Topic

# Ellipse

About: Ellipse is a research topic. Over the lifetime, 4742 publications have been published within this topic receiving 49208 citations. The topic is also known as: ⬭.

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TL;DR: In this article, the Stokeslet strength density of a rigid body is estimated to be independent of the body shape and is of order μUe, where U is a measure of the undisturbed velocity and e = (log 2l/R0)−1.

Abstract: A rigid body whose length (2l) is large compared with its breadth (represented by R0) is straight but is otherwise of arbitrary shape. It is immersed in fluid whose undisturbed velocity, at the position of the body and relative to it, may be either uniform, corresponding to translational motion of the body, parallel or perpendicular to the body length, or a linear function of distance along the body length, corresponding to an ambient pure straining motion or to rotational motion of the body. Inertia forces are negligible. It is possible to represent the body approximately by a distribution of Stokeslets over a line enclosed by the body; and then the resultant force required to sustain translational motion, the net stresslet strength in a straining motion, and the resultant couple required to sustain rotational motion, can all be calculated. In the first approximation the Stokeslet strength density F(x) is independent of the body shape and is of order μUe, where U is a measure of the undisturbed velocity and e = (log 2l/R0)−1. In higher approximations, F(x) depends on both the body cross-section and the way in which it varies along the length. From an investigation of the ‘inner’ flow field near one section of the body, and the condition that it should join smoothly with the ‘outer’ flow which is determined by the body as a whole, it is found that a given shape and size of the local cross-section is equivalent, in all cases of longitudinal relative motion, to a circle of certain radius, and, in all cases of transverse relative motion, to an ellipse of certain dimensions and orientation. The equivalent circle and the equivalent ellipse may be found from certain boundary-value problems for the harmonic and biharmonic equations respectively. The perimeter usually provides a better measure of the magnitude of the effect of a non-circular shape of a cross-section than its area. Explicit expressions for the various integral force parameters correct to the order of e2 are presented, together with iterative relations which allow their determination to the order of any power of e. For a body which is ‘longitudinally elliptic’ and has uniform cross-sectional shape, the force parameters are given explicitly to the order of any power of e, and, for a cylindrical body, to the order of e3.

965 citations

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23 Jun 1998TL;DR: An algorithm that is able to track a person's head with enough accuracy to automatically control the camera's pan, tilt, and zoom in order to keep the person centered in the field of view at a desired size is presented.

Abstract: An algorithm for tracking a person's head is presented. The head's projection onto the image plane is modeled as an ellipse whose position and size are continually updated by a local search combining the output of a module concentrating on the intensity gradient around the ellipse's perimeter with that of another module focusing on the color histogram of the ellipse's interior. Since these two modules have roughly orthogonal failure modes, they serve to complement one another. The result is a robust, real-time system that is able to track a person's head with enough accuracy to automatically control the camera's pan, tilt, and zoom in order to keep the person centered in the field of view at a desired size. Extensive experimentation shows the algorithm's robustness with respect to full 360-degree out-of-plane rotation, up to 90-degree tilting, severe but brief occlusion, arbitrary camera movement, and multiple moving people in the background.

871 citations

01 Jan 1998

TL;DR: This paper presents a numerically stable non-iterative algorithm for fitting an ellipse to a set of data points based on a least squares minimization which leads to a simple, stable and robust fitting method which can be easily implemented.

Abstract: This paper presents a numerically stable non-iterative algorithm for fitting an ellipse to a set of data points. The approach is based on a least squares minimization and it guarantees an ellipse-specific solution even for scattered or noisy data. The optimal solution is computed directly, no iterations are required. This leads to a simple, stable and robust fitting method which can be easily implemented. The proposed algorithm has no computational ambiguity and it is able to fit more than 100,000 points in a second.

520 citations

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TL;DR: In this paper, a technique is presented for polarization analysis of three-component seismic array data, where the polarization ellipse is computed within sliding time windows by solving the eigenproblem for the covariance matrix.

Abstract: A technique is presented for polarization analysis of three-component seismic array data. The process is applied to a large suite of regional events recorded on the three-component sensors in the NORESS array in southern Norway. Polarization properties of the regional seismic phases P,, S,, and Lg are examined in detail. The analysis technique is based on a time-domain algorithm originally proposed by Flinn (1965). The polarization ellipse is computed within sliding time windows by solving the eigenproblem for the covariance matrix. Various attributes characterizing the particle motions are extracted from the motion ellipse. This technique is extended to multiple three-component sensors in an array configuration by averaging covariance matrices for the different sensors. In this case a 1/M reduction in the estimation variance is obtained (M is the number of sensors), when the noise and local scattering effects are uncorrelated. An important feature of this approach is that the phase velocities of coherent wavefronts across the array are not required to a high degree of accuracy. Significant results of the data analysis are the well-defined polarization of P, and Sn waves across the entire short-period band, the source azimuth estimates obtained from P, and Lg motions, and the distinct polarization for S, and Lg waves allowing these phases to be distinguished in most cases.

502 citations

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TL;DR: In this article, a generalization of surface photometry to the higher-order moments of the line-of-sight velocity distribution of galaxies observed with integral-field spectrographs is presented.

Abstract: We present a generalization of surface photometry to the higher-order moments of the line-of-sight velocity distribution of galaxies observed with integral-field spectrographs. The generalization follows the approach of surface photometry by determining the best-fitting ellipses along which the profiles of the moments can be extracted and analysed by means of harmonic expansion. The assumption for the odd moments (e.g. mean velocity) is that the profile along an ellipse satisfies a simple cosine law. The assumption for the even moments (e.g. velocity dispersion) is that the profile is constant, as it is used in surface photometry. We test the method on a number of model maps and discuss the meaning of the resulting harmonic terms. We apply the method to the kinematic moments of an axisymmetric model elliptical galaxy and probe the influence of noise on the harmonic terms. We also apply the method to SAURON observations of NGC 2549, NGC 2974, NGC 4459 and NGC 4473 where we detect multiple co- and counter-rotating (NGC 2549 and NGC 4473, respectively) components. We find that velocity profiles extracted along ellipses of early-type galaxies are well represented by the simple cosine law (with 2 per cent accuracy), while possible deviations are carried in the fifth harmonic term which is sensitive to the existence of multiple kinematic components, and has some analogy to the shape parameter of photometry. We compare the properties of the kinematic and photometric ellipses and find that they are often very similar, but a study on a larger sample is necessary. Finally, we offer a characterization of the main velocity structures based only on the kinemetric parameters which can be used to quantify the features in velocity maps.

489 citations