Abstract:

An Affine matrix which maps an initial and final pose can be computed by solving a system of linear equations.
Then there exists an interesting problem of finding a time varying affinity which maps the given set of poses
and if it exists is always unique and should hold some interesting properites such as affineinvariant, reversible,
preserve rigidity, similarities and volume. The Steady Affine Motions and Morphs (SAM) introduced by Jarek
Rossignac and Alvar Vinacua solved this problem of time varying affinity and defines the quality of such affinity
by the term steadiness. Until SAM, no mathematical definition of steadiness was available and intuitively SAM
defined a steady animation to be continuous, to vary dimensions and angles monotonically and rather uniformly,
and to move points along pleasing arcs that are free of unnecessary kinks or loops. The authors defined the term
”Steady” as a constant velocity motion in the local moving frame. SAM creates pleasing inbetweening motions
that interpolates between an initial and final pose, B and C and the derived equation of beauty was At B with
A = C B·1. SAM is affineinvariant, reversible, preserves isometries (i.e., rigidity), similarities and volume.
Previously proposed approaches came up with a solution for the time varying affinity problem, but there was no
proper definition of how beautiful or how good the motion was. With the advent of SAM, the beauty of a motion
can now be measured by the unsteadiness and Steady Affine motions and morphs is the one solution which comes
to have a value of zero for the unsteadiness term.
Uniform Accelerated Motions (UAM) carries forward the above definition of steadiness into a constant acceleration
motion in the local moving frame. The time varying affinity At is computed using closed form expressions
and some of its interesting properties are studied. The constant acceleration motion (in local frame) in UAM is
then compared with the constant velocity motion (in local frame) of SAM and the resuls are discussed. 