Make many creases in a circle and you can form an ellipse.
Draw a circle and place a dot anywhere inside the circle.
Fold the circle so a point on the edge touches the dot. It doesn't matter which point on the edge you pick. Open.
Fold many different points along the edge of the circle to the dot.
Notice that the creases define an ellipse with one focus at the dot and the other at the center of the circle.
If the dot is inside the circle, your folds produce an ellipse as above. If your dot is outside the circle, your folds produce a hyperbola. What happens when your dot is on the circle? If you try it, you will discover the answer after only one fold. You will realize that each fold creates a crease that is a diameter of the circle. Your creases no longer define a conic section. If you animate the creases as your dot moves from inside a circle to outside, you will get an intuitive sense as to why this is the case. As your dot moves toward the edge of the circle, the ellipse gets skinnier. The ellipse seems to turn inside out to form the hyperbola as the dot moves outside the circle.