Tracing a planet’s orbit with a straight edge and a compass with the help of the hodograph and the Hamilton vector

Abstract

We describe a geometrical method for tracing a planet's orbit using its velocity hodograph, that is, the path of the planet's velocity. The method requires only a straight edge, a compass, and the help of the hodograph. We also obtain analitically the hodograph and some of the features of the motion that can be obtained from it.

Vectorial Regularization and Temporal Means in Keplerian Motion

“…He found a replica of vector e with different interesting properties (see [2]). This result was found again identically by various authors 35 years after (see [6], [9]- [11], [17]- [19]). This vector e can be computed only from the initial conditions and gives a complete overview to the entire motion, containing all the informations about it.…”

“…Using complex numbers, professor A. Braier found in 1965 a replica of vector e with specific consequences (see [1,2,3]). He deduced a vectorial eccentricity (see [1]) that was again discovered by various authors in the same way between 1996-2004 (see [6], [9]- [11], [17]- [19]).…”

A Hamilton-like vector for the special-relativistic Coulomb problem

“…It reflects a special property-often referred to as a dynamical symmetry-of a radial inverse-square-law force within the context of non-relativistic mechanics and allows for a rather straightforward derivation of the orbit of a particle moving under the influence of such a force. An alternative [6][7][8] is to use the Hamilton vector, a quantity related in a simple manner to the Laplace-Runge-Lenz vector whose conservation was first discovered by W R Hamilton in 1845 [9].…”

“…Given the remarkable efficiency introduced by the Hamilton vector into the derivation of the orbit of the particle [6][7][8], it is only natural to ask whether the relativistic problem might not also benefit from a similar development. At first sight, the answer seems rather trivially no, since special-relativistic effects break the degeneracy that ensures conservation of the Hamilton and Laplace-Runge-Lenz vectors.…”

Orbits of the Kepler problem via polar reciprocals

“…A completely different way of tracing all three kinds of orbits with the help of hodographs has been successfully developed by Salas-Brito and co-workers in a series of publications culminating in Ref. 10.…”