On the geometry of the smallest circle enclosing a finite set of points (original) (raw)
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Tribute. We would like to dedicate this paper to Elijah Polak. Professor Polak has made substantial contributions to a truly broad spectrum of topics in nonlinear optimization, including optimization for engineering design centering, multi-criteria optimization, optimal control, feasible directions methods, quasi-Newton and Newton methods, non-differential optimization, semi-infinite optimization, conjugate directions methods, gradient projection and reduced gradient methods, and barrier methods, among many other topics. His many and varied contributions to our field are important today and will influence the research in our field well into the future.
Fitting a set of points by a circle Jesus Garca-Lopez y
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Given a set of points S = {p 1 ,. .. , p n } in Euclidean d-dimensional space, we address the problem of computing the d-dimensional annulus of smallest width containing the set. We give a complete characterization of the centers of annuli which are locally minimal in arbitrary dimension and we show that, for d = 2, a locally minimal annulus has two points on the inner circle and two points on the outer circle that interlace anglewise as seen from the center of the annulus. Using this characterization, we show that, given a circular order of the points, there is at most one locally minimal annulus consistent with that order and it can be computed in time O(n log n) using a simple algorithm. Furthermore, when points are in convex position, the problem can be solved in optimal (n) time.
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The smallest enclosing circle is a well-known problem. In this paper, we propose modifications to speed-up the existing Weltzl’s algorithm. We perform the preprocessing to reduce as many input points as possible. The reduction step has lower computational complexity than the Weltzl’s algorithm and thus speed-ups its computation. Next, we propose some changes to Weltzl’s algorithm. In the end are summarized results, that show the speed-up for 106{10^{6}}106 input points up to 100 times compared to the original Weltzl’s algorithm. Even more, the proposed algorithm is capable to process significantly larger data sets than the standard Weltzl’s algorithm.
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Given a set P with n points in R li, its diameter d, is the maximum of the Euclidean distances between its points. We describe an algorithm that in m < n iterations obtains r, < rs < . . < r,,, < d,, < min ( fir,, d-r,,, ). For k fixed, the cost of each iteration is O(n). In particular, the first approximation r, is within fi of dp, independent of the dimension k.
2010
Given DeltaABC\Delta ABCDeltaABC and angles alpha,beta,gammain(0,pi)\alpha,\beta,\gamma\in(0,\pi)alpha,beta,gammain(0,pi) with alpha+beta+gamma=pi\alpha+\beta+\gamma=\pialpha+beta+gamma=pi, we study the properties of the triangle DEFDEFDEF which satisfies: (i) DinBCD\in BCDinBC, EinACE\in ACEinAC, FinABF\in ABFinAB, (ii) aangleD=alpha\aangle D=\alphaaangleD=alpha, aangleE=beta\aangle E=\betaaangleE=beta, aangleF=gamma\aangle F=\gammaaangleF=gamma, (iii) DeltaDEF\Delta DEFDeltaDEF has the minimal area in the class of triangles satisfying (i) and (ii). In particular, we show that minimizer DeltaDEF\Delta DEFDeltaDEF, exists, is unique and is a pedal triangle, corresponding to a certain pedal point PPP. Permuting the roles played by the angles alpha,beta,gamma\alpha,\beta,\gammaalpha,beta,gamma in (ii), yields a total of six such area-minimizing triangles, which are pedal relative to six pedal points, say, P1,....,P6P_1,....,P_6P_1,....,P_6. The main result of the paper is the fact that there exists a circle which contains all six points.