scholarly journals Degeneracy locus of critical points of the distance function on a holomorphic foliation

2014 ◽  
Vol 66 (1) ◽  
pp. 123-137
Author(s):  
Toshikazu ITO ◽  
Bruno SCÁRDUA ◽  
Yoshikazu YAMAGISHI
Keyword(s):  
Distance Function ◽  
Critical Points ◽  
Holomorphic Foliation ◽  
Degeneracy Locus

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.

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 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 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 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.

The relationship of the scleractinian corals to the rugose corals

Paleobiology ◽  
1980 ◽  
Vol 6 (02) ◽  
pp. 146-160 ◽  
Author(s):  
William A. Oliver
Keyword(s):  
Critical Points ◽  
Scleractinian Corals ◽  
Convincing Evidence ◽  
Early Triassic ◽  
Rugose Corals ◽  
History Of ◽  
The Subject ◽  
Relationship Of ◽  
The Relationship ◽  
Biologic Factors

The Mesozoic-Cenozoic coral Order Scleractinia has been suggested to have originated or evolved (1) by direct descent from the Paleozoic Order Rugosa or (2) by the development of a skeleton in members of one of the anemone groups that probably have existed throughout Phanerozoic time. In spite of much work on the subject, advocates of the direct descent hypothesis have failed to find convincing evidence of this relationship. Critical points are:(1) Rugosan septal insertion is serial; Scleractinian insertion is cyclic; no intermediate stages have been demonstrated. Apparent intermediates are Scleractinia having bilateral cyclic insertion or teratological Rugosa.(2) There is convincing evidence that the skeletons of many Rugosa were calcitic and none are known to be or to have been aragonitic. In contrast, the skeletons of all living Scleractinia are aragonitic and there is evidence that fossil Scleractinia were aragonitic also. The mineralogic difference is almost certainly due to intrinsic biologic factors.(3) No early Triassic corals of either group are known. This fact is not compelling (by itself) but is important in connection with points 1 and 2, because, given direct descent, both changes took place during this only stage in the history of the two groups in which there are no known corals.

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