Distance Between Two Keplerian Orbits

In this paper, constrained minimization for the point of closest approach of two conic sections is developed. For this development, we considered the nine cases of possible conics, namely, (elliptic–elliptic), (elliptic–parabolic), (elliptic–hyperbolic), (parabolic–elliptic), (parabolic–parabolic), (parabolic–hyperbolic), (hyperbolic–elliptic), (hyperbolic–parabolic), and (hyperbolic–hyperbolic). The developments are considered from two points of view, namely, analytical and computational. For the analytical developments, the literal expression of the minimum distance equation (S) and the constraint equation (G), including the first and second derivatives for each case, are established. For the computational developments, we construct an efficient algorithm for calculating the minimum distance by using the Lagrange multiplier method under the constraint on time. Finally, we compute the closest distance S between two conics for some orbits. The accuracy of the solutions was checked under the conditions that L| solution ≤ ɛ1; G| solution ≤ ɛ2, where ɛ1,2 < 10−10. For the cases of (parabolic–parabolic), (parabolic–hyperbolic), and (hyperbolic–hyperbolic), we studied thousands of comets, but the condition of the closest approach was not met.

ON THE VISUALIZATION OF POSITIONAL PRECISION

Tasks such as image registration or pose estimation require the determination of transformations based on uncertain observations. Hence, the position of any geometric object transformed according to this estimate is also uncertain, at least in terms of precision. Often the knowledge of uncertainty changes the judgment of individuals. Thus, the visualization of this information is crucial whenever a human decision-maker is involved. In the absence of error-free reference data, we consider the estimated precision as the probably most important quantity characterizing the uncertainty. This contribution focuses on the visualization of positional precision as provided by estimated covariance matrices. Basic design principles such as coloration and contouring in 2D and 3D are presented and discussed in the context of practical applications, e.g., the superimposition of distance information as seen nowadays in sports broadcasts. As a novel contribution, we propose quartic plane curves to represent the confidence regions of the loci of conic sections.

Taking a Cue from Conic Sections

First graders learn about artist Wayne Thiebaud and explore estimation, counting, and probability while creating an artistic masterpiece.

Pedal equation and Kepler kinematics

Kepler's laws is an appropriate topic which brings out the significance of pedal equation in Physics. There are several articles which obtain the Kepler's laws as a consequence of the conservation and gravitation laws. This can be shown more easily and ingeniously if one uses the pedal equation of an Ellipse. In fact the complete kinematics of a particle in a attractive central force field can be derived from one single pedal form. Though many articles use the pedal equation, only in few the classical procedure (without proof) for obtaining the pedal equation is mentioned. The reason being the classical derivations can sometimes be lengthier and also not simple. In this paper using elementary physics we derive the pedal equation for all conic sections in an unique, short and pedagogical way. Later from the dynamics of a particle in the attractive central force field we deduce the single pedal form, which elegantly describes all the possible trajectories. Also for the purpose of completion we derive the Kepler's laws.

Operationalism: An Interpretation of the Philosophy of Ancient Greek Geometry

I present a systematic interpretation of the foundational purpose of constructions in ancient Greek geometry. I argue that Greek geometers were committed to an operationalist foundational program, according to which all of mathematics—including its entire ontology and epistemology—is based entirely on concrete physical constructions. On this reading, key foundational aspects of Greek geometry are analogous to core tenets of 20th-century operationalist/positivist/constructivist/intuitionist philosophy of science and mathematics. Operationalism provides coherent answers to a range of traditional philosophical problems regarding classical mathematics, such as the epistemic warrant and generality of diagrammatic reasoning, superposition, and the relation between constructivism and proof by contradiction. Alleged logical flaws in Euclid (implicit diagrammatic reasoning, superposition) can be interpreted as sound operationalist reasoning. Operationalism also provides a compelling philosophical motivation for the otherwise inexplicable Greek obsession with cube duplication, angle trisection, and circle quadrature. Operationalism makes coherent sense of numerous specific choices made in this tradition, and suggests new interpretations of several solutions to these problems. In particular, I argue that: Archytas’s cube duplication was originally a single-motion machine; Diocles’s cissoid was originally traced by a linkage device; Greek conic section theory was thoroughly constructive, based on the conic compass; in a few cases, string-based constructions of conic sections were used instead; pointwise constructions of curves were rejected in foundational contexts by Greek mathematicians, with good reason. Operationalism enables us to view the classical geometrical tradition as a more unified and philosophically aware enterprise than has hitherto been recognised.

Some mathematical appreciations in the physical sciences

According to reports in the media, there is a dearth of practical examples that students of mathematics en route to their qualification can feast upon, at either sixth form level or an undergraduate level. Despite these alleged shortages, it is this author’s opinion that there are numerous examples that can be drawn from the literature and it is the purpose of this article to address the issue by providing examples from the realms of antenna reflector theory and the use therein of conic sections. Some students will be familiar with conic sections and others might not, but the numerous instances of their manifestation in the real world would suggest that they are a force to be reckoned with, and this is certainly true from a mathematical perspective.

A geometric study of different curves for the rotor profiles of a twin-screw compressor

This paper presents the results of comparisons among some patented solutions for profiling the contours of the rotors in twin-screw compressors. Referring to a base case where all the generating curves are circumferences, patents suggesting to replace arcs of circumference with arcs of conic sections, i.e. parabola, ellipse and hyperbola, but even a straight line segment, are presented and guidelines for rotor profile construction are reported. After setting the size of the compressor, attention is paid to the inter-lobe area, as the sum of the area between two consecutive lobes in the male rotor and of the area of the groove in the female rotor. Actually, this area is strictly related to the volume displacement. Limited to the current case study, the profile including an elliptic segment seems to be the preferable solution for higher inter-lobe area, then for higher displacement, though a number of considerations should be necessary for a broader context.