Skip to content

uv_docs.md

Latest commit

 

History

History
66 lines (48 loc) · 2.37 KB

uv_docs.md

File metadata and controls

66 lines (48 loc) · 2.37 KB
Raw

TODO: (obselete) Update the documentation

Algorithm: Isometric Triangle Flattening (UV Unwrapping)

This algorithm takes a 3D triangle ($T_2$ defined by points $B, C, D$) and "unfolds" it into a 2D plane (where $y = 0$) relative to an already flattened neighbor triangle ($T_1$ defined by $A, B, C$).

1. Analyze the 3D Geometry (The "Source")

To preserve the true shape of the mesh, we extract the geometric relationships from the 3D coordinates.

  • Define 3D Vectors from vertex $B$:
    • $\vec{V}_{BC} = C - B$
    • $\vec{V}_{BD} = D - B$
  • Calculate $\cos(\beta)$ using the Dot Product: $$\cos(\beta) = \frac{\vec{V}{BC} \cdot \vec{V}{BD}}{|\vec{V}{BC}| \cdot |\vec{V}{BD}|}$$
  • Derive $\sin(\beta)$: Using the Pythagorean identity: $$\sin(\beta) = \sqrt{1 - \cos^2(\beta)}$$

2. Establish the 2D Basis (The "Target")

In the UV plane ($y=0$), we use the shared edge $bc$ to create a local coordinate system. This ensures the triangles stay connected.

  • Edge Length: $L = |c - b|$
  • Local X-axis ($\vec{u}$): $$\vec{u} = \frac{c - b}{L}$$
  • Local Z-axis ($\vec{v}$): Rotate $\vec{u}$ by 90° in the 2D plane to create an orthogonal (perpendicular) axis: $$\vec{v} = (-u_z, u_x)$$

3. Calculate Local 2D Coordinates

We determine the position of $d$ relative to $b$ in our new 2D "grid" using the true 3D distance $d_1 = |\vec{V}_{BD}|$.

  • $x_{local} = d_1 \cdot \cos(\beta)$
  • $z_{local} = d_1 \cdot \sin(\beta)$

4. The Flip Logic (Overlap Prevention)

To ensure the triangles "unfold" like a piece of paper rather than folding back on top of the previous triangle, we use point $a$ as a reference.

  1. Reference Vector: $\vec{w} = a - b$
  2. Alignment Test: Calculate the dot product of $\vec{w}$ and our perpendicular basis vector $\vec{v}$:
    • $Side_A = \vec{w} \cdot \vec{v}$
  3. Final Placement:
    • If $Side_A$ is positive: Point $a$ lies in the direction of $+\vec{v}$. Therefore, we must place $d$ in the opposite direction ($-\vec{v}$): $$d = b + (x_{local} \cdot \vec{u}) - (z_{local} \cdot \vec{v})$$
    • If $Side_A$ is negative: Point $a$ lies in the direction of $-\vec{v}$. Therefore, we place $d$ in the positive direction ($+\vec{v}$): $$d = b + (x_{local} \cdot \vec{u}) + (z_{local} \cdot \vec{v})$$