Contact detection between a small ellipsoid and another quadric

[Abstract] We analyze the characteristic polynomial associated to an ellipsoid and another quadric in the context of the contact detection problem. We obtain a necessary and sufficient condition for an efficient method to detect contact. This condition, named smallness condition, is a feature on the size and the shape of the quadrics and can be checked directly from their parameters. Under this hypothesis, contact can be noticed by means of the expressions in a discriminant system of the characteristic polynomial. Furthermore, relative positions can be classified through the sign of the coefficients of this polynomial. As an application of these results, a method to detect contact between a small ellipsoid and a combination of quadrics is given

Cosmology: macro and micro

A new approach to cosmology and space-time is developed, which emphasizes the description of the matter degrees of freedom of Einstein's theory of gravity by a family of K\"ahler-Einstein Fano manifolds

Classification of the relative positions between a small ellipsoid and an elliptic paraboloid

©2019. This manuscript version is made available under the CC-BY-NC-ND 4.0 license https://creativecommons.org/licenses/by-nc-nd/4.0/. This version of the article Brozos-Vázquez, M., Pereira-Sáez, M. J., Souto-Salorio, M. J., & Tarrío-Tobar, A. D. (2019). “Classification of the relative positions between a small ellipsoid and an elliptic paraboloid” has been accepted for publication in Computer Aided Geometric Design, 72, 34–48. The Version of Record is available online at https://doi.org/10.1016/j.cagd.2019.05.002.[Abstract]: We classify all the relative positions between an ellipsoid and an elliptic paraboloid when the ellipsoid is small in comparison with the paraboloid (small meaning that the two surfaces cannot be tangent at two points simultaneously when one is moved with respect to the other). This provides an easy way to detect contact between the two surfaces by a direct analysis of the coefficients of a fourth degree polynomial.The authors wish to thank the referees for extremely valuable comments and suggestions, which were essential to improve the final version of the paper. Supported by Projects ED431F 2017/03, TIN2017-85160-C2-1-R, MTM2016-75897-P and MTM2016-78647-P (AEI/FEDER, UE).Xunta de Galicia; ED431F 2017/0

Enumerating the morphologies of non-degenerate Darboux cyclides

International audienceWe provide an enumeration of all possible morphologies of non-degenerate Darboux cyclides. Based on the fact that every Darboux cyclide in R 3 is the stereographic projection of the intersection surface of a sphere and a quadric in R 4 , we transform the enumeration problem of morphologies of Darboux cyclides to the enumeration of the algebraic sequences that characterize the intersection of a sphere and a quadric in R 4

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

Kinematics and Robot Design I, KaRD2018

This volume collects the papers published on the Special Issue “Kinematics and Robot Design I, KaRD2018” (https://www.mdpi.com/journal/robotics/special_issues/KARD), which is the first issue of the KaRD Special Issue series, hosted by the open access journal “MDPI Robotics”. The KaRD series aims at creating an open environment where researchers can present their works and discuss all the topics focused on the many aspects that involve kinematics in the design of robotic/automatic systems. Kinematics is so intimately related to the design of robotic/automatic systems that the admitted topics of the KaRD series practically cover all the subjects normally present in well-established international conferences on “mechanisms and robotics”. KaRD2018 received 22 papers and, after the peer-review process, accepted only 14 papers. The accepted papers cover some theoretical and many design/applicative aspects

Algorithms for determining relative position between spheroids and hyperboloids with one sheet

©2019. This manuscript version is made available under the CC-BY-NC-ND 4.0 license https:// creativecommons.org/licenses/by-nc-nd/4.0/. This version of the article: Castro, P. M., Dapena, A., Souto- Salorio, M. J., & Tarrío-Tobar, A. D. (2019). 'Algorithms for determining relative position between spheroids and hyperboloids with one sheet', has been accepted for publication in Mathematics and Computers in Simulation, 160, 168–179.The Version of Record is available online at https://doi.org/10.1016/j.matcom.2018.12.006.[Abstract]: In this work we present a new method for determining relative positions between one moving object, modelled by a bounding spheroid, and surrounding static objects, modelled by circular hyperboloids of one sheet. The proposed strategy is based on the real-time calculation of the coefficients of degree three polynomial. We propose several algorithms for two real applications of this geometric problem: the first one, oriented to the design of video games, and the second one, devoted to surveillance tasks of a quadcopter in industrial or commercial environments.This work has been funded by the Xunta de Galicia, Spain (ED431C 2016-045, ED341D R2016/012, ED431G/01), the Agencia Estatal de Investigación of Spain (TEC2013-47141-C4-1-R, TEC2015-69648-REDC, TEC2016-75067-C4-1-R) and ERDF funds of the EU (AEI/FEDER, UE). The authors wish to thank José Sanmartín for his helpful work in the graphic design of the pictures included in this paper.Xunta de Galicia; ED431C 2016-045Xunta de Galicia; ED341D R2016/012Xunta de Galicia; ED431G/0

Vector Geometry and Applications to Three-Dimensional Computer Graphics

The mathematics behind algorithms involved in generating three-dimensional images on a computer has stemmed from the analysis of the processes of sight and vision. These processes have been modeled to provide methods of visualising three-dimensional data sets. The applications of such visualisations are varied. This project will study some of the mathematics that IS used in three-dimensional graphics application