One dimensional metrical geometry

One dimensional metrical geometry may be developed in either an affine or projective setting over a general field using only algebraic ideas and quadratic forms. Some basic results of universal geometry are already present in this situation, such as the Triple quad formula, the Triple spread formula and the Spread polynomials, which are universal analogs of the Chebyshev polynomials of the first kind. Chromogeometry appears here, and the related metrical and algebraic properties of the projective line are brought to the fore.Comment: 19 page

Pell's equation without irrational numbers

We solve Pell's equation in a simple way without continued fractions or irrational numbers, and relate the algorithm to the Stern Brocot tree.Comment: 10 pages, 3 figures added some references, fixed typos, added remarks on Speeding up the algorith

Neuberg cubics over finite fields

The framework of universal geometry allows us to consider metrical properties of affine views of elliptic curves, even over finite fields. We show how the Neuberg cubic of triangle geometry extends to the finite field situation and provides interesting potential invariants for elliptic curves, focussing on an explicit example over $\mathbb{F}_{23}$. We also prove that tangent conics for a Weierstrass cubic are identical or disjoint.Comment: 16 pages, 6 figure