Dan wrote:
> On Thu, 10 Feb 2011, Barton Willis wrote:
>
>> Apparently, Maxima sometimes tries a z =tan(x/2) substitution (expunged
>> from calculus books in the 1980s?)
>
> I don't know if you count
> <http://www.phy.cam.ac.uk/teaching/teachingfiles/formulae.pdf>; as a
> "calculus book", but it's right there on page 14 of the 2005 edition.
It would be unfortunate to expunge this substitution from books since this
is the one which reveal that the circle is a genus 0 curve, isomorphic to a
line. Geometrically this rational parametrisation is obtained by cutting the
circle by a line through one point of the circle, so the second point
depends rationally on the slope. This slope is obviously the tangent of the
half angle, and the fixed point becomes the point at infinity on the line,
hence the "spurious" infinities in question. Hence showing this
parametrization is an introduction to most interesting subjects about curves
and also points to the futility of classical trigonometry.
--
Michel Talon