ABOUT THE MINKOWSKI DIFFERENCE OF SQUARES ON A PLANE

Introduction. As you know, the concept of a set is a basic concept in mathematics, and many mathematical problems have been solved using the theory developed around it. Building a set theory apparatus begins with defining the operations that can be performed on sets. Most of us know about operations on sets, such as joining, intersecting, subtracting, symmetric subtraction, and we also have an understanding of the practical problems they can solve. With the development of mathematics, including the science of geometry, the idea of adding other operations to the sets in addition to the above operations arose, and there was a need to enrich the content of set theory and apply them to new practical problems.

On Minkowski difference-based contact detection in discrete/discontinuous modelling of convex polygons/polyhedra

Purpose Contact detection for convex polygons/polyhedra has been a critical issue in discrete/discontinuous modelling, such as the discrete element method (DEM) and the discontinuous deformation analysis (DDA). The recently developed 3D contact theory for polyhedra in DDA depends on the so-called entrance block of two polyhedra and reduces the contact to evaluate the distance between the reference point to the corresponding entrance block, but effective implementation is still lacking. Design/methodology/approach In this paper, the equivalence of the entrance block and the Minkowski difference of two polyhedra is emphasised and two well-known Minkowski difference-based contact detection and overlap computation algorithms, GJK and expanding polytope algorithm (EPA), are chosen as the possible numerical approaches to the 3D contact theory for DDA, and also as alternatives for computing polyhedral contact features in DEM. The key algorithmic issues are outlined and their important features are highlighted. Findings Numerical examples indicate that the average number of updates required in GJK for polyhedral contact is around 6, and only 1 or 2 iterations are needed in EPA to find the overlap and all the relevant contact features when the overlap between polyhedra is small. Originality/value The equivalence of the entrance block in DDA and the Minkowski difference of two polyhedra is emphasised; GJK- and EPA-based contact algorithms are applied to convex polyhedra in DEM; energy conservation is guaranteed for the contact theory used; and numerical results demonstrate the effectiveness of the proposed methodologies.

An Application of Mathematics to Computer Programming: Connecting Translation Vectors, the Minkowski Difference, and Collision Detection

Translation by a vector in the coordinate plane is first introduced in precalculus and connects to the basic theory of vector spaces in linear algebra. In this article, we explore the topic of collision detection in which the idea of a translation vector plays a significant role. Because collision detection has various applications in video games, virtual simulations, and robotics (Garcia-Alonso, Serrano, and Flaquer 1994; Rodrigue 2012), using it as a motivator in the study of translation vectors can be helpful. For example, students might be interested in the question, “How does the computer recognize when a player's character gets hit by a fireball?” Computer science provides a rich context for real-life applications of mathematics-programmers use mathematics for coding an algorithm in which the computer recognizes two objects nearing each other or colliding. The Minkowski difference, named after the nineteenth century German mathematician Hermann Minkowski, is used to solve collision detection problems (Ericson 2004). Applying the Minkowski difference to collision detection is based on translation vectors, and programmers use the algorithm as a method for detecting collision in video games.

Lower Bounds of the Allowable Motions of One N-Dimensional Ellipsoid Contained in Another

This paper studies the representations of a subset of the allowable motions for an N-dimensional ellipsoid inside another slightly larger ellipsoid without collision based on the idea of the Kinematics of Containment. As an extension to the previous work on the closed-form lower bounds, this paper proposes another two lower bounds based on the first-order algebraic condition of containment and the closed-form Minkowski difference between two ellipsoids respectively. Querying processes for a specific configuration of the moving ellipsoid and the calculations of the volume of the proposed lower bounds in configuration space (C-space) are introduced. Examples for the proposed lower bounds in 2D and 3D Euclidean space are implemented and the corresponding motion volumes in C-space are compared with different shapes of the ellipsoids. Finally a case study of the application on automated assembly is introduced.

Geometric Study of Minkowski Differences of Plane Convex Bodies

In the Euclidean plane ℝ2, we define the Minkowski difference 𝒦–𝓛 of two arbitrary convex bodies 𝒦, 𝓛 as a rectifiable closed curve ℋh⊂ ℝ2that is determined by the differenceh=h𝒦–h𝓛of their support functions. This curve ℋhis called the hedgehog with support functionh. More generally, the object of hedgehog theory is to study the Brunn–Minkowski theory in the vector space of Minkowski differences of arbitrary convex bodies of Euclidean space ℝn+1, defined as (possibly singular and self-intersecting) hypersurfaces of ℝn+1. Hedgehog theory is useful for: (i) studying convex bodies by splitting them into a sum in order to reveal their structure; (ii) converting analytical problems into geometrical ones by considering certain real functions as support functions. The purpose of this paper is to give a detailed study of plane hedgehogs, which constitute the basis of the theory. In particular: (i) we study their lengthmeasures and solve the extension of the Christoffel–Minkowski problemto plane hedgehogs; (ii) we characterize support functions of plane convex bodies among support functions of plane hedgehogs and support functions of plane hedgehogs among continuous functions; (iii) we study the mixed area of hedgehogs in ℝ2and give an extension of the classical Minkowski inequality (and thus of the isoperimetric inequality) to hedgehogs.