envelope

Two plane curves are said to touch each other or have a tangency at a point if they have a common tangent line at that point.

The envelope of a family of plane curves is a curve which touches in each of its points one of the curves of the family.

For example, the envelope of the family  $y=mx-\sqrt{1+m^2}$,  with $m$ the parameter, may be justified geometrically.  It is the open (http://planetmath.org/OpenSet) lower semicircle of the unit circle.  Indeed, the distance of any line

$$mx-y-\sqrt{1+m^2}=0$$

of the family from the center of the unit circle is

$$\frac{|m\cdot 0-1\cdot 0-\sqrt{1+m^2}|}{\sqrt{m^2+{(-1)}^2}}=1,$$

whence the line is the tangent to the circle.

Below, the red curve is the lower semicircle of the unit circle, the black lines belong to the family  $y=mx-\sqrt{1+m^2}$,  and the equation of each line is given.