Rossignac, Jarek

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

2021 , Rossignac, Jarek , Vinacua, Àlvar

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.

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SAM: Steady Affine Motions

2009-11-23 , Rossignac, Jarek , Vinacua, Àlvar

An affine motion is a continuous map from time value t to an affinity A subscript t. It is a SAM (Steady Affine Motion), when A subscript t = A superscript t. Although the beauty of a motion is subjective, the above equation provides one mathematical characterization and includes the screw ("universal instantaneous") motion and the golden ("mirabilis") spiral. Although a real matrix, A superscript t, may not exist, we show that it does for a dense set of affinities A covering a significant range of rotations and shears around the identity and that it may be computed efficiently and robustly in two and three dimensions using closed form expressions. SAMs have remarkable properties. For example, the velocity of any point remains constant, both in the global (fixed) and local (moving) frames, which facilitates the exact computation of derived entities, such as the envelope surfaces used to define the boundary of a swept volume. We say that a pattern of features F subscript i is steady when there exists an affinity M such that F subscript i = M superscript i F subscript 0. Each M superscript i is a frame of a SAM and may be computed as A superscript (i/n), where A is the afiine relation F subscript n = A F subscript 0 between the first and the last feature. This option makes it possible to edit directly the feature count n or the cumulative transformation A.