scholarly journals The proximities of asteroids and critical points of the distance function

2010 ◽  
pp. 91-102 ◽  
Author(s):  
Slavisa Milisavljevic
Keyword(s):  
Solar System ◽  
Distance Function ◽  
Critical Points ◽  
Efficient Method ◽  
Polynomial Solutions ◽  
Elliptical Orbits

The proximities are important for different purposes, for example to evaluate the risk of collisions of asteroids or comets with the Solar-System planets. We describe a simple and efficient method for finding the asteroid proximities in the case of elliptical orbits with a common focus. In several examples we have compared our method with the recent excellent algebraic and polynomial solutions of Gronchi (2002, 2005).

scholarly journals Tracking the critical points of curves evolving under planar curvature flows

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Eszter Fehér ◽  
Gábor Domokos ◽  
Bernd Krauskopf
Keyword(s):  
Distance Function ◽  
Critical Points ◽  
Broad Class ◽  
Computational Cost ◽  
Radial Distance ◽  
Test Case ◽  
Case Examples ◽  
Planar Curve ◽  
Special Cases ◽  
Two Parameters

<p style='text-indent:20px;'>We are concerned with the evolution of planar, star-like curves and associated shapes under a broad class of curvature-driven geometric flows, which we refer to as the Andrews-Bloore flow. This family of flows has two parameters that control one constant and one curvature-dependent component for the velocity in the direction of the normal to the curve. The Andrews-Bloore flow includes as special cases the well known Eikonal, curve-shortening and affine shortening flows, and for positive parameter values its evolution shrinks the area enclosed by the curve to zero in finite time. A question of key interest has been how various shape descriptors of the evolving shape behave as this limit is approached. Star-like curves (which include convex curves) can be represented by a periodic scalar polar distance function <inline-formula><tex-math id="M1">\begin{document}$ r(\varphi) $\end{document}</tex-math></inline-formula> measured from a reference point, which may or may not be fixed. An important question is how the numbers and the trajectories of critical points of the distance function <inline-formula><tex-math id="M2">\begin{document}$ r(\varphi) $\end{document}</tex-math></inline-formula> and of the curvature <inline-formula><tex-math id="M3">\begin{document}$ \kappa(\varphi) $\end{document}</tex-math></inline-formula> (characterized by <inline-formula><tex-math id="M4">\begin{document}$ dr/d\varphi = 0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ d\kappa /d\varphi = 0 $\end{document}</tex-math></inline-formula>, respectively) evolve under the Andrews-Bloore flows for different choices of the parameters.</p><p style='text-indent:20px;'>We present a numerical method that is specifically designed to meet the challenge of computing accurate trajectories of the critical points of an evolving curve up to the vicinity of a limiting shape. Each curve is represented by a piecewise polynomial periodic radial distance function, as determined by a chosen mesh; different types of meshes and mesh adaptation can be chosen to ensure a good balance between accuracy and computational cost. As we demonstrate with test-case examples and two longer case studies, our method allows one to perform numerical investigations into subtle questions of planar curve evolution. More specifically — in the spirit of experimental mathematics — we provide illustrations of some known results, numerical evidence for two stated conjectures, as well as new insights and observations regarding the limits of shapes and their critical points.</p>

On the Distance Between two Ellipses in \(\mathbb{R}^3\)

2020 ◽  
Vol 57 ◽  
pp. 111-122
Author(s):  
Ivaylo Tounchev ◽  
Keyword(s):  
Distance Function ◽  
Critical Points ◽  
Global Minimum ◽  
Saddle Points ◽  
Global Maximum

We prove that in the general case the number of critical points of the distance function between two ellipses in \(\mathbb{R}^3\) equals to 4, 6, 8, 10, 12, 14 or 16. As an example, the distance between the nowadays orbits of Neptune and Pluto has six critical points: one maximum two minima and three saddle points. The global minimum is 2.503 au (astronomical units), while the global maximum is 79.111 au. If we ignore the perturbations, then in year 21103 AD the distance between Neptune and Pluto would be 2.527 au.

XV. Account of the changes that have happened, during the last twenty-five years, in the relative situation of double-stars; with an investigation of the cause to which they are owing

1803 ◽  
Vol 93 ◽  
pp. 339-382 ◽  
Keyword(s):  
Solar System ◽  
The Other ◽  
The Sun ◽  
Double Stars ◽  
Celestial Bodies ◽  
Planets And Satellites ◽  
The Subject ◽  
Elliptical Orbits ◽  
Asteroids And Comets

In the Remarks on the Construction of the Heavens, contained in my last Paper on this subject, I have divided the various objects which astronomy has hitherto brought to our view, into twelve classes. The first comprehends insulated stars. As the solar system presents us with all the particulars that may be known, respecting the arrangement of the various su­bordinate celestial bodies that are under the influence of stars which I have called insulated, such as planets and satellites, asteroids and comets, I shall here say but little on that subject. It will, however, not be amiss to remark, that the late addition of two new celestial bodies, has undoubtedly enlarged our know­ledge of the construction of the system of insulated stars. Whatever may be the nature of these two new bodies, we know that they move in regular elliptical orbits round the sun. It is not in the least material whether we call them asteroids, as I have proposed; or planetoids, as an eminent astronomer, in a letter to me, suggested; or whether we admit them at once into the class of our old seven large planets. In the latter case, however, we must recollect, that if we would speak with precision, they should be called very small, and exzodiacal; for, the great inclination of the orbit of one of them to the ecliptic, amounting to 35 degrees, is certainly remarkable. That of the other is also considerable; its latitude, the last time I saw it, being more than 15 degrees north. These circumstances, added to their smallness, show that there exists a greater variety of arrange­ment and size among the bodies which our sun holds in subor­dination, than we had formerly been acquainted with, and extend our knowledge of the construction of the solar, or insulated sidereal system. It will not be required that I should add any thing farther on the subject of this first article of my clas­sification; I may therefore immediately go to the second, which treats of binary sidereal systems, or real double stars.

CRITICAL POINTS OF DISTANCE TO AN ε-SAMPLING OF A SURFACE AND FLOW-COMPLEX-BASED SURFACE RECONSTRUCTION

2008 ◽  
Vol 18 (01n02) ◽  
pp. 29-61 ◽  
Author(s):  
TAMAL K. DEY ◽  
JOACHIM GIESEN ◽  
EDGAR A. RAMOS ◽  
BARDIA SADRI
Keyword(s):  
Surface Reconstruction ◽  
Distance Function ◽  
Critical Points ◽  
Geometric Modeling ◽  
Medial Axis ◽  
Reconstruction Algorithm ◽  
Three Dimensions ◽  
Distance Functions ◽  
Curve Reconstruction ◽  
Discrete Sample

The distance function to surfaces in three dimensions plays a key role in many geometric modeling applications such as medial axis approximations, surface reconstructions, offset computations and feature extractions among others. In many cases, the distance function induced by the surface can be approximated by the distance function induced by a discrete sample of the surface. The critical points of the distance functions are known to be closely related to the topology of the sets inducing them. However, no earlier theoretical result has found a link between topological properties of a geometric object and critical points of the distance to a discrete sample of it. We provide this link by showing that the critical points of the distance function induced by a discrete sample of a surface fall into two disjoint classes: those that lie very close to the surface and those that are near its medial axis. This closeness is precisely quantified and is shown to depend on the sampling density. It turns out that critical points near the medial axis can be used to extract topological information about the sampled surface. Based on this, we provide a new flow-complex-based surface reconstruction algorithm that, given a tight ε-sample of a surface, approximates the surface geometrically, both in distance and normals, and captures its topology. Furthermore, we show that the same algorithm can be used for curve reconstruction.

scholarly journals Distance between Hermitian operators in Schatten classes

1996 ◽  
Vol 39 (2) ◽  
pp. 377-380 ◽  
Author(s):  
Rajendra Bhatia ◽  
Peter Šemrl
Keyword(s):  
Distance Function ◽  
Critical Points ◽  
Hermitian Operator ◽  
Schatten Classes ◽  
Perturbation Bound ◽  
Class 1 ◽  
Unitary Orbit

We consider the distance between a fixed Hermitian operator B and the unitary orbit of another Hermitian operator A and show that in each Schatten p-class, 1<p<∞, critical points of this distance function are at operators commuting with B. As a consequence we obtain a perturbation bound for the eigenvalues of Hermitian operators in these Schatten classes.

Inscribed rectangle coincidences

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Richard Evan Schwartz
Keyword(s):  
Distance Function ◽  
Critical Points ◽  
Integral Formula ◽  
Piecewise Smooth ◽  
Total Multiplicity

Abstract We prove an integral formula for continuous paths of rectangles inscribed in a piecewise smooth loop. We use this integral formula to prove the inequality M(γ) ≥ Δ(γ)/2 – 1, where M(γ) denotes the total multiplicity of rectangle coincidences, i.e. pairs, triples, etc. of isometric rectangles inscribed in γ, and Δ(γ) denotes the number of stable diameters of γ, i.e. critical points of the distance function on γ.

scholarly journals Collisional Focusing of Particles in Space Causing Jetstreams

1971 ◽  
Vol 12 ◽  
pp. 327-335 ◽  
Author(s):  
Jan Trulsen
Keyword(s):  
Mass Spectrum ◽  
Solar System ◽  
Kinetic Energy ◽  
Internal Structure ◽  
Impact Velocity ◽  
Intermediate Stage ◽  
Collision Processes ◽  
Elliptical Orbits

Jetstreams probably played an important role at an intermediate stage of the formation of the solar system (Alfvén and Arrhenius, 1970). A Jetstream is defined here as a collection of grains moving in neighboring elliptical orbits around a central gravitating body and with the dynamics modified by the action of complicated collision processes among the grains themselves.Three main types of collisions will take place in such a stream. Hyper-velocity impacts will lead to fragmentation of the grains involved. At lower impact velocities, the particles will retain their identities after the collision even if they might be deformed to some degree depending on impact velocity and internal structure of the grains. With still lower impact velocity, accretion can take place, the grains sticking together after collision to form larger grains. A common feature of these collision processes are that they will be partially inelastic. A certain fraction of the kinetic energy of the colliding particles will be spent on changing their internal structure. The internal kinetic energy in an isolated Jetstream thus will tend to decrease with time. The mass spectrum of the grains also will vary during the lifetime of a stream, the probability for accretive processes increasing with time.

7. Moons of small bodies

2015 ◽  
pp. 123-136
Author(s):  
David A. Rothery
Keyword(s):  
Solar System ◽  
The Sun ◽  
Small Bodies ◽  
Icy Bodies ◽  
Elliptical Orbits

The small bodies of our Solar System consist of: asteroids (rocky or carbonaceous objects that are mainly concentrated in the space between the orbits of Mars and Jupiter); trans-Neptunian objects (icy bodies beyond Neptune’s orbit); comets (small icy bodies with strongly elliptical orbits that can come close to the Sun); and centaurs (asteroid-like bodies of dominantly icy rather than rocky composition, whose orbits lie beyond that of Jupiter, but inside Neptune’s). Of these, only comets are devoid of known moons. ‘Moons of small bodies’ describes some of the 165 asteroids known to have moons, as well as the moons of centaurs and trans-Neptunian objects, including the dwarf planet Pluto.

scholarly journals THE EULER NUMBER FROM THE DISTANCE FUNCTION

2013 ◽  
Vol 32 (3) ◽  
pp. 175 ◽  
Author(s):  
Ximo Gual-Arnau
Keyword(s):  
Distance Function ◽  
Critical Points ◽  
Euler Number ◽  
New Method ◽  
Concentric Spheres ◽  
Hopf Theorem

We present a new method to obtain the Euler number of a domain based on the tangent counts of concentric spheres in ℝ³ (or circles in ℝ², with respect to the center O, that sweeps the domain. This method is derived from the Poincaré-Hopf Theorem, when the index of critical points of the square of the distance function with respect to the origin O are considered.

Sign in / Sign up

Export Citation Format

Share Document