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General-purpose object placement is a fundamental capability of an intelligent generalist robot: being capable of rearranging objects following precise human instructions even in novel environments. This work is dedicated to achieving…
In this work we define the trajectory coset of an affine map and use it to study the similarity classes of affine maps. We use the trajectory coset, a tool which allows us to gain a deeper understanding of the interplay between geometry…
We show how to use geometric arguments to prove that the terminal solution to a rough differential equation driven by a geometric rough path can be obtained by driving the same equation by a piecewise linear path. For this purpose, we…
Trajectory Prediction of dynamic objects is a widely studied topic in the field of artificial intelligence. Thanks to a large number of applications like predicting abnormal events, navigation system for the blind, etc. there have been many…
Geographical research was successfully quantified through the quantitative revolution of geography. However, the succeeding theorization of geography encountered insurmountable difficulties. The largest obstacle of geography's theorization…
Surfaces and curves play an important role in geometric design. In recent years, problem of finding a surface passing through a given curve have attracted much interest. In the present paper, we propose a new method to construct a surface…
A projective variety whose Gauss map has positive dimensional fibres corresponds to a special kind of scroll called \emph{Gaussian}. A Gaussian scroll is a member of a canonical derived \emph{ Gaussian flag}. We introduce a duality in the…
The classical Galois theory deals with certain finite algebraic extensions and establishes a bijective order reversing correspondence between the intermediate fields and the subgroups of a group of permutations called the Galois group of…
Looking to the history of mathematics one could find out two outer approaches to Geometry. First one (algebraic) is due to Descartes and second one (group-theoretic)--to Klein. We will see that they are not rivalling but are tied (by…
This paper is devoted to presenting a new approach to determine the intersection of two quadrics based on the detailed analysis of its projection in the plane (the so called cutcurve) allowing to perform the corresponding lifting correctly.…
We introduce the Primary Gasing Triangle, a right triangle with a hypotenuse of 1 unit, to define the primary trigonometric functions: sine and cosine. This triangle serves as the foundational element in a new approach to learning…
This article is devoted to the study of classical and new results concerning equidistant sets, both from the topological and metric point of view. We start with a review of the most interesting known facts about these sets in the euclidean…
We give a relationship that yields an effective geometric way of evaluating mean curvature of surfaces. The approach is reminiscent of the Gauss's contour based evaluation of intrinsic curvature. The presented formula may have a number of…
In this work, we introduce a new geometry based on the difference angle, an angle defined as the difference of slopes of two lines, together with an axiomatic system for angles. This framework provides a constructive approach to the…
Enumerative geometry, the art and science of counting geometric objects satisfying geometric conditions, has seen a resurgence of activity in recent years due to an influx of new techniques that allow for enriched computations. This paper…
For lateral vehicle dynamics, planning trajectories for lane-keeping and lane-change can be generalized as a path planning task to stabilize a vehicle onto a target lane, which is a fundamental element in nowadays autonomous driving…
Learning from demonstration allows robots to acquire complex skills from human demonstrations, but conventional approaches often require large datasets and fail to generalize across coordinate transformations. In this paper, we propose…
The development of precision agriculture has gradually introduced automation in the agricultural process to support and rationalize all the activities related to field management. In particular, service robotics plays a predominant role in…
Presenting systems of differential equations in the form of diagrams has become common in certain parts of physics, especially electromagnetism and computational physics. In this work, we aim to put such use of diagrams on a firm…
Nicolas-Auguste Tissot (1824--1897) published a series of papers on cartography in which he introduced a tool which became known later on, among geographers, under the name of the "Tissot indicatrix." This tool was broadly used during the…