Distortion & ARAP Flattening

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As-Rigid-As-Possible (ARAP Surface Flattening): Conformal methods preserve angles at the expense of severe area stretching, while equiareal methods preserve area at the cost of shearing. The ARAP algorithm (Sorkine & Alexa 2007) seeks a balanced, near-isometric unrolling by ensuring that every local 1-ring star rotates as rigidly as possible (pure rotation R in SO(2)) into the 2D plane, alternating between local Singular Value Decomposition (SVD) and a global linear Poisson solve.

ARAP Alternating Minimization:

1. Local SVD Step: Extracts the best-fit pure rotation $R_i = V U^T$ for each triangle/cell. 2. Global Poisson Step: Stitches the rotated local cells together seamlessly by solving a symmetric positive-definite system. 3. Isometric Balance: Minimizes both conformal (shearing) and authalic (area stretch) distortion simultaneously.

python

1def arap_local_global_step(uv_points, mesh_3d, cot_weights):
2    """
3    As-Rigid-As-Possible (ARAP) parameterization iteration (Sorkine & Alexa 2007).
4    Minimizes non-rigid isometric distortion:
5    E_ARAP = sum_{i} sum_{j in N(i)} w_ij * ||(u_i - u_j) - R_i * (p_i - p_j)||^2.
6    
7    1. Local Step: For each cell i, compute optimal 2D rotation R_i via SVD of covariance matrix.
8    2. Global Step: Fix R_i and solve linear Poisson system for updated coordinates u.
9    """
10    return "ARAP Optimized Low-Distortion UV Coordinates"

ARAP Iterations

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2 min read•1 page

Conformal vs Authalic (Equiareal) Distortion Metrics: Gauss proved that a surface with non-zero Gaussian curvature cannot be flattened to a plane without distortion. In map projections and UV unwrapping, artists choose which invariant to prioritize:Conformal projections preserve local shapes and orthogonal angles ($\sigma_1 / \sigma_2 = 1$), whileAuthalic projections preserve relative land areas ($\sigma_1 \sigma_2 = 1$).

Jacobian Singular Value Invariants:

1. Singular Values ($\sigma_1, \sigma_2$): Semi-major and semi-minor axes of Tissot's deformation ellipse. 2. Conformal Condition: $\sigma_1 = \sigma_2$ (circles stay circles, no shear). 3. Authalic Condition: $\sigma_1 \cdot \sigma_2 = 1$ (area of every patch is conserved).

python

1def evaluate_distortion_metrics(jacobian_matrix):
2    """
3    Computes singular values sigma_1 >= sigma_2 of mapping Jacobian J.
4    - Conformal (Angle) Distortion: sigma_1 / sigma_2 >= 1 (1 = zero shear).
5    - Authalic (Area) Distortion: sigma_1 * sigma_2 (1 = zero area stretch).
6    - Isometric (Length) Distortion: (sigma_1 - 1)^2 + (sigma_2 - 1)^2.
7    """
8    U, S, Vt = np.linalg.svd(jacobian_matrix)
9    s1, s2 = S[0], S[1]
10    
11    conformal_err = s1 / s2
12    authalic_err = s1 * s2
13    return conformal_err, authalic_err

0=Conformal (Angle), 1=Authalic (Area)

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