Tweaking the Expanding Circle Construction

In last month’s blog post, I described a parabola construction technique dating back to the work of Persian polymath  Ibn Sina  (c. 970 – 1037). After I published the post, my colleague Scott noted that my construction could be more robust to allow for parabolas that are downward facing as well as upward facing. Such flexibility requires a clever construction technique—one that I’ll demonstrate here.

First, let’s review the original construction that appears on page 1 of the Web Sketchpad model below (and here). Points B and C sit on the y-axis with point C as the center of a circle passing through point B. Points E and F mark where the circle intersects the x-axis. Points G and H sit at the intersection points of the tangent to the circle through point D with the lines passing through points E and F that a parallel to the y-axis. As you drag point C along the y-axis, observe the traces of points G and H. These points (as you can prove) trace an upward-facing parabola.

The problem occurs when you move points C and B, dragging point C below point B, to trace a downward-facing parabola. In the video at the end of this post, I demonstrate the problem that arises and then rebuild the construction from scratch to fix the issue.