Lines, Circles, Planes and Spheres

Purdy, George B.; Smith, Justin W.

doi:10.1007/s00454-010-9270-3

Cited by 6 publications

(6 citation statements)

References 18 publications

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“…Purdy and Smith [21] considered instead finite non-coplanar point sets in 3-space with no three points collinear, and provided a lower bound on the number of planes containing exactly three points of the set. Referring to such a plane as an ordinary plane, Ball [1] proved a 3-dimensional analogue of Green and Tao's [8] structure theorem, and found the exact minimum number of ordinary planes spanned by sufficiently large non-coplanar point sets in real projective 3-space with no three points collinear.…”

Section: Introductionmentioning

confidence: 99%

On Sets Defining Few Ordinary Circles

Lin

Makhul

Mojarrad

et al. 2017

Discrete Comput Geom

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An ordinary circle of a set P of n points in the plane is defined as a circle that contains exactly three points of P. We show that if P is not contained in a line or a circle, then P spans at least ordinary circles. Moreover, we determine the exact minimum number of ordinary circles for all sufficiently large n and describe all point sets that come close to this minimum. We also consider the circle variant of the orchard problem. We prove that P spans at most circles passing through exactly four points of P. Here we determine the exact maximum and the extremal configurations for all sufficiently large n. These results are based on the following structure theorem. If n is sufficiently large depending on K, and P is a set of n points spanning at most ordinary circles, then all but O(K) points of P lie on an algebraic curve of degree at most four. Our proofs rely on a recent result of Green and Tao on ordinary lines, combined with circular inversion and some classical results regarding algebraic curves.

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Section: Introductionmentioning

confidence: 99%

On Sets Defining Few Ordinary Circles

Lin

Makhul

Mojarrad

et al. 2017

Discrete Comput Geom

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“…On a related note, Purdy and Smith [10] considered ordinary spheres in 3-space in the slightly more restricted setting of a finite set of points with no four concyclic and no three collinear. We include hyperplanes as degenerate spheres because the collection of all hyperspheres and hyperplanes is closed under inversion (see Section 2).…”

Section: Introductionmentioning

confidence: 99%

Ordinary hyperspheres and spherical curves

Lin

Swanepoel

2021

Advances in Geometry

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An ordinary hypersphere of a set of points in real d-space, where no d + 1 points lie on a (d - 2)-sphere or a (d - 2)-flat, is a hypersphere (including the degenerate case of a hyperplane) that contains exactly d + 1 points of the set. Similarly, a (d + 2)-point hypersphere of such a set is one that contains exactly d + 2 points of the set. We find the minimum number of ordinary hyperspheres, solving the d-dimensional spherical analogue of the Dirac–Motzkin conjecture for d ⩾ 3. We also find the maximum number of (d + 2)-point hyperspheres in even dimensions, solving the d-dimensional spherical analogue of the orchard problem for even d ⩾ 4.

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“…Purdy and Smith [25] considered instead finite non-coplanar point sets in 3-space with no three points collinear, and provided a lower bound on the number of planes containing exactly three points of the set. Referring to such a plane as an ordinary plane, Ball [1] proved a 3-dimensional analogue of Green and Tao's [9] structure theorem, and found the exact minimum number of ordinary planes spanned by sufficiently large non-coplanar point sets in real projective 3-space with no three points collinear.…”

Section: Introductionmentioning

confidence: 99%

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Lin¹,

Swanepoel²

2020

discrete_Anal

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Let P be a set of n points in real projective d-space, not all contained in a hyperplane, such that any d points span a hyperplane. An ordinary hyperplane of P is a hyperplane containing exactly d points of P. We show that if d 4, the number of ordinaryif n is sufficiently large depending on d. This bound is tight, and given d, we can calculate the exact minimum number for sufficiently large n. This is a consequence of a structure theorem for sets with few ordinary hyperplanes: For any d 4 and K > 0, if n C d K 8 for some constant C d > 0 depending on d, and P spans at most K n−1 d−1 ordinary hyperplanes, then all but at most O d (K) points of P lie on a hyperplane, an elliptic normal curve, or a rational acnodal curve. We also find the maximum number of (d + 1)-point hyperplanes, solving a d-dimensional analogue of the orchard problem. Our proofs rely on Green and Tao's results on ordinary lines, our earlier work on the 3-dimensional case, as well as results from classical algebraic geometry.

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Lines, Circles, Planes and Spheres

Cited by 6 publications

References 18 publications

On Sets Defining Few Ordinary Circles

On Sets Defining Few Ordinary Circles

Ordinary hyperspheres and spherical curves

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