the given pencil: (2p) Hence the differential equation of the orthogonal trajectories is: (3p) After solving this differential equation by separation of variables one gets which describes the pencil of cardioids (red in diagram), symmetric to the given pencil. Isogonal trajectory A curve, which intersects any curve of a given pencil of (planar) curves by a fixed angle is called isogonal trajectory. Between the slope of an isogonal trajectory and the slope of the curve of the pencil at a point the following relation holds: This relation is due to the formula for . For one gets the condition for the orthogonal trajectory. For the determination of the isogonal trajectory one has to adjust the 3. step of the instruction above: 3. step (isog. traj.) The differential equation of the isogonal trajectory is: (3i) For the 1. example (concentric circles) and the angle one gets (3i) This is a special kind of differential equation, which can be transformed by the substitution into a differential equation, that can be solved by separation of variables. After reversing the substitution one gets the equation of the solution: Introducing polar coordinates leads to the simple equation which describes logarithmic spirals (s. diagram). Numerical methods In case that the differential equation of the trajectories can not be solved by theoretical methods, one has to solve it numerically, for example by Runge–Kutta methods. See also Cassini oval Confocal conic sections Trajectory Apollonian circles, pairs of families of circles that are all orthogonal to each other References A. Jeffrey: Advanced Engineering Mathematics, Hartcourt/Academic Press, 2002, , p. 233. S. B. Rao: Differential Equations, University Press, 1996, , p. 95. R. M. Redheffer, D. Port: Differential Equations: Theory and Applications, Jones & Bartlett, 1991, , p. 63. H. Heuser: Gewöhnliche Differentialgleichungen, Vieweg+Teubner, 2009, , p. 120. . External links Exploring orthogonal trajectories - applet allowing user to draw families of curves and their orthogonal trajectories. mathcurve: FIELD LINES, ORTHOGONAL LINES, DOUBLE ORTHOGONAL SYSTEM Curves