This page is intended as place for newer texts related with quadratic forms, in particular of dimension 2. Another text in this area is
by Markus Rost (Notes, August 2026, 8 pages)
The text presents the solution to the Newton differential equation in the case of negative energy and nonzero angular momentum.
The point of view is algebraic with no emphasis on geometry or physics.
The material is directly inspired by Appendix 1. Conic sections in: Milnor, John. On the geometry of the Kepler problem. Amer. Math. Monthly 90 (1983), no. 6, 353-365. MR 707149, Zbl 0518.70008.
Full text (September 1, 2026): [pdf]
The final conclusion on one page. The description is essentially the same as in Milnor, ibid., p.360.
Full text (August 31, 2026): [tex] [pdf]
The ellipse with a:b:c=5:4:3.
For two symmetric bilinear forms h, p on a 2-dimensional space, with h playing the role of a metric and p providing the equation of a conic section, one may define two quadratic forms q, f.
The form q yields the common diagonalization (the axes of the conic section), the form f yields on the axes the 4 focus points (2 per axis). In the classical real case, 2 of the focus points are real, the other 2 are imaginary.
The constructions can be carried out in a coordinate free way over any ring.
Caveat: The ⨯-product of symmetric bilinear forms in dimension 2 is a quadratic form, with underlying (indefinite) scalar product given by the determinant. In the formula, Bq denotes the symmetric bilinear form of the quadratic form q.
The text is in delayed production since September 2026.
Appetizer (September 2, 2026): [tex] [pdf]