their radii mark the ray directions, and their common tangent surface is a general wavefront. Thus the rays are normal (orthogonal) to the wavefronts. Because much of the teaching of optics concentrates on isotropic media, treating anisotropic media as an optional topic, the assumption that the rays are normal to the wavefronts can become so pervasive that even Fermat's principle is explained under that assumption, although in fact Fermat's principle is more general. Homogeneous media: Rectilinear propagation In a homogeneous medium (also called a uniform medium), all the secondary wavefronts that expand from a given primary wavefront in a given time are congruent and similarly oriented, so that their envelope may be considered as the envelope of a single secondary wavefront which preserves its orientation while its center (source) moves over . If is its center while is its point of tangency with , then moves parallel to , so that the plane tangential to at is parallel to the plane tangential to at . Let another (congruent and similarly orientated) secondary wavefront be centered on , moving with , and let it meet its envelope at point . Then, by the same reasoning, the plane tangential to at is parallel to the other two planes. Hence, due to the congruence and similar orientations, the ray directions and are the same (but not necessarily normal to the wavefronts, since the secondary wavefronts are not necessarily spherical). This construction can be repeated any number of times, giving a straight ray of any length. Thus a homogeneous medium admits rectilinear rays. Modern version Formulation in terms of refractive index Let a path extend from point to point . Let be the arc length measured along the path from , and let be the time taken to traverse that arc length at the ray speed (that is, at the radial speed of the local secondary wavefront, for each location and direction on the path). Then the traversal time of the entire path is (where and simply denote the endpoints and are not to be construed as values of or ). The condition for to be a ray path is that the first-order change in due to a change in is zero; that is, Now let us define the optical length of a given path (optical path length, OPL) as the distance traversed by a ray in a homogeneous isotropic reference medium (e.g., a