In mathematics, the term "envelope" can refer to a variety of concepts depending on the context in which it is used. Here are a few common interpretations: 1. **Envelope of a Family of Curves**: The envelope of a family of curves is a curve that is tangent to each member of the family at some point.
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A geometric envelope is a curve tangent to the members of a smoothly varying family of curves. For a regular implicit family , candidate contact points also satisfy the partial derivative condition ; regularity is needed to turn these candidates into an actual envelope. The Mach cone is a three-dimensional analogue for expanding spheres.