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ItemPlanar similarity-motion 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 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.
ItemSQUINT Fields, Maps, Patterns, and Lattices(Georgia Institute of Technology, 2018-07-23) 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, log-spiral isocurves decompose R into tiles that are similar to each other and, hence, each have equal angles at opposite corners. We provide closed-form 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 constant-cost query of procedural models of arbitrarily complex lattices.
ItemPermutation Classifier(Georgia Institute of Technology, 2018-04-24) Zhou, Xinrui ; Guerra, Concettina ; Rossignac, Jarek ; Rossignac-Milon, 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 PQ-tree 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 PQ-trees). (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.
ItemRangeFinder: Accelerating ball-interference 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 Ball-Interference 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 multi-level lattices that we call Lattice-in-Lattice (LiL).
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 Programmed-Lattice 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.
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 wedge-based 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 vertex-to-incident-corner 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.
ItemSAM: Steady Affine Motions(Georgia Institute of Technology, 2009-11-23) Rossignac, Jarek ; Vinacua, ÀlvarAn 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.
ItemITR/PE+SY digital clay for shape input and display(Georgia Institute of Technology, 2007-11-30) Book, Wayne J. ; Rossignac, Jarek ; Mynatt, Elizabeth D. ; Allen, Mark G. ; Goldthwaite, John Randall ; Rosen, David W. ; Glezer, Ari
ItemOptimized Blist Form (OBF)(Georgia Institute of Technology, 2007-05-23) Rossignac, JarekAny Boolean expressions may be converted into positive-form, which has only union and intersection operators. Let E be a positive-form expression with n literals. Assume that the truth-values of the literals are read one at a time. The numbers s(n) of steps (operations) and b(n) of working memory bits (footprint) needed to evaluate E depend on E and on the evaluation technique. A recursive evaluation performs s(n)=n–1 steps but requires b(n)=log(n)+1 bits. Evaluating the disjunctive form of E uses only b(n)=2 bits, but may lead to an exponential growth of s(n). We propose a new Optimized Blist Form (OBF) that requires only s(n)=n steps and b(n)=⌈log2j⌉ bits, where j=⌈log2(2n/3+2)⌉. We provide a simple and linear cost algorithm for converting positive-form expressions to their OBF. We discuss three applications: (1) Direct CSG rendering, where a candidate surfel stored at a pixel is classified against an arbitrarily complex Boolean expression using a footprint of only 6 stencil bits; (2) the new Logic Matrix (LM), which evaluates any positive form logical expression of n literals in a single cycle and uses a matrix of at most n×j wire/line connections; and (3) the new Logic Pipe (LP), which uses n gates that are connected by a pipe of ⌈log2j⌉ lines and when receiving a staggered stream of input vectors produces a value of a logical expression at each cycle.
ItemSimulation of Bubbles in Foam With The Volume Control Method(Georgia Institute of Technology, 2007) Kim, Byungmoon ; Liu, Yingjie ; Llamas, Ignacio ; Jiao, Xiangmin ; Rossignac, JarekLiquid and gas interactions often produce bubbles that stay for a long time without bursting on the surface, making a dry foam structure. Such long lasting bubbles simulated by the level set method can suffer from a small but steady volume error that accumulates to a visible amount of volume change. We propose to address this problem by using the volume control method. We track the volume change of each connected region, and apply a carefully computed divergence that compensates undesired volume changes. To compute the divergence, we construct a mathematical model of the volume change, choose control strategies that regulate the modeled volume error, and establish methods to compute the control gains that provide robust and fast reduction of the volume error, and (if desired) the control of how the volume changes over time.