##
Person:
Rossignac,
Jarek

Rossignac,
Jarek

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#### SQUINT Fields, Maps, Patterns, and Lattices

2018-07-23
,
Rossignac, Jarek

The 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.

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#### RangeFinder: Accelerating ball-interference queries against steady lattices

2018
,
Kurzeja, Kelsey
,
Rossignac, Jarek

Advances 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).

Item

#### Permutation Classifier

2018-04-24
,
Zhou, Xinrui
,
Guerra, Concettina
,
Rossignac, Jarek
,
Rossignac-Milon, Leo

We 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.

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#### Designing and processing parametric models of steady lattices

2018
,
Gupta, Ashish
,
Kurzeja, Kelsey
,
Rossignac, Jarek
,
Allen, George
,
Kumar, Pranav Srinivas
,
Musuvathy, Suraj

Our 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.