Second-Order Curves as Envelopes

Let us consider a circle and a point that does not belong to it. Connect all points of a circle with the given point by line segments and draw perpendicular lines through their midpoints. These straight lines will fi ll a part of the plane as if bending round the untouched areas. In the case when the selected point lies inside the circle, the envelope boundary will be an ellipse, and if the point is outside the circle — a hyperbola. A similar operation can be performed with a straight line and a point that lies outside it. In this case, the boundary will be a parabola.