169 research outputs found
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 . We also prove that tangent conics for
a Weierstrass cubic are identical or disjoint.Comment: 16 pages, 6 figure
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