Related papers: Computational Euclid
Representing graphs as sets of node embeddings in certain curved Riemannian manifolds has recently gained momentum in machine learning due to their desirable geometric inductive biases, e.g., hierarchical structures benefit from hyperbolic…
Matrix geometric means between two positive definite matrices can be defined from distinct perspectives - as solutions to certain nonlinear systems of equations, as points along geodesics in Riemannian geometry, and as solutions to certain…
Abstraction plays a key role in concept learning and knowledge discovery; this paper is concerned with computational abstraction. In particular, we study the nature of abstraction through a group-theoretic approach, formalizing it as…
What is the best representation for doing euclidean geometry on computers? These notes from a SIGGRAPH 2019 short course entitled "Geometric algebra for computer graphics" introduce projective geometric algebra (PGA) as a modern framework…
This paper studies binary quadratic programs in which the objective is defined by a Euclidean distance matrix, subject to a general polyhedral constraint set. This class of nonconcave maximisation problems includes the capacitated,…
Regions-based theories of space aim -- among others -- to define points in a geometrically appealing way. The most famous definition of this kind is probably due to Whitehead. However, to conclude that the objects defined are points indeed,…
We present alternative postulates for Euclidean geometry whose merit is that they lead to a new class of invariants and associated geometries for real finite-dimensional unital associative algebras.
Euclid uses an undefined notion of "equal figures", to which he applies the common notions about equals added to equals or subtracted from equals. When (in previous work) we formalized Euclid Book~I for computer proof-checking, we had to…
Many interesting phenomena are characterized by the complex interaction of different physical processes, each often best modeled numerically via a specific approach. In this paper, we present the design and implementation of an…
These are lecture notes for a course on the theory of Clifford algebras, with special emphasis on their wide range of applications in mathematics and physics. Clifford algebra is introduced both through a conventional tensor algebra…
Manifold optimization is ubiquitous in computational and applied mathematics, statistics, engineering, machine learning, physics, chemistry and etc. One of the main challenges usually is the non-convexity of the manifold constraints. By…
Though many safety-critical software systems use floating point to represent real-world input and output, programmers usually have idealized versions in mind that compute with real numbers. Significant deviations from the ideal can cause…
In the first part of this review we introduce the basics theory behind geometric phases and emphasize their importance in quantum theory. The subject is presented in a general way so as to illustrate its wide applicability, but we also…
We introduce cilantro, an open-source C++ library for geometric and general-purpose point cloud data processing. The library provides functionality that covers low-level point cloud operations, spatial reasoning, various methods for point…
We define an abstract framework for object-oriented programming and show that object-oriented languages, such as C++, can be interpreted as parallel programming languages. Parallel C++ code is typically more than ten times shorter than the…
The proper Euclidean geometry is considered to be metric space and described in terms of only metric and finite metric subspaces (sigma-immanent description). Constructing the geometry, one does not use topology and topological properties.…
This paper introduces abstractions that are meaningful for computers and that can be built and used according to computers' own criteria, i.e., computable abstractions. It is analyzed how abstractions can be seen to serve as the building…
This paper considers an extension of origami geometry to the case of "folding" a three dimensional (3D) space along a plane. First, all possible incidence constraints between given points, lines and planes are analyzed by using the geometry…
The distributional analysis of Euclidean algorithms was carried out by Baladi and Vall\'{e}e. They showed the asymptotic normality of the number of division steps and associated costs in the Euclidean algorithm as a random variable on the…
We offer a view of mathematics as an experimental science where axioms play the role of foundational theories like general relativity and quantum mechanics in physics. Under this view, axioms are provisional and inferred from experience…