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Lines, Circles, Planes and Spheres.

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Title: Lines, Circles, Planes and Spheres.
Authors: Purdy, George1 george.purdy@uc.edu, Smith, Justin1 smith5jw@mail.uc.edu
Source: Discrete & Computational Geometry. Dec2010, Vol. 44 Issue 4, p860-882. 23p.
Subjects: Line geometry, Plane geometry, Spheres, Boundary value problems, Set theory, Number theory, Fixed point theory, Circle
Abstract: Let S be a set of n points in ℝ, no three collinear and not all coplanar. If at most n− k are coplanar and n is sufficiently large, the total number of planes determined is at least $1+k\binom{n-k}{2}-\binom{k}{2}(\frac{n-k}{2})$. For similar conditions and sufficiently large n, (inspired by the work of P.D.T.A. Elliott in Acta Math. Sci. Hung. 18:181-188, ) we also show that the number of spheres determined by n points is at least $1+\binom{n-1}{3}-t_{3}^{\mathrm{orchard}}(n-1)$, and this bound is best possible under its hypothesis. (By $t_{3}^{\mathrm{orchard}}(n)$, we are denoting the maximum number of three-point lines attainable by a configuration of n points, no four collinear, in the plane, i.e., the classic Orchard Problem.) New lower bounds are also given for both lines and circles. [ABSTRACT FROM AUTHOR]
Copyright of Discrete & Computational Geometry is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: <searchLink fieldCode="DE" term="%22Line+geometry%22">Line geometry</searchLink><br /><searchLink fieldCode="DE" term="%22Plane+geometry%22">Plane geometry</searchLink><br /><searchLink fieldCode="DE" term="%22Spheres%22">Spheres</searchLink><br /><searchLink fieldCode="DE" term="%22Boundary+value+problems%22">Boundary value problems</searchLink><br /><searchLink fieldCode="DE" term="%22Set+theory%22">Set theory</searchLink><br /><searchLink fieldCode="DE" term="%22Number+theory%22">Number theory</searchLink><br /><searchLink fieldCode="DE" term="%22Fixed+point+theory%22">Fixed point theory</searchLink><br /><searchLink fieldCode="DE" term="%22Circle%22">Circle</searchLink>
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  Data: Let S be a set of n points in ℝ, no three collinear and not all coplanar. If at most n− k are coplanar and n is sufficiently large, the total number of planes determined is at least $1+k\binom{n-k}{2}-\binom{k}{2}(\frac{n-k}{2})$. For similar conditions and sufficiently large n, (inspired by the work of P.D.T.A. Elliott in Acta Math. Sci. Hung. 18:181-188, ) we also show that the number of spheres determined by n points is at least $1+\binom{n-1}{3}-t_{3}^{\mathrm{orchard}}(n-1)$, and this bound is best possible under its hypothesis. (By $t_{3}^{\mathrm{orchard}}(n)$, we are denoting the maximum number of three-point lines attainable by a configuration of n points, no four collinear, in the plane, i.e., the classic Orchard Problem.) New lower bounds are also given for both lines and circles. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of Discrete & Computational Geometry is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1007/s00454-010-9270-3
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        Text: English
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    Subjects:
      – SubjectFull: Line geometry
        Type: general
      – SubjectFull: Plane geometry
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      – SubjectFull: Spheres
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      – SubjectFull: Boundary value problems
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      – SubjectFull: Number theory
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              M: 12
              Text: Dec2010
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