You can create a hyperbola from the creases made by folding a dot outside a circle onto the circle.
Draw a circle on a light piece of paper and place a dot somewhere outside the circle. You will need to be able to see the circle and the dot through the paper, so draw the circle with a dark line. Alternately, you can cut a small hole where the dot is so you can see the circle underneath as you fold. Or you can place the dot on the edge of the paper.
Fold the paper so the dot lands on the circle. Open.
Fold the paper again so the dot lands on another spot on the circle. Open
Repeat several more times. Each time the dot should land somewhere else on the circle. You will see a hyperbola emerge from the envelope of lines.