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Rossignac,
Jarek
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Jarek
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ItemPlanar similaritymotion interpolating three keyframes: Comparative assessment of prior and novel solutions(Georgia Institute of Technology, 2021) Rossignac, Jarek ; Vinacua, ÀlvarWe 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 locallyperseverant 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 startingpoint p of the edge, its length m, and its winding (arbitrary angle) w. Hence, an arrow defines an integer windingcount 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 logspiral 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 usecases: Keyframeanimation, Variablewidth stroke design, Banner deformation, Pattern animation, Motion prediction, and Curve design.

ItemSQUINT Fields, Maps, Patterns, and Lattices(Georgia Institute of Technology, 20180723) Rossignac, JarekThe proposed Steady QUad INTerpolating (SQUINT) map is formulated in terms of a SQUINT Field of Similarities (FoS). It is controlled by four coplanar points. It maps the unit square onto a curved planar quad, R, which has these points as corners. Uniformly spaced, logspiral isocurves decompose R into tiles that are similar to each other and, hence, each have equal angles at opposite corners. We provide closedform expressions for computing the representation of the SQUINT map and for evaluating the map and its inverse. We discuss extensions and potential applications to texture maps and field warps and to the design, display, and constantcost query of procedural models of arbitrarily complex lattices.

ItemPermutation Classifier(Georgia Institute of Technology, 20180424) Zhou, Xinrui ; Guerra, Concettina ; Rossignac, Jarek ; RossignacMilon, LeoWe consider permutations of a given set of n different symbols. We are given two unordered training sets, T1 and T2, of such permutations that are each assumed to contain examples of permutations of the corresponding type, t1 and t2. Our goal is to train a classifier, C(q), by computing a statistical model from T1 and T2, which, when given a candidate permutation, q, decides whether q is of type t1 or t2. We discuss two versions of this problem. The ranking version focuses on the order of the symbols. Our Separation Average Distance Matrix (SADiM) solution expands on previously proposed ranking aggregation formulations. The grouping version focuses on contiguity of symbols and hierarchical grouping. We propose and compare two solutions: (1) The Population Augmentation Ratio (PAR) solution computes a PQtree for each training set and uses a novel measure of distance between these and q that is based on ratios of population counts (i.e., of numbers of permutations explained by specific PQtrees). (2) The Difference of Positions (DoP) solution is computationally less expensive than PAR and is independent of the absolute population counts. Although DoP does not have the simple statistical grounding of PAR, our experiments show that it is consistently effective.

ItemDesigning and processing parametric models of steady lattices( 2018) Gupta, Ashish ; Kurzeja, Kelsey ; Rossignac, Jarek ; Allen, George ; Kumar, Pranav Srinivas ; Musuvathy, SurajOur goal is to facilitate the design, analysis, optimization, and additive manufacturing of a specific class of 3D lattices that may comprise an extremely large number of elements. We target curved lattices that exhibit periodicity and uniform geometric gradations in three directions, along possibly curved axes. We represent a lattice by a simple computer program with a carefully selected set of exposed control parameters that may be used to adjust the overall shape of the lattice, its repetition count in each direction, its microstructure, and its gradation. In our ProgrammedLattice Editor (PLE), a typical lattice is represented by a short program of 10 to 50 statements. We propose a simple API and a few rudimentary GUI tools that automate the creation of the corresponding expressions in the program. The overall shape and gradation of the lattice is controlled by three similarity transformations. This deliberate design choice ensures that the gradation in each direction is regular (i.e., mathematically steady), that each cell can be evaluated directly, without iterations, and that integral properties (such as surface area, volume, center of mass and spherical inertia) can be obtained rapidly without having to calculate them for each individual element of the lattice.

ItemRangeFinder: Accelerating ballinterference queries against steady lattices(Georgia Institute of Technology, 2018) Kurzeja, Kelsey ; Rossignac, JarekAdvances in additive manufacturing techniques are enabling the fabrication of new microstructures and materials. These may often be defined in terms of a set of balls and of beams that each connects two balls. To support application needs, we must support lattices with billions of such elements. To address this problem, we focus on architected and periodic structures in which the connectivity pattern repeats in three directions, and in which the positions and radii of the balls evolve through the structure in a prescribed and steady way that is defined by three similarity transforms. We propose here an algorithm that accelerates the BallInterference Query (BIQ), which establishes which elements of the lattice interfere with a query ball Q. Our RangeFinder (RF) solution reduces the asymptotic complexity of BIQs, which, in our tests, reduced the query time by a factor of between 45 and 5500. RF does not use any spatial occupancy data structure and can be trivially parallelized. We demonstrate the effectiveness of RangeFinder through the generation of multilevel lattices that we call LatticeinLattice (LiL).

ItemSOT: Compact Representation for Triangle and Tetrahedral Meshes(Georgia Institute of Technology, 2010) Rossignac, Jarek ; Gurung, ToprajThe Corner Table (CT) represents a triangle mesh by storing 6 integer references per triangle (3 vertex references in the Vertex table and 3 references to opposite corners in the Opposite table, which accelerate access to adjacent triangles). The Compact Half Face (CHF) representation extends CT to tetrahedral meshes, storing 8 references per tetrahedron (4 in the Vertex table and 4 in the Opposite table). We use the term Vertex Opposite Table (VOT) to refer to both CT and CHF and propose a sorted variation, SVOT, which is inspired by tetrahedral mesh encoding techniques and which works for both triangle and tetrahedral meshes. The SVOT does not require additional storage and yet provides, for each vertex, a reference to an incident corner from which the star (incident cells) of the vertex may be traversed at a constant cost per visited element. We use the corner operators for querying and traversing the triangle meshes while for tetrahedral meshes, we propose a set of powerful wedgebased operators. Improving on the SVOT, we propose our Sorted Opposite Table (SOT) variation, which eliminates the Vertex table completely and hence reduces storage requirements by 50% to only 3 references per triangle for triangle meshes and 4 references and 9 bits per tetrahedron for tetrahedral meshes, while preserving the vertextoincidentcorner references and supporting the corner operators and our wedge operators with a constant average cost. The SVOT and SOT representation work on manifold meshes with boundaries.