Update README.md

README.md

@@ -31,7 +31,7 @@ This document explains how the Python pipeline creates the **500 aircraft-style
   - 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 $\alpha$**.
+  - 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.
 
 ---
@@ -40,15 +40,15 @@ This document explains how the Python pipeline creates the **500 aircraft-style
 
 We generate **500** wings. Airfoils are cycled round-robin (≈50 wings per foil). For each wing we sample:
 
-- **Half-span** $s$ (inches):  
-  $s \sim \mathcal{U}[60,\;120]$.
+- **Half-span** \\(s\\) (inches):  
+  \\(s \sim \mathcal{U}[60,\;120]\\).
 - **Root chord** \\(c_{root}\\) (inches):  
-  $c_{\text{root}} \sim \mathcal{U}[18,\;36]$.
-- **Taper ratio** $\lambda$:  
-  $\lambda \sim \mathcal{U}[0.25,\;0.50]$, so $c_{\text{tip}}=\lambda\,c_{\text{root}}$.
+  \\(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 **$0^{\circ}$** so the wing “hinges” at the root plane (legacy convention).
+  \\(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^{\circ}\\)** so the wing “hinges” at the root plane (legacy convention).
 
 We use **20 stations** (indices or 'slices') along the half-span:
 
@@ -92,23 +92,23 @@ 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:
+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).
+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** $x_{\text{pivot}}=0.25$:
+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 $\theta_j=\text{Twi}[j]\cdot\pi/180$:
+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\\):
 
 $$
 \begin{bmatrix} Y \\ Z \end{bmatrix}
@@ -116,15 +116,15 @@ $$
 \begin{bmatrix} X_s \\ Y_s \end{bmatrix}.
 $$
 
-4. Spanwise coordinate for the whole perimeter at station $j$: $S=y_j$ (inches).
+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.
+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.
+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):
 
@@ -134,7 +134,7 @@ S_{\tfrac{1}{2}} = \int_0^s c(y)\,dy \;\approx\; \operatorname{trapz}(\text{Dis}
 S_{\text{full}} = 2\,S_{\tfrac{1}{2}}.
 $$
 
-Convert: $S_{\text{full (m}^2)} = S_{\text{full}}\cdot(0.0254)^2$.
+Convert: \\(S_{\text{full (m}^2)} = S_{\text{full}}\cdot(0.0254)^2\\).
 
 - **Mean Aerodynamic Chord** (full wing):
 
@@ -145,17 +145,17 @@ $$
 
 We integrate on the half-span (inches), then convert MAC to meters.
 
-- **Aspect Ratio** (full span $b=2s\cdot 0.0254$ 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,\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_\text{deg} + b
@@ -183,10 +183,10 @@ If a polar is not found, these fields are **NaN**; geometry is still fully popul
 
 ## 8) Image Preview Column
 
-Each wing includes a 3D wireframe PNG (`render_png`) created from the lofted $S,Y,Z$ arrays:
+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 $35^{\circ}$),
+- 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.
@@ -196,7 +196,7 @@ 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.
+- **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.
 
@@ -204,10 +204,10 @@ This renders directly in the Hugging Face Dataset viewer.
 
 ## 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 $\text{Twi}[1]=0^{\circ}$.
+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:
+Mid-span chord \\(y=50\\) in:
 
 $$
 c_{\text{trap}}(50) = 30 + (10.5-30)\cdot 0.5 = 20.25\text{ in},\quad
@@ -218,22 +218,22 @@ $$
 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:
+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.
+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.
+- **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.
 
 ---