In 7th grade I learned something considered 'magical' whenever I have shown it since. We constructed all of the basic plane geometric shapes by "advancing a point through a line" of some sort -- physically demonstrating the mathematical equation. We folded wax paper with a line on it and a point. By folding the point over the line visible through the paper and creasing a new line, you form a vector that meets the formula definition. Make enough crease lines and you form the shape.

Folding a single point through a straight line forms a parabola. A point (inside) folded through a circle forms an ellipse, with the circle center being the other foci -- and so on.

The real 'magic' is that we were taught these things back then ...

So how do you get a hyperbola?
DW

A point outside the circle, I think, to draw one half -- then the center of the cicle uses that as a line -- not sure though

Outside the circle seems reasonable. But I haven't been able to prove it works yet!
DW