ORTHOGONAL TRAJECTORY

family of curves that intersect another family of curves at right angles (see illustration). Such families of mutually orthogonal curves occur in such branches of physics as electrostatics, in which the lines of force and the lines of constant potential are orthogonal; and in hydrodynamics, in which the streamlines and the lines of constant velocity are orthogonal. In two dimensions, a family of curves is given by y = f (x, k), in which the value of k, called the parameter, determines the particular member of the family. Two lines are perpendicular if their slopes are negative reciprocals of each other. To apply this to two curves, their tangents at the point of intersection must be perpendicular. The slope of the tangent to a curve at a point, called its derivative, can be found using calculus. This derivative, written as y, will also be a function of x and k. Solving the original equation for k in terms of x and y and substituting this expression into the equation for y will give y in terms of x and y, as some function y = g(x, y). As noted above, a member of the family of orthogonal trajectories, y1, must have a slope satisfying y1 = -1/y = -1/g(x, y), resulting in a differential equation that will have the orthogonal trajectory as its solution. To illustrate, if y = kx2 represents a family of parabolas, then y = 2kx, and, because k = y/x2, y = 2y/x. For the orthogonal curve, y1 = -1/y = -x/2y, which is a differential equation that can be solved to give the solution y2 + (x2/2) = k, which represents a family of ellipses orthogonal to the family of parabolas (see illustration).