Planar similarity-motion interpolating three keyframes: Comparative assessment of prior and novel solutions

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Rossignac, Jarek
Vinacua, Àlvar
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We compare 8 solutions for defining the planar motion of an oriented edge that interpolates 3 keyframes. One contribution is the discovery of several novel solutions, one of which produces what we call a locally-perseverant motion, for which the acceleration of a moving point remains constant in the local (moving) frame. The other contribution is to demonstrate that: (a) many interesting solutions exist, (b) the mathematical and perceived differences between the animations they produce are significant, and (c) these differences may matter for designers and applications. To allow motions that rotate by more than 2π, we represent the 3 keyframes and the moving edge by arrows, each storing the starting-point p of the edge, its length m, and its winding (arbitrary angle) w. Hence, an arrow defines an integer winding-count k (with |w − 2kπ| ≤ π) and a similarity transformation that combines dilation by m, rotation by w − 2kπ, and translation from the origin to p. Our chosen PITA (Planar Interpolation of Three Arrows) solutions are formulated using compositions of linear, polar, or log-spiral interpolations, or using ODEs or logarithms of matrices. We compare these solutions in terms of 11 mathematical properties and also in terms of subjective attributes that may be important for designers. We illustrate differences between our 8 chosen PITAs in 6 use-cases: Keyframe-animation, Variable-width stroke design, Banner deformation, Pattern animation, Motion prediction, and Curve design.
DARPA and FEDER Grant TIN2017-88515-C2-1-R
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