Angle Measures and Bisectors in Minkowski Planes

2005 ◽  
Vol 48 (4) ◽  
pp. 523-534 ◽  
Author(s):  
Nico Düvelmeyer
Keyword(s):  
Unit Circle ◽  
Minkowski Plane ◽  
Arc Length ◽  
Area Measure ◽  
Angular Measure ◽  
Minkowski Planes ◽  
Length Measure

AbstractWe prove that a Minkowski plane is Euclidean if and only if Busemann's or Glogovskij's definitions of angular bisectors coincide with a bisector defined by an angular measure in the sense of Brass. In addition, bisectors defined by the area measure coincide with bisectors defined by the circumference (arc length) measure if and only if the unit circle is an equiframed curve.

scholarly journals A Novel Comprehensive Evaluation Method of Forest State Based on Unit Circle

Forests ◽  
2018 ◽  
Vol 10 (1) ◽  
pp. 5
Author(s):  
Ganggang Zhang ◽  
Gangying Hui ◽  
Gongqiao Zhang ◽  
Yanbo Hu ◽  
Zhonghua Zhao
Keyword(s):  
Forest Management ◽  
Unit Circle ◽  
Comprehensive Evaluation ◽  
Mixed Forest ◽  
Circle Method ◽  
Indicator System ◽  
Gansu Province ◽  
Pinus Koraiensis ◽  
Arc Length ◽  
Closed Graph

Comprehensive evaluation of forest state is the precondition and critical step for forest management. To solve the problem that the radar plot and unit circle only focus on the value of each the evaluation index, this paper proposes a novel method for comprehensively and simultaneously evaluating the functionality and inhomogeneity of forest state based on the modified unit circle method. We evaluated the forest state of the Quercus aliena BL. var. acuteserrata Maxim. ex Wenz. broad-leaved mixed forest in the Xiaolong Mountains Forest Area of Gansu Province and the Pinus koraiensis Sieb. et Zucc. broad-leaved mixed forest in Jilin Province in China. According to the principle of comprehensive, scientific and operability, 10 evaluation indices on forest structure and vitality were selected to construct the evaluation indicator system. Each index was normalized based on the assignment method and ensured to be strictly positive based on reciprocal transformation method. The areas and arc length of the closed graph, formed by connecting every two adjacent indicators, in the radar plot and unit circle were extracted. Based on the isoperimetric theorem (isoperimetric inequality), a comprehensive evaluation model was constructed. Compared with radar chart and unit circle method, each index in the newly proposed unit circle method is represented by an independent sector region, reflecting the contribution of the index to the overall evaluation result. Each index has the same relative importance weight, contributing to the estimation the relative sizes of each aspect of forest state. The unique area and arc length of the closed graph help summarize the overall performance with a global score. The expression effect of improved unit circle has been enhanced, and as an English proverb put it, “A picture is worth a thousand words.” The new proposed method simultaneously evaluates the functionality and inhomogeneity of the forest state and it is a powerful tool for the diagnosis of forest state problems and the decision-making of forest management.

Swept Arc Length Measure of Abrasive Wear

2010 ◽  
Author(s):  
Nessa Johnson ◽  
Michael Hennessey ◽  
John Abraham
Keyword(s):  
Initial Conditions ◽  
Single Point ◽  
Geometric Theory ◽  
Arc Length ◽  
Abrasive Wheel ◽  
Length Measure ◽  
Abrasive Disk ◽  
Part Surface ◽  
Dynamic Simulation Model ◽  
Industry Professionals

A geometric theory is presented to enhance understanding of part wear and lay patterns generated during surface grinding, such as when a part is fed through a machine with a rotating circular abrasive disk. This theory uses the integrated relative speed between the part and the abrasive wheel, the swept arc length, as a wear measure. A dynamic simulation model was created in SIMULINK™ to evaluate the swept arc length computation at a single point, and a MATLAB™ script was used to provide initial conditions, spatially loop through the entire part surface, and evaluate the resultant part wear. Three common machine configurations have been dynamically modeled and evaluated using the swept arc length theory. Additionally, simulation results have been validated through comparison with industry professionals using the swept arc length as a wear measure.

On some arc length properties in Minkowski planes

2015 ◽  
Vol 38 (3) ◽  
pp. 327-339
Author(s):  
M. Baronti ◽  
C. Franchetti
Keyword(s):  
Arc Length ◽  
Minkowski Planes

scholarly journals The Class of (p,q)-spherical Distributions

2017 ◽  
Author(s):  
Wolf-Dieter Richter
Keyword(s):  
Simulation Method ◽  
Probability Measures ◽  
Two Dimensional ◽  
Arc Length ◽  
Length Measure ◽  
Random Events ◽  
Spherical Distributions ◽  
Multivariate Probability Distribution ◽  
Elliptically Contoured ◽  
Stochastic Representations

For evaluating probabilities of arbitrary random events with respect to a given multivariate probability distribution, specific techniques are of great interest. An important two-dimensional high risk limit law is the Gauss-exponential distribution whose probabilities can be dealt with based upon the Gauss-Laplace law. The latter will be considered here as an element of the newly introduced family of (p,q)-spherical distributions. Based upon a suitably defined non-Euclidean arc-length measure on (p,q)-circles we prove geometric and stochastic representations of these distributions and correspondingly distributed random vectors, respectively. These representations allow dealing with the new probability measures similarly like with elliptically contoured distributions and more general homogeneous star-shaped ones. This is demonstrated at hand of a generalization of the Box-Muller simulation method. En passant, we prove an extension of the sector and circle number functions.

scholarly journals An affine-invariant inequality for rational functions and applications in harmonic analysis

2010 ◽  
Vol 53 (3) ◽  
pp. 639-655 ◽  
Author(s):  
Spyridon Dendrinos ◽  
Magali Folch-Gabayet ◽  
James Wright
Keyword(s):  
Harmonic Analysis ◽  
Rational Functions ◽  
Affine Invariant ◽  
Arc Length ◽  
Length Measure

AbstractWe extend an affine-invariant inequality for vector polynomials established by Dendrinos and Wright to general rational functions. As a consequence we obtain sharp universal estimates for various problems in Euclidean harmonic analysis defined with respect to the so-called affine arc-length measure.

scholarly journals Curves in the Lorentz-Minkowski plane with curvature depending on their position

2020 ◽  
Vol 18 (1) ◽  
pp. 749-770
Author(s):  
Ildefonso Castro ◽  
Ildefonso Castro-Infantes ◽  
Jesús Castro-Infantes
Keyword(s):  
Angular Momentum ◽  
Minkowski Plane ◽  
Plane Curve ◽  
Linear Momentum ◽  
Arc Length ◽  
Elementary Functions ◽  
Grim Reaper ◽  
Uniqueness Results ◽  
Elastic Curves ◽  
Euclidean Setting

Abstract Motivated by the classical Euler elastic curves, David A. Singer posed in 1999 the problem of determining a plane curve whose curvature is given in terms of its position. We propound the same question in the Lorentz-Minkowski plane, focusing on spacelike and timelike curves. In this article, we study those curves in {{\mathbb{L}}}^{2} whose curvature depends on the Lorentzian pseudodistance from the origin, and those ones whose curvature depends on the Lorentzian pseudodistance through the horizontal or vertical geodesic to a fixed lightlike geodesic. Making use of the notions of geometric angular momentum (with respect to the origin) and geometric linear momentum (with respect to the fixed lightlike geodesic), respectively, we get two abstract integrability results to determine such curves through quadratures. In this way, we find out several new families of Lorentzian spiral, special elastic and grim-reaper curves whose intrinsic equations are expressed in terms of elementary functions. In addition, we provide uniqueness results for the generatrix curve of the Enneper surface of second kind and for Lorentzian versions of some well-known curves in the Euclidean setting, like the Bernoulli lemniscate, the cardioid, the sinusoidal spirals and some non-degenerate conics. We are able to get arc-length parametrizations of them and they are depicted graphically.

scholarly journals Curves in the Lorentz-Minkowski plane: elasticae, catenaries and grim-reapers

2018 ◽  
Vol 16 (1) ◽  
pp. 747-766 ◽  
Author(s):  
Ildefonso Castro ◽  
Ildefonso Castro-Infantes ◽  
Jesús Castro-Infantes
Keyword(s):  
Minkowski Plane ◽  
Linear Momentum ◽  
Arc Length ◽  
Grim Reaper ◽  
Uniqueness Results ◽  
Elastic Curves

AbstractThis article is motivated by a problem posed by David A. Singer in 1999 and by the classical Euler elastic curves. We study spacelike and timelike curves in the Lorentz-Minkowski plane 𝕃2 whose curvature is expressed in terms of the Lorentzian pseudodistance to fixed geodesics. In this way, we get a complete description of all the elastic curves in 𝕃2 and provide the Lorentzian versions of catenaries and grim-reaper curves. We show several uniqueness results for them in terms of their geometric linear momentum. In addition, we are able to get arc-length parametrizations of all the aforementioned curves and they are depicted graphically.

scholarly journals Steinhardt's inequality in the Minkowski plane

1992 ◽  
Vol 45 (2) ◽  
pp. 261-266 ◽  
Author(s):  
Mostafa Ghandehari
Keyword(s):  
Convex Body ◽  
Unit Circle ◽  
Minkowski Plane ◽  
Plane Convex ◽  
Euclidean Length ◽  
Plane Convex Body

In a Minkowski plane with unit circle E, the product of the positive circumference of a plane convex body K and that of its polar dual is greater than or equal to the square of the Euclidean length of the polar dual of E. Equality holds if and only if K is a Euclidean unit circle.

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