A massive particle under the influence of a constant gravitational force that is bouncing inside an ideal reflecting mirror described by some function f(x) is considered. For the associated flight ...trajectories we derive the parametric curves, named foci curves. All foci points of the parabolas for a given initial position and energy lie on these curves. From these foci curves the associated flight parabola envelopes are derived resulting, together with the mirror surface, in a confined domain for all possible particle trajectories in the non-periodic orbit case. The general results are briefly discussed and visualized for three concrete mirror surfaces.
•Foci curves of flight parabolas.•Confined domains.•Application to various examples.
Introduction: In this work, we use simulated data to quantify the different failure mechanisms of a previously presented low-cost jump height measurement system, based on widely available consumer ...smartphone technology. Methods: In order to assess the importance of the different preconditions of the jump height measurement algorithm, we generate a synthetic dataset of 2000 random jump parabolas for 2000 randomly generated persons without real-world artifacts. We then selectively add different perturbations to the parabolas and reconstruct the jump height using the evaluated algorithm. The degree to which the manipulations influence the reconstructed jump height gives us insights into how critical each precondition is for the method’s accuracy. Results: For a subject-to-camera distance of 2.5 meters, we found the most important influences to be tracking inaccuracies and distance changes (non-vertical jumps). These are also the most difficult factors to control. Camera angle and lens distortion are easier to handle in practice and have a very low impact on the reconstructed jump height. The intraclass correlation value ICC(3,1) between true jump height and the reconstruction from distorted data ranges between 0.999 for mild and 0.988 for more severe distortions. Conclusion: Our results support the design of future studies and tools for accurate and affordable jump height measurement, which can be used in individual fitness, sports medicine, and rehabilitation applications.
Currently, there is a significant growth of interest in the practical problems of mathematically modeling the placement of geometric objects of various physical natures in given areas. When solving ...such problems, there is a need to build their mathematical models, which are implemented through the construction of analytical conditions for the relations of the objects being placed and placement regions. The problem of constructing conditions for the mutual non-intersection of arbitrarily oriented objects whose boundaries are formed by second-order curves is widely used in practice and, at the same time, much less studied than a similar problem for simpler objects. A fruitful and worked out method of representing such conditions is the construction of Stoyan’s Φ-functions (further referred to as phi-functions) and quasi-phi-functions. In this article, considered as geometric objects are an ellipse and a parabola-bounded region. The boundaries of the objects under study allow both implicit and parametric representations. The proposed approach to modeling the geometric relationships of ellipses and parabola-bounded regions is based on coordinate transformation, reduction of an ellipse equation to a circle equation with the use of a canonical transformation. In particular, constructed are the conditions for the inclusion of an ellipse in a parabola-bounded region, as well as the conditions for their mutual non-intersection. The conditions for the relationships between the geometric objects under study are constructed on the basis of the canonical equations of the ellipse and parabola, taking into account their placement parameters, including rotations. These conditions are presented in the form of a system of inequalities, as well as in the form of a single analytical expression. The presented conditions can be used in constructing adequate mathematical models of optimization problems of placing corresponding geometric objects for an analytical description of feasible regions. These models can be used further in the formulation of mathematical models of packing and cutting problems, expanding the range of objects and / or increasing solution accuracy and decreasing time to solution.
The quintics are the lowest–order planar Pythagorean–hodograph (PH) curves suitable for free–form design, since they can exhibit inflections. A quintic PH curve r(t) may be constructed from a complex ...quadratic pre–image polynomial w(t) by integration of r′(t)=w2(t), and it thus incorporates (modulo translations) six real parameters — the real and imaginary parts of the coefficients of w(t). Within this 6–dimensional space of planar PH quintics, a 5–dimensional hypersurface separates the inflectional and non–inflectional curves. Points of the hypersurface identify exceptional curves that possess a tangent–continuous point of infinite curvature, corresponding to the fact that the parabolic locus specified by the quadratic pre–image polynomial w(t) passes through the origin of the complex plane. Correspondingly, extreme curvatures and tight loops on r(t) are incurred by a close proximity of w(t) to the origin of the complex plane. These observations provide useful insight into the disparate shapes of the four distinct PH quintic solutions to the first–order Hermite interpolation problem.
Left: The four canonical–form quintic PH curves r(t) determined by the two end derivatives di=2.0+1.5i, df=1.5–4.5i. Right: the corresponding pre–image polynomials w(t), showing the footpoints of the origin on them.
•The presence or absence of inflections on planar quintic PH curves is characterized.•The space of planar PH quintics is partitioned by curves that possess points of infinite curvature.•Proximity of the pre-image parabola to the origin of the hodograph plane incurs tight loops.•These observations provide insight into the behavior of planar PH quintic Hermite interpolants.
The primary fuel used for cooking in most of the hostels in India is LPG which costs around 700-1500 rupees per cylinder. The temperature requirement for most of the meals to be prepared is in the ...range of 60-100°C, and majorly all of them require heating in the form of boiled water or steam approximately ≥ (100°C) for cooking. In the present case study, a sample of 1000 students is taken, consuming meals of two times. The integration of conventional LPG system with parabolic trough solar collector is studied and detailed cost analysis is carried out. Based on the above case study, the parabolic trough solar collector integrated system with LPG leads to a reduction in 15 tonnes of CO
2
/year with a suggestable payback period of 2 years.
Purpose:
The purpose of this study is the correction of the lateral scanner artifact, i.e., the effect that, on a large homogeneously exposed EBT3 film, a flatbed scanner measures different optical ...densities at different positions along thex axis, the axis parallel to the elongated light source. At constant dose, the measured optical densitiy profiles along this axis have a parabolic shape with significant dose dependent curvature. Therefore, the effect is shortly called the parabola effect. The objective of the algorithm developed in this study is to correct for the parabola effect. Any optical density measured at given position x is transformed into the equivalent optical density c at the apex of the parabola and then converted into the corresponding dose via the calibration of c versus dose.
Methods:
For the present study EBT3 films and an Epson 10000XL scanner including transparency unit were used for the analysis of the parabola effect. The films were irradiated with 6 MV photons from an Elekta Synergy accelerator in a RW3 slab phantom. In order to quantify the effect, ten film pieces with doses graded from 0 to 20.9 Gy were sequentially scanned at eight positions along thex axis and at six positions along the z axis (the movement direction of the light source) both for the portrait and landscape film orientations. In order to test the effectiveness of the new correction algorithm, the dose profiles of an open square field and an IMRT plan were measured by EBT3 films and compared with ionization chamber and ionization chamber array measurement.
Results:
The parabola effect has been numerically studied over the whole measuring field of the Epson 10000XL scanner for doses up to 20.9 Gy and for both film orientations. The presented algorithm transforms any optical density at positionx into the equivalent optical density that would be measured at the same dose at the apex of the parabola. This correction method has been validated up to doses of 5.2 Gy all over the scanner bed with 2D dose distributions of an open square photon field and an IMRT distribution.
Conclusions:
The algorithm presented in this study quantifies and corrects the parabola effect of EBT3 films scanned in commonly used commercial flatbed scanners at doses up to 5.2 Gy. It is easy to implement, and no additional work steps are necessary in daily routine film dosimetry.
L’anello che manca. E quello che non torna in Nathans Tod di Georg Tabori propose a modern rereading of Lessing’s play Nathan the Wise (Nathan der Weise): Georg Tabori’s Nathans Death (Nathans Tod). ...The paper aims for a comparative analysis of the famous ring parable. Asked by the Sultan to decide which of the three religions owns the truth, Nathan narrates the fable of a magic ring that has been passed from generation to generation and finally lost, when a father with three sons makes two copies of the original one. The ring symbolises the tyranny of a unique truth that has to be overcome, in order to achieve tolerance. My interest focuses on two moments of Tabori’s rewriting of this fable. On the one hand, I analyse the way in which Tabori glosses over the ring parable in the dialogue between Nathan and Saladin thanks to Horkheimer’s and Adorno’s Dialectic of Enlightement (Dialektik der Aufklärung) and to Arendt’s Man in Dark Times (Von der Menschlichkeit in finsteren Zeiten). In the economic and social paradigm of the modern Totalitarianism there is no more place for tolerance. On the other hand, I reflect on the new context in which Tabori places the ring fable narrated by Nathan just before his death in front of the bodies of his dead sons. I consider that choice to be a parody: The contrast between the original context of enunciation and the modern one is the way in which Lessing’s tolerance is finally checkmated.
Consider the class ${\bf QS}$ of all non-degenerate planar quadratic differential systems and its subclass ${\bf QSP}$ of all its systems possessing an invariant parabola. This is an interesting ...family because on one side it is defined by an algebraic geometric property and on the other, it is a family where limit cycles occur. Note that each quadratic differential system can be identified with a point of ${\mathbb R}^{12}$ through its coefficients. In this paper, we provide necessary and sufficient conditions for a system in ${\bf QS}$ to have at least one invariant parabola. We give the global “bifurcation” diagram of the family ${\bf QS}$ which indicates where a parabola is present or absent and in case it is present, the diagram indicates how many parabolas there could be, their reciprocal position and what kind of singular points at infinity (simple or multiple) as well as their multiplicities are the points at infinity of the parabolas. The diagram is expressed in terms of affine invariant polynomials and it is done in the 12-dimensional space of parameters.
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