In geometry, an envelope of a family of curves in the plane is a curve that is tangent to each member of the family at some point, and these points of tangency together form the whole envelope. Classically, a point on the envelope can be thought of as the intersection of two “infinitesimally adjacent” curves, meaning the limit of intersections of nearby curves. This idea can be generalized to an envelope of surfaces in space, and so on to higher dimensions.

To have an envelope, the individual members of the family of curves need to be differentiable curves, for otherwise the concept of tangency does not apply, and there has to be a smooth transition proceeding through the members. But even if these conditions are satisfied, a given family may fail to have an envelope. A simple example of this is given by a family of concentric circles of expanding radius.

Have you ever seen this, a little bright curve formed when light rays reflect off the circular walls of your cup and overlap?

That curve so happens to be the curve of a Cardioid, and is the caustic envelope of a circle.

cardioid (from the Greek καρδία “heart”) is a plane curve traced by a point on the perimeter of a circle that is rolling around a fixed circle of the same radius. It can also be defined as an epicycloid having a single cusp. It is also a type of sinusoidal spiral, and an inverse curve of the parabola with the focus as the center of inversion.

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