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. But how do they know this? In this wiki, we aim to show a method of deriving such shapes, or for finding the envelope of a function.
Compactly it can be said that 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. The envelope can also be seen as a set of points which a family of curves, , intersects , as n approaches 0.
The black point is a point on the envelope. The red line is for a particular value of and the blue line is as approaches 0.
The envelope is defined as the set of points for which
The intersection between and is given by
Limiting to 0 gives the definition above.
Find the area under which a ladder of length 1 occupies while sliding down a wall.
Our first step is to define , which in this case models the behavior of the ladder.
The red line between the -axis and the -axis represents the ladder:
Solving the above equations, we get the envelope in parametric form as