Hodograph

It appears to have been used by James Bradley but for its practical development we are mainly indebted to Sir William Rowan Hamilton, who published an account of it in the "Proceedings of the Royal Irish Academy", 1846.

If a point be in motion in any orbit and with any velocity, and if, at each instant, a line be drawn from a fixed point parallel and equal to the velocity of the moving point at that instant, the extremities of these lines will lie on a curve called the hodograph.

Let P-P1-P2-...-Pn-... be the path of the moving point, and let OT, OT1, OT2, be drawn from the fixed point O parallel and equal to the velocities at P, P1, P2 respectively, then the locus of T (T-T1-T2-...-Tn-...) is the hodograph of the orbit described by P.

From this definition we have the following important fundamental property which belongs to all hodographs, viz., that at any point the tangent to the OT hodograph is parallel to the direction — and the velocity in the hodograph equal to the magnitude — of the resultant acceleration at the corresponding point of the orbit.

This will be evident if we consider that, since radii vectores of the hodograph represent velocities in the orbit, the elementary arc between two consecutive radii vectores of the hodograph represents the velocity which must be compounded with the velocity of the moving point at the beginning of any short interval of time to get the velocity at the end of that interval, that is to say, represents the change of velocity for that interval. Hence the elementary arc divided by the element of time is the rate of change of velocity of the moving-point, or in other words, the velocity in the hodograph is the acceleration in the orbit.

Further reading