Hodograph

Hodograph (Gr. hodos, 'a way') of a moving particle is the curve passing through the extremities of those lines which, drawn from a fixed point as origin, represent in direction and magnitude the velocities of the particle at the different points of its path. It is a velocity diagram of a particular kind. Just as the tangent to the path at any point gives the direction of motion of the particle at that point, so the tangent to the hodograph at the corresponding point gives the direction in which the velocity is changing—i.e. the direction of the acceleration. Thus, if the hodograph is a straight line with origin anywhere outside it, we see that the acceleration is constant in direction, for a straight line is its own tangent. Another conclusion at once deducible is that the velocity resolved perpendicular to the direction of the acceleration is always the same, being given by the perpendicular from the origin upon the line. If, in this case, the acceleration is also constant in amount, we obtain the hodograph of the parabolic motion of a projectile. As another simple case, let the hodograph be a circle, centre the origin. Here the speed of the particle in its path must be constant; and further, the acceleration is perpendicular to the velocity, having the effect of changing the direction only of motion. If, in this case also, the acceleration is given as constant in amount, then the line representing the velocity in direction must rotate uniformly. Hence the path must be such that the angle between the tangents at two points must be proportional to the length of the arc joining them. In technical language, the path must be a plane curve of constant curvature—i.e. either a straight line or a circle, obviously the latter in this case. Thus, under an acceleration constant in amount and always perpendicular to the direction of motion, and to a fixed direction in space, the particle will describe a circle with constant speed, the radius of the circle being a third proportional to the magnitude of the acceleration and the speed. The name hodograph was invented by Sir W. R. Hamilton, who made many elegant applications of its properties to dynamics. In virtue of the aberration of light, every star describes a projection of the hodograph of the earth's motion in its orbit—i.e. the projection of a circle. The properties of the hodograph are treated in all modern treatises on the dynamics or kinematics of a particle.