Affine interpolation in a lie group framework

Abstract

Affine transformations are of vital importance in many tasks pertaining to motion design and animation. Interpolation of affine transformations is non-trivial. Typically, the given affine transformation is decomposed into simpler components which are easier to interpolate. This may lead to unintuitive results, while in some cases, such solutions may not work. In this work, we propose an interpolation framework which is based on a Lie group representation of the affine transformation. The Lie group representation decomposes the given transformation into simpler and meaningful components, on which computational tools like the exponential and logarithm maps are available in closed form. Interpolation exists for all affine transformations while preserving a few characteristics of the original transformation. A detailed analysis and several experiments of the proposed framework are included.