a conjugate point to P. Caustics In geometrical optics, a caustic is the envelope of a family of light rays. In this picture there is an arc of a circle. The light rays (shown in blue) are coming from a source at infinity, and so arrive parallel. When they hit the circular arc the light rays are scattered in different directions according to the law of reflection. When a light ray hits the arc at a point the light will be reflected as though it had been reflected by the arc's tangent line at that point. The reflected light rays give a one-parameter family of lines in the plane. The envelope of these lines is the reflective caustic. A reflective caustic will generically consist of smooth points and ordinary cusp points. From the point of view of the calculus of variations, Fermat's principle (in its modern form) implies that light rays are the extremals for the length functional among smooth curves γ on [a,b] with fixed endpoints γ(a) and γ(b). The caustic determined by a given point P (in the image the point is at infinity) is the set of conjugate points to P. Huygens's principle Light may pass through anisotropic inhomogeneous media at different rates depending on the direction and starting position of a light ray. The boundary of the set of points to which light can travel from a given point q after a time t is known as the wave front after time t, denoted here by Φq(t). It consists of precisely the points that can be reached from q in time t by travelling at the speed of light. Huygens's principle asserts that the wave front set is the envelope of the family of wave fronts for q ∈ Φq0(t). More generally, the point q0 could be replaced by any curve, surface or closed set in space. See also Caustic (mathematics) Envelope theorem Ruled surface References External links "Envelope of a family of plane curves" at MathCurve. Differential geometry Analytic geometry