Topic

# General position

About: General position is a research topic. Over the lifetime, 1077 publications have been published within this topic receiving 12681 citations.

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TL;DR: If the points are added one by one, then flipping in a topological order will succeed in constructing this triangulation and the algorithm takes expected time at mostO(nlogn+n[d/2]).

Abstract: A set ofn weighted points in general position in źd defines a unique regular triangulation. This paper proves that if the points are added one by one, then flipping in a topological order will succeed in constructing this triangulation. If, in addition, the points are added in a random sequence and the history of the flips is used for locating the next point, then the algorithm takes expected time at mostO(nlogn+n[d/2]). Under the assumption that the points and weights are independently and identically distributed, the expected running time is between proportional to and a factor logn more than the expected size of the regular triangulation. The expectation is over choosing the points and over independent coin-flips performed by the algorithm.

341 citations

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TL;DR: In this paper, the local Langlands correspondence for pure inner forms of unramied p-adic groups and tame Langlands parameters in general position has been verified, and it is shown that the appropriate sum of characters of the representations in the L-packet is stable.

Abstract: In this paper we verify the local Langlands correspondence for pure inner forms of unramied p-adic groups and tame Langlands parameters in \general position". For each such parameter, we explicitly construct, in a natural way, a nite set (\ L-packet") of depth-zero supercuspidal representations of the appropriate p-adic group, and we verify some expected properties of this L-packet. In particular, we prove, with some conditions on the base eld, that the appropriate sum of characters of the representations in our L-packet is stable; no proper subset of our L-packets can form a stable combination. Our L-packets are also consistent with the conjectures of B. Gross and D. Prasad on restriction from SO2n+1 to SO2n [24]. These L-packets are, in general, quite large. For example, Sp2n has an L-packet containing 2 n representations, of which exactly two are generic. In fact, on a quasi-split form, eachL-packet contains exactly one generic representation for every rational orbit of hyperspecial vertices in the reduced BruhatTits building. When the group has connected center, every depth-zero generic supercuspidal representation appears in one of these L-packets. We emphasize that there is nothing new about the representations we construct. They are induced from Deligne-Lusztig representations on subgroups of nite

172 citations

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01 Jul 1992TL;DR: If the points are added one by one, then flipping in a topological order will succeed in constructing this triangulation and the algorithm takes expected time at mostO(nlogn+n[d/2]).

Abstract: A set ofn weighted points in general position in ℝ d defines a unique regular triangulation. This paper proves that if the points are added one by one, then flipping in a topological order will succeed in constructing this triangulation. If, in addition, the points are added in a random sequence and the history of the flips is used for locating the next point, then the algorithm takes expected time at mostO(nlogn+n[d/2]). Under the assumption that the points and weights are independently and identically distributed, the expected running time is between proportional to and a factor logn more than the expected size of the regular triangulation. The expectation is over choosing the points and over independent coin-flips performed by the algorithm.

171 citations

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TL;DR: It is proved that any triangulation of a set of n points in general position contains at least $\lceil (n-4)/2 \rceil$ edges that can be flipped.

Abstract: In this paper we study the problem of flipping edges in triangulations of polygons and point sets. One of the main results is that any triangulation of a set of n points in general position contains at least $\lceil (n-4)/2 \rceil$ edges that can be flipped. We also prove that O(n + k
2
) flips are sufficient to transform any triangulation of an n -gon with k reflex vertices into any other triangulation. We produce examples of n -gons with triangulations T and T' such that to transform T into T' requires Ω(n
2
) flips. Finally we show that if a set of n points has k convex layers, then any triangulation of the point set can be transformed into any other triangulation using at most O(kn) flips.

165 citations

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TL;DR: It is established that the minimum number of images for computing these invariants is three, and the computation of invariants of six points from three images can have as many as three solutions.

Abstract: There are three projective invariants of a set of six points in general position in space. It is well known that these invariants cannot be recovered from one image, however an invariant relationship does exist between space invariants and image invariants. This invariant relationship is first derived for a single image. Then this invariant relationship is used to derive the space invariants, when multiple images are available. This paper establishes that the minimum number of images for computing these invariants is three, and the computation of invariants of six points from three images can have as many as three solutions. Algorithms are presented for computing these invariants in closed form. The accuracy and stability with respect to image noise, selection of the triplets of images and distance between viewing positions are studied both through real and simulated images. Applications of these invariants are also presented. Both the results of Faugeras (1992) and Hartley et al. (1992) for projective reconstruction and Sturm's method (1869) for epipolar geometry determination from two uncalibrated images with at least seven points are extended to the case of three uncalibrated images with only six points. >

144 citations