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.