In my prior weblog posts, I've described how to construct ellipses using linkages, concentric circles, congruent triangles, and tangent circles. These are all groovy methods, but I think I got alee of myself: There's a elementary ellipse structure technique described in nearly every precalculus volume that I've bypassed in my excitement to show y'all the more than exotic approaches. Say hullo to the pins-and-string technique:

Screen Shot 2022-01-28 at 1.59.38 PMThe moving picture higher up comes from the 17th-century manuscriptSive de Organica Conicarum Sectionum in Plano Descriptione, Tractatus (A Treatise on De vic es for Drawing Conic Sections)  past the Dutch mathematician Frans van Schooten. The ellipse construction in the illustration is quite simple. Press two pins into a corkboard, place a loop of cord effectually the pins, pull the string tight with a pencil, and trace the pencil tip's path as yous pull the pencil effectually the taut cord. Guaranteeing that the traced path is an ellipse is this definition of an ellipse: An ellipse is the fix of points P such that PF1 + PF2 is constant for two stock-still points, Fi and Fii.

I don't think any introduction to ellipses is complete without students making their own physical model of the pins-and-string structure and experimenting with how the distance between Fane and F2 too equally the length of the string affects the shape of the ellipse. But I also call back there is value in edifice a Sketchpad version of the ellipse construction. Below is a pre-congenital Web Sketchpad model that your students can investigate. In my upcoming post, I'll evidence how y'all can construct this model from scratch using Web Sketchpad.

You lot'll find this construction, as well every bit many others, in my book Exploring Conic Sections with The Geometer'south Sketchpad.