Related papers: Elementary Constructions of conic sections
Here we briefly describe some topics along the lines of projective spaces and related geometric constructions connected to linear algebra, which provide fundamental examples in classical geometry and analysis.
We present a criterion when six points chosen on the sides of a triangle belong to the same conic. Using this tool we show how the two geometrical gems - celebrated Poncelet's theorem of projective geometry and incredible Morley's theorem…
In this paper, we highlight that the point group structure of elliptic curves over finite or infinite fields, may be also observed on singular cubics with a quadratic component. Starting from this, we are able to introduce in a very general…
As it is well known, classical mechanics consists of several basic features like determinism, reductionism, completeness of knowledge and mechanicism. In this article the basic assumptions are discussed which underlie those features. It is…
The classical theory of plane projective geometry is examined constructively, using both synthetic and analytic methods. The topics include Desargues's Theorem, harmonic conjugates, projectivities, involutions, conics, Pascal's Theorem,…
This article explores the limits of geometric construction using various tools, both classical and modern. Starting with ruler and compass constructions, we examine how adding methods such as origami, marked rulers (neusis), conic sections,…
Geometry is essentially a global language, which is fully understood in different times, countries and cultures. The proof of a geometric theorem (e.g. the Pythagorean Theorem) or a geometric construction (e.g. the construction of an…
Angular momentum is taught in every classical mechanics course. It is a difficult topic with misconceptions commonly forming significant barriers to student success. My intention in writing this paper is to combat some of the most common…
This paper gives a complete classification of conics in $PE_2(\mathbb{R})$. The classification has been made earlier (Reveruk [5]), but it showed to be incomplete and not possible to cite and use in further studies of properties of conics,…
A problem that is simple to state in the context of spherical geometry, and that seems rather interesting, appears to have been unexamined to date in the mathematical literature. The problem can also be recast as a problem in the real…
For a given triangle $\triangle ABC$, we define two sequences of points on line $BC$ and provide their generalizations to real functions such that centers of circumscribed circles around $A$ and adjacent points in subsequences generate a…
We formulate a theory of shape valid for objects of arbitrary dimension whose contours are path connected. We apply this theory to the design and modeling of viable trajectories of complex dynamical systems. Infinite families of…
In this paper we will develop an axiomatic foundation for the geometric study of straight edge, protractor, and compass constructions, which while being related to previous foundations, will be the first to have all axioms written and all…
We give geometric ways to determine the cross ratio of conics in a pencil using elementary projective geometry.
The field of structural engineering is vast, spanning areas from the design of new infrastructure to the assessment of existing infrastructure. From the onset, traditional entry-level university courses teach students to analyse structural…
Typically, when we are given the section (or projection) function of a convex body, it means that in each direction we know the size of the central section (or projection) perpendicular to this direction. Suppose now that we can only get…
In this paper, we introduce discrete conics, polygonal analogues of conics. We show that discrete conics satisfy a number of nice properties analogous to those of conics, and arise naturally from several constructions, including the…
In this paper we overview the theory of conics and roulettes in four non-Euclidean planes. We collect the literature about these classical concepts, from the eighteenth century to the present, including papers available only on arXiv. The…
In this Phd. thesis, a structural analysis of construction schemes is developed. The importance of this study will be justified by constructing several distinct combinatorial objects which have been of great interest in mathematics. We then…
The problem of construction of the surfaces with given sets of the points with horizontal tangential planes is considered. Such considerations are of interest in the problem of computer simulations of the waved ocean surfaces.