Asrigidaspossible

asrigidaspossible is a concept used in computer graphics and geometric modeling to describe a deformation principle that aims to keep local regions of a shape as rigid as possible during a global transformation. In practice, it is commonly realized through the As-Rigid-As-Possible (ARAP) deformation framework, a widely used approach for mesh editing and shape manipulation.

The core idea behind asrigidaspossible is that, during deformation, each point should move as if its immediate neighborhood undergoes a rigid transformation. This minimizes changing local geometry and preserves features such as angles and edge lengths, while still allowing the overall object to bend, twist, or stretch in a controlled way. The method typically enforces that local transformations are rotations (or near-rotations) and blends them to produce a coherent global deformation.

Mathematically, for a mesh with original vertex positions p_i and deformed positions q_i, the problem is formulated by assigning a rotation R_i to each vertex i and minimizing a sum over adjacent pairs that measures how well the local neighborhood matches R_i applied to the original neighborhood. A common solution alternates between estimating the rotations R_i (a Procrustes alignment step) and solving a sparse linear system to update the vertex positions q_i, iterating until convergence.

Applications of asrigidaspossible include interactive modeling, character animation, shape interpolation, and medical image deformation, where preserving local rigidity leads to more natural and predictable deformations. Variants adapt the framework to different dimensions, weightings, or non-rigid penalties, broadening its use in design and analysis.

See also: As-Rigid-As-Possible, mesh editing, Procrustes analysis, rigidity, optimization.