Bibliographic Details
| 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] |
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| Database: |
Engineering Source |
