Geometrical definition of a continuous family of time transformations on the hyperbolic two-body problem

In this paper, we obtain the link between hyperbolic functions of hyperbolic argument H to trigonometric functions for each temporal variable in the new family, including also the inverse relations. We also carry out the study, in the attractive case, of the minimization of the errors due to the choice of a temporal variable included in our family in the numerical integration by an appropriate choice of parameters. This study includes the analysis of the dependence on the parameter of integration errors in a great time span for several eccentricities as well as the study of local truncation errors along the region with true anomaly contained in the interval [−π/2,π/2] around the primary for several values of the parameter.