Let PP1P2 be the path of the moving point, and let OT, OT 1, OT2, be drawn from the fixed point 0 parallel and equal to the velocities at P, P 1, respectively, then the locus of T is the hodograph of the orbits described by P (see figure).
The pole 0 of the hodograph is inside on or outside the circle, according as the orbit is an ellipse, parabola or hyperbola.
Hence the hodograph is similar and similarly situated to the locus of Z (the Flo.
In any case of a central orbit the hodograph (when turned through a right angle) is similar and similarly situated to the reciprocal polar of the orbit with respect to the centre of force.
In the motion of a projectile under gravity the hodograph is a vertical line described with constant velocity.
Thus for a circular orbit with the centre of force at an excentric point, the hodograph is a conic with the pole as focus.
Every orbit must clearly have a hodograph, and, conversely, every hodograph a corresponding orbit; and, theoretically speaking, it is possible to deduce the one from the other, having given the other circumstances of the motion.
For applications of the hodograph to the solution of kinematical problems see Mechanics.
In elliptic harmonic motion the velocity of P is parallel and proportional to the semi-diameter CD which is conjugate to the radius CP; the hodograph is therefore an ellipse similar to the actual orbit.
That at any point the tangent to the 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.