Near-Optimal Parameterization of the Intersection of Quadrics: Theory and Implementation

Colloque avec actes et comité de lecture. internationale.International audienceWe present an algorithm that computes an exact parametric form of the intersection of two real quadrics in projective three-space given by implicit equations with rational coefficients. This algorithm represents the first complete and robust solution to what is perhaps the most basic problem of solid modeling by implicit curved surfaces

Near-Optimal Parameterization of the Intersection of Quadrics: III. Parameterizing Singular Intersections

In Part II [3] of this paper, we have shown, using a classification of pencils of quadrics over the reals, how to determine quickly and efficiently the real type of the intersection of two given quadrics. For each real type of intersection, we design, in this third part, an algorithm for computing a near-optimal parameterization. We also give here examples covering all the possible situations, in terms of both the real type of intersection and the number and depth of square roots appearing in the coefficients

Complete Classification and Efficient Determination of Arrangements Formed by Two Ellipsoids

International audienceArrangements of geometric objects refer to the spatial partitions formed by the objects and they serve as an underlining structure of motion design, analysis and planning in CAD/CAM, robotics, molecular modeling, manufacturing and computer-assisted radio-surgery. Arrangements are especially useful to collision detection, which is a key task in various applications such as computer animation , virtual reality, computer games, robotics, CAD/CAM and computational physics. Ellipsoids are commonly used as bounding volumes in approximating complex geometric objects in collision detection. In this paper we present an in-depth study on the arrangements formed by two ellipsoids. Specifically, we present a classification of these arrangements and propose an efficient algorithm for determining the arrangement formed by any particular pair of ellipsoids. A stratification diagram is also established to show the connections among all the arrangements formed by two ellipsoids. Our results for the first time elucidate all possible relative positions between two arbitrary ellipsoids and provides an efficient and robust algorithm for determining the relative position of any two given ellipsoids, therefore providing the necessary foundation for developing practical and trustworthy methods for processing ellipsoids for collision analysis or simulation in various applications

A Unified Algorithm for Analysis and Simulation of Planar Four-Bar Motions Defined With R- and P-Joints,”

ABSTRACT This paper presents a single, unified and efficient algorithm for animating the motions of the coupler of all four-bar mechanisms formed with revolute (R) and prismatic (P) joints. This is achieved without having to formulate and solve the loop closure equation associated with each type of four-bar linkages separately. In our previous paper on four-bar linkage synthesis, we map the planar displacements from Cartesian to image space using planar quaternion. Given a set of image points that represent planar displacements, the problem of synthesizing a planar fourbar linkage is reduced to finding a pencil of Generalized-or Gmanifolds that best fit the image points in the least squares sense. The three planar dyads associated with Generalized G-manifolds are RR, PR and RP which could construct six types of four-bar mechanisms. In this paper, we show that the same unified formulation for linkage synthesis leads to a unified algorithm for linkage analysis and simulation as well. Both the unified synthesis and analysis algorithms have been implemented on Apple's iOS platform

Розробка алгоритму з покращеною релевантністю локалізації координат вектора для інтелектуальних сенсорів

There are sensors of vector quantities whose field characteristics are described by the equations of quadrics. These sensors have improved sensitivity and smaller dimensions, the "payment" for which is, in fact, the non-linearity of field characteristics. In order to use such sensors, one has to solve the system of three quadric equations. Given the labor-intensity of this process, the sensors are designed intelligent – a finished device includes microcontrollers or other units that are able to process results of measurements. These devices are characterized by "curtailed" software and hardware capacities that necessitate the development of algorithms and their implementations with regard to these constraints.Classic algorithms for solving the systems of polynomial equations are not appropriate because of their violating the requirements to minimal resource consumption. The search for solutions of the systems of quadric equations is carried out in two stages – first, numerical fields that potentially contain intersections are localized, and then in these regions the search is conducted for accurate solutions by numerical methods. The success of using numerical methods depends on the quality of the conducted localization of solutions. The means of localization are not sufficiently worked out. Earlier, an algorithm was developed using interval arithmetic, the implementation of which by a microcontroller of the ARM Cortex-M4 architecture proved its capacity to find all, without exception, regions with the sought solutions of a system of equations, however, in addition to the "useful" regions, this algorithm finds as well the regions that do not actually contain the solutions. This fact led to unnecessary waste of time trying to find exact solutions in the regions where they do not exist.Thus, there was a need to search for alternative ways to the localization of solutions for the systems of quadrics equations. It is natural to search for these methods, based on the properties of quadrics in particular and continuous differentiable functions in general. A foundation of the proposed algorithm is the fact that a function in a closed region acquires its maximum or minimum value either at the borders or in critical points. If we accept, as a closed region, a rectangular parallelepiped, then its boundaries are its six sides, the boundaries of sides are its edges, and the boundaries of edges are the tops of parallelepiped. On the sides of the parallelepiped, function of three variables is reduced to function of two variables, on the edges – to functions of one variable. In the case of quadrics, finding the critical points of function on the edges comes down to solving a linear equation, and the critical points of function on the sides – to solving a system of two linear equations with two unknowns. Therefore, it is sufficient to check the signs of function at the tops of rectangular parallelepiped and those critical points of function that belong to the examined region. If in all these points the signs of function are the same, then there is no any point inside where function takes the 0 value. Thus, checking the signs of all functions that represent the left parts of the quadrics equation allows us to "reject" the regions, where there possible may not be any points of intersection. Instead of remembering the values of functions (valid numbers), it is sufficient to keep the signs of functions (one bit), which provides for the less consumption of operative memory. The tests proved that the proposed new algorithm is applicable for the implementation in the micro programming software, thus providing for a higher relevance of the found regions in comparison with the algorithm-analogue. An increase in relevancy is explained by the fact that interval arithmetic always implies overstated evaluations because, as the lower boundaries of intervals, the minimum permissible values are accepted, and as the upper limits – maximum permissible values. Checking the signs of functions in the selected points is free from the revaluation of results. The new algorithm somewhat deteriorated performance of permanent memory and execution time, however, these costs are compensated for by the further search for the solutions for a smaller number of irrelevant regions.Исследована проблема недостаточного развития алгоритмов, применимых для реализации в микропрограммном обеспечении для поиска точек пересечения квадрик. Разработан, реализован и изучен алгоритм локализации точек пересечения квадрик на основе свойств непрерывных дифференцируемых функций в замкнутой области. Новым алгоритмом достигается релевантность результатов выше, чем у его единственного аналога. Результаты предназначены для интеллектуальных сенсоровДосліджено проблему недостатнього розвитку алгоритмів, реалізованих у мікропрограмному забезпеченні для пошуку точок перетину квадрик. Розроблено, реалізовано та досліджено алгоритм локалізації точок перетину квадрик на основі властивостей неперервних диференційовних функцій у замкнутій області. Новим алгоритмом досягається вища релевантність результатів, ніж його єдиним аналогом. Результати призначені для інтелектуальних сенсорів векторних величи

Signature Sequence of Intersection Curve of Two Quadrics for Exact Morphological Classification

We present an efficient method for classifying the morphology of the intersection curve of two quadrics (QSIC) in PR3, 3D real projective space; here, the term morphology is used in a broad sense to mean the shape, topological, and algebraic properties of a QSIC, including singularity, reducibility, the number of connected components, and the degree of each irreducible component, etc. There are in total 35 different QSIC morphologies with non-degenerate quadric pencils. For each of these 35 QSIC morphologies, through a detailed study of the eigenvalue curve and the index function jump we establish a characterizing algebraic condition expressed in terms of the Segre characteristics and the signature sequence of a quadric pencil. We show how to compute a signature sequence with rational arithmetic so as to determine the morphology of the intersection curve of any two given quadrics. Two immediate applications of our results are the robust topological classification of QSIC in computing B-rep surface representation in solid modeling and the derivation of algebraic conditions for collision detection of quadric primitives