Envelope

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, \(F(x,y,k)=0\), intersects \(F(x,y,k+n)=0\), as n approaches 0.

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The black point is a point on the envelope. The red line is \(F(x,y,k)=0\) for a particular value of \(k,\) and the blue line is \(F(x,y,k+n)=0\) as \(n\) approaches 0.

The envelope is defined as the set of points for which

\[\frac { \partial }{ \partial k } F(x,y,k)=F(x,y,k)=0.\]

The intersection between \(F(x,y,k)\) and \(F(x,y,k+n)\) is given by

\[\begin{align} F(x,y,k)&=F(x,y,k+n)=0\\ \Rightarrow F(x,y,k)&=\frac{F(x,y,k+n)-F(x,y,k)}{n}. \end{align}\]

Limiting \(n\) to 0 gives the definition above.

Find the area under which a ladder of length 1 occupies while sliding down a wall.

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Our first step is to define \(F(x,y,k)\), which in this case models the behavior of the ladder.

The red line between the \(y\)-axis and the \(x\)-axis represents the ladder:

\[\begin{align} F(x,y,k) &=-\frac{\sqrt{1-k^2}}{k}x+\sqrt{1-k^2}-y\\ &=0 \\ \frac { \partial }{ \partial k } F(x,y,k) &=\frac{x-k^3}{k^2\sqrt{1-k^2}}\\ &=0. \end{align}\]

Solving the above equations, we get the envelope in parametric form as

\[\left(k^3,\left(1-k^2\right)\sqrt{1-k^2}\right).\]