envelope of the corresponding family of line segments (that is, the hypotenuses of the triangles), and has Cartesian equation Notice that, in particular, the value gives the arc of parabola of the Example 2, and the value (meaning that all hypotenuses are unit length segments) gives the astroid. Example 5 We consider the following example of envelope in motion. Suppose at initial height 0, one casts a projectile into the air with constant initial velocity v but different elevation angles θ. Let x be the horizontal axis in the motion surface, and let y denote the vertical axis. Then the motion gives the following differential dynamical system: which satisfies four initial conditions: Here t denotes motion time, θ is elevation angle, g denotes gravitational acceleration, and v is the constant initial speed (not velocity). The solution of the above system can take an implicit form: To find its envelope equation, one may compute the desired derivative: By eliminating θ, one may reach the following envelope equation: Clearly the resulted envelope is also a concave parabola. Envelope of a family of surfaces A one-parameter family of surfaces in three-dimensional Euclidean space is given by a set of equations depending on a real parameter a. For example, the tangent planes to a surface along a curve in the surface form such a family. Two surfaces corresponding to different values a and a' intersect in a common curve defined by In the limit as a' approaches a, this curve tends to a curve contained in the surface at a This curve is called the characteristic of the family at a. As a varies the locus of these characteristic curves defines a surface called the envelope of the family of surfaces. Generalisations The idea of an envelope of a family of smooth submanifolds follows naturally. In general, if we have a family of submanifolds with codimension c then we need to have at least a c-parameter family of such submanifolds. For example: a one-parameter family of curves in three-space (c = 2) does not, generically, have an envelope. Applications Ordinary differential equations Envelopes are connected to the study of ordinary differential equations (ODEs), and in particular singular solutions of ODEs. Consider, for example, the one-parameter family of tangent lines to the parabola y = x2. These are given by the generating family . The zero level set gives the equation of the tangent line