Newton’s Hyperbola Observed from Newton’s Evolute (1687), Gudermann’s Circle (1833), the Auxiliary Circle (Pedal Curve and Inversion Curve), the Lemniscate of Bernoulli (1694) (Pedal Curve and Inversion Curve) (09.01.2019)

Johannes Kepler and Isaac Newton inspired generations of researchers to study properties of elliptic, hyperbolic, and parabolic paths of planets orbiting around the Sun. After the intensive study of those conic sections during the last four hundred years it is believed that this topic is practically closed and the 21st Century cannot bring anything new to this subject. Can we add to those visible orbits from the Aristotelian World some curves from the Plato’s Realm that might bring to us new information about those conic sections? Isaac Newton in 1687 discovered one such curve - the evolute of the hyperbola - behind his famous gravitation law. In our model we have been working with Newton’s Hyperbola in a more complex way. We have found that the interplay of the empty focus M (= Menaechmus - the discoverer of hyperbola), the center of the hyperbola A (= Apollonius of Perga - the Great Geometer), and the occupied focus N (= Isaac Newton - the Great Mathematician) together form the MAN Hyperbola with several interesting hidden properties of those hyperbolic paths. We have found that the auxiliary circle of the MAN Hyperbola could be used as a new hodograph and we will get the tangent velocity of planets around the Sun and their moment of tangent momentum. We can use the lemniscate of Bernoulli as the pedal curve of that hyperbola and we will get the normal velocities of those orbiting planets and their moment of normal momentum. The first derivation of this moment of normal momentum will reveal the torque of that hyperbola and we can estimate the precession of hyperbolic paths and to test this model for the case of the flyby anomalies. The auxiliary circle might be used as the inversion curve of that hyperbola and the Lemniscate of Bernoulli could help us to describe the Kepler’s Equation (KE) for the hyperbolic paths. Have we found the Arriadne’s Thread leading out of the Labyrinth or are we still lost in the Labyrinth?