We propose to measure the beauty of an affine motion by the inverse of its steadiness, which we define as its Average Relative Acceleration (ARA). Steady affine motions, for which ARA=0, include translations, rotations, rigid body screws, and the golden spiral. To facilitate the design of beautiful in-betweening motions that interpolate between an initial and a final pose (affine transformation), B and C, we propose the Steady Affine Morph (SAM), defined as A^tB with A=CB^–1. A SAM is affine-invariant and reversible. It preserves isometries (i.e., rigidity), similarities, and volume. Its velocity field is stationary both in the global and the local (moving) frames. Given a copy count, n, the series of uniformly sampled poses, A^i/nB, of a SAM form a regular pattern, which may be easily controlled by changing B, C, or n, and where consecutive poses are related by the same affinity A^1/n. Although a real matrix A^t does not always exist, we show that it does for a convex and large subset of orientation-preserving affinities A. Our fast and accurate Extraction of Affinity Roots (EAR) algorithm computes A^t, when it exists, using closed form expressions in two or in three dimensions. We discuss SAM applications to pattern design and animation and to key-frame interpolation. In particular, we present a new curve subdivision scheme in the space of affine motions, which produces an apparently smooth, piecewise-steady motion that interpolates a given series of poses.

The talk describes joint work with Prof. Alvar Vinacua from UPC in Barcelona Spain. Reference: “Steady Affine Motions and Morphs”, J. Rossignac and A. Vinacua. ACM Transactions on Graphics. SIGGRAPH 2012. Videos are posted at http://www.gvu.gatech.edu/~jarek/SAM