The flame from the first slide in the review had the parametrization 
r(t,s) = (cosh(t) sin(3 A s),
          sinh(t) sin(3 A s), 
          cosh(s))
where A is a parameter which is changed for seeing the movie. The cosh(x) function is defined as cosh(x) = (exp(x)+exp(-x))/2 and sinh(x) = (exp(x) - exp(-x))/2.