A hyperbolic poynomial is
defined in the following way: ?k=0makcosh(k)+?k=0mbksinh(k),
where ak???R and
bk???R. A
hyperbolic curve is a real plane curve where each coordinate is
given parametrically by a hyperbolic poynomial:
x=?k=0makcosh(k)+?k=0mbksinh(k)
y=?k=0mckcosh(k)+?k=0mdksinh(k)
By adapting to hyperbolic curves the algorithms presented in
[Hong and Schicho 98] for the trigonometric case, we give
algorithms for simplifying a given parametric representation and
for computing an implicit representation from a given parametric
representation.
We show moreover that some of the algebraic curves arising
from the implicitization of a hyperbolic curve have a very
special structure containing both one hyperbolic part and one
trigonometric part. For example:
2752x2-32x2y+2x4+310632-172y-130y2-y3=0
contains two curves, one trigonometric and the other
hyperbolic. |