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 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 $$\frac{|m\cdot0-1\cdot0-\sqrt{1+m^2}|}{\sqrt{m^2+(-1)^2}} = 1,$$ 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.