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Source. Theorem 4.4, p. 27, of George B. Purdy and Justin W. Smith, Lines, circles, planes and spheres, arXiv:0907.0724 (2009); Discrete Comput. Geom. 44 (2010), no. 4, 860--882, doi:10.1007/s00454-010-9270-3. Labels and pages here are those of arXiv v1 (3 July 2009), whose pagination differs from the journal's; the edition is named on the source card.

Read depth. Claims checked: the statement and its hypotheses were read clause by clause on the page image of the print; the proof was not checked.

Statement

Let $S$ be a configuration of $n$ points, no four collinear, on a plane $\pi$. Then there exist a set $S'$ of $n$ points, all cospherical but no four cocircular, and a point of projection $p$ such that projecting the points of $S'$ onto $\pi$ from $p$ produces $S$. In the proof, $p$ is the centre of a unit sphere tangent to $\pi$, and $S'$ lies on that sphere.

Orchard numbers (p. 25). For a finite planar set $S$ let $t_3(S)$ be the number of lines through exactly three of its points, and let $t_3^{orchard}(n)$ be the maximum of $t_3(S)$ over $n$-point planar sets with no four collinear. For $n$ cospherical points with no four cocircular, let $\mathcal M_3^{max}(n)$ be the maximum number of planes they determine that pass through a common point not in the set.

Consequence (p. 29). The paper concludes from the theorem that

$$t_3^{orchard}(n)=\mathcal M_3^{max}(n).$$

The inequality $\mathcal M_3^{max}(n)\le t_3^{orchard}(n)$ is the easy direction (p. 26): projecting from the common point turns such planes into three-point lines of a planar set with no four collinear. The theorem supplies the converse.

Proof pointer

Pages 27--29. The proof places $S$ in the plane at height one and the sphere's centre on a fixed line in the plane below it, and shows that, for each four points of $S$ with no three collinear, the positions of the centre on that line for which their lifts are cocircular lie among the real roots of a polynomial that is not identically zero. Since there are finitely many four-point subsets, some position avoids all of them.

Dependencies

None among the paper's numbered results.

Bears on

None of the problem pages directly. The orchard numbers $t_3^{orchard}(n)$ are the planar three-point-line maxima under the condition that no four points are collinear; the paper quotes known bounds on them on p. 26.