Quadratic primas of rays
Abstract
In the article on linear aggregates of rays, I showed that the system of rays is not an independent system, but that for it one must adopt a parameter in the form of an extra-ray, necessarily entering into the composition of linear aggregates. Then a linear prima is completely and uniquely determined by two, a linear second by three, and a linear tertia by four arbitrary rays. If three arbitrary rays are given, then together with the constant fourth extra-ray we obtain the necessary and sufficient data for determining the rays of a complete linear secund. Since four arbitrary rays are in the general case intersected by a pair of transversals, real or imaginary, it is clear that we can define a linear second only as the aggregate of rays intersecting a given pair of lines a and b. If, consequently, we intersect the lines a and b with three mutually perpendicular rays, then by these are determined a linear straight tetraprima.
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References
- Bennо Klein. Ueber die geradlinige Fläche dritter Ordnung und ihre Abbildung an einer Ebene. Strassburg 1876.
- R. Schumacher, Mathem. Annalen 37 102.
- Reye Geom. d. Lage II 185.