Von Staudt recognized that projective geometric constructions on points and lines can operate as sums and products of “real” numbers. The most readable continuation (including conics) appears in Coxeter’s book The Real Projective Plane. Further developments on other systems and models, e.g., in Benz’s Vorlesungen über Geometrie der Algebren.
Current project: Classification of 2-D numerical structures (e.g., complex, bicomplex, split-complex, dual) based on polarity induced by quadric surfaces. Creating geometric models of these structures.
Previous:
- In the reviews is the paper on complex numbers connecting von Staudt’s constructions with the Riemann sphere. An analogy to Matiyasevich’s parabola is in the video of the geometric algorithm to construct Gaussian complex numbers.
- Extending to imaginary elements according to Poncelet and Hatton: A ruler construction of imaginary numbers on a conic. In Proceedings of the Czech-Slovak Conference on Geometry and Graphics 2024 (pp. 143–148).
- Starting with real numbers on a parabola and developing the arithmetic of quadrics according to von Staudt, the paper is rather lightweight: Calculating on a parabola. In Proceedings of the 9th Slovak-Czech Conference on Geometry and Graphics 2023 (pp. 203–210)
- Some bits (analysis of von Staudt’s Geometrie der Lage, appendix: Field of cows) in my Ph. D. thesis: Synthetic projective geometry