A hyperbolic poynomial is
defined in the following way: ?_{k=0}^{m}a_{k}cosh(k)+?_{k=0}^{m}b_{k}sinh(k),
where a_{k}???R and
b_{k}???R. A
hyperbolic curve is a real plane curve where each coordinate is
given parametrically by a hyperbolic poynomial:
x=?_{k=0}^{m}a_{k}cosh(k)+?_{k=0}^{m}b_{k}sinh(k)
y=?_{k=0}^{m}c_{k}cosh(k)+?_{k=0}^{m}d_{k}sinh(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:
2752x^{2}-32x^{2}y+2x^{4}+310632-172y-130y^{2}-y^{3}=0
contains two curves, one trigonometric and the other
hyperbolic. |