Does this method of curve stitching make a parabola, or just something that looks kind of parabola-ish?
Geometrically, a parabola is the set of points in the plane that are equidistant from a point (the focus) and a line (the directrix).
The parabola whose focus is the point (0,1), and whose directrix is y= -1 is algebraically defined as y=¼ x².
One way to illustrate the geometric definition is with paper folding. Mark the focus and directrix, and then fold your paper so that one point on the directrix matches up with the focus. Make a crease. Choose another point on the directrix and repeat many times.
Each of those folds is the perpendicular bisector of a segment connecting (1) the focus, with (2) some point on the directrix. Here is what one of those segments looks like.
And here is what one of those segments looks like, with its perpendicular bisector (in black).
That perpendicular bisector is a fold line when you’re folding paper.
When you draw in a large collection of these fold lines, you see the parabola emerge.
The fold lines come from the green segments, and I chose those the endpoints of those segments to be evenly spaced along the directrix—each is a half-unit from its neighbors.
Surprisingly (surprising to me, anyway!) each black fold line crosses the others at constant intervals. This could be a lemma worth proving: When drawing segments from the focus to evenly spaced points on the directrix, the perpendicular bisectors of these segments will cross each other at regular intervals. But that proof will not be a part of this blog post.
We’ll take the outermost of these fold lines and make it the frame (in orange) for our curve stitching. The holes in the frame are at the regularly-spaced intersection with the other fold lines.
The upshot is that our curve stitching model makes a parabola because the threads of the model are the fold lines on the tracing paper, which together record the points that are equidistant from the focus and the directrix, which is the geometric definition of a parabola.
More lines, more closely spaced, get us a more perfect parabola.
You can play with the Desmos file that generated these images, and you can read more about the hinged parabola stretcher in our Make and Take blog post.