Ellipse Constructions

Solutions for the four ellipse challenges can be found at the links below.

Ellipse 1
Ellipse 2
Ellipse 3
Ellipse 4

The biggest challenge of the ellipse constructions follows from the fact that an ellipse is intersected by two axes, with two distinct pairs of axial vertices. This typically is not an issue with the construction of the curve itself, but when it comes to construction of the foci and directrices, that distinction will have to be made. It usually is plain to see, but my goal is to have the construction take care of itself. When given objects are dragged, the minor axis may become the major axis. That part of the solution is not shown here. It involves judicious use of rays rather than lines. No measurements are employed.

All of the ellipse constructions came down to this. Circles c₁ and c₂ are constructed using the corresponding axes, a₁ and a₂ as diameters. Let O be the center, and let T be the traveling point, attached to circle c₁. (It does not matter whether it is the major or the minor.) Construct ray OT, and let it intersect circle c₂ at A. Through T construct the line perpendicular to a₁, and through A construct the line perpendicular to a₂. Let these lines intersect at U, and construct the locus of U as T travels on its path.

And so, given the four axial vertices, the curve itself is a can of corn. There is still the matter of constructing those vertices, along with the foci and directrices.

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