As Rigid As Possible (ARAP)

• The Laplacian formulation tries to preserve the surface orientation.
• • While we could expect surface rotation

• Idea: Include rotation transformation \(\color{red}{\mathrm{R}}\) in the formulation

• • \(\displaystyle E = \sum_{i=0}^{N-1} \|\delta(q_i)- {\color{red}{R(q_i)}} \,\delta(p_i)\|^2 + \omega\,\sum_{i\in C} \|q_i-c_i\|^2\) \(\;\;\)
  • \(R(q_i)\): best rotation at point \(q_i\)

• Finding the best rotation at a given point
• • 1- Look at surrounding edges at the point \(q_i\)/\(p_i\)
  • • \(e_j=q_j-q_i\), \(e_j^0=p_j-p_i\)
  • 2- Compute covariance matrix \( \sigma = \sum_j e_j\,(e_j^0)^T \)
  • 3- Compute rotation using polar decomposition: \(R=polar(\sigma)\)