}, i_{\text{tip}})\) → different load tendency.
+- **Performance summaries**: objective‑style scores (\(\min C_d\), \(\max C_l\), \(\max C_l/C_d\)) and the \(\alpha\) at which they occur differ per wing.
+
+---
+
+### Notes for Objective‑Conditioned Training
+
+To bias toward a given objective at training time:
+- **Condition** on `objective ∈ {min Cd, max Cl, max Cl/Cd}` and **airfoil** (name or perimeter).
+- Use the scalar **scores** as targets (e.g., regress `score_min_cd`) or form **ranking** pairs within the same airfoil.
+- Optionally **post‑select** the top‑k wings per airfoil by the chosen objective as exemplar targets.