Wednesday, August 26, 2009
Rational parameterization with certain conics
While number theorists consider Pell's Equation only with integers, it can be considered with rationals, parameterized and related then to all the conic sections except the parabola:
Given x^2 - Dy^2 = 1, in rationals:
y = 2t/(D - t^2)
and
x = (D + t^2)/(D - t^2)
and you get hyperbolas with D>0, the circle with D=-1, and ellipses in general with D<0.
You can see the D=-1 case from a well-known mainstream source at the following link:
See: http://mathworld.wolfram.com/Circle.html eqns. 16 & 17
Given x^2 - Dy^2 = 1, in rationals:
y = 2t/(D - t^2)
and
x = (D + t^2)/(D - t^2)
and you get hyperbolas with D>0, the circle with D=-1, and ellipses in general with D<0.
You can see the D=-1 case from a well-known mainstream source at the following link:
See: http://mathworld.wolfram.com/Circle.html eqns. 16 & 17
# posted by James Harris @ 20:38 
