Drawing Multi-focal Ellipses: The Gardener’s Method

Common-or-Garden Ellipses

In an earlier post we saw how a gardener may set out oval flower-beds using a well-known property of ellipses: the sum of the distances from any point on the ellipse to the two foci is always the same value, {2a}, the length of the major axis. The gardener puts down two stakes and loops a piece of rope around them. Using a stick, he pulls the loop taut, marking the points around a curve. This is illustrated here.

Ellipse-GardenersMethod

Gardener’s method of drawing an ellipse [Image Wikimedia].

Definition of an n-Ellipse

A circle with centre at {\mathbf{p}_0} and radius {k} is defined by the `level set’ {|\mathbf{r}-\mathbf{p}_0| = k,} where {\mathbf{r}} is an arbitrary point on the circumference. For an ellipse with foci at {\mathbf{p}_1} and {\mathbf{p}_2} we have

\displaystyle |\mathbf{r}-\mathbf{p}_1| + |\mathbf{r}-\mathbf{p}_2| = k

More generally, we consider a set of {n} foci and define the distance sum

\displaystyle d(\mathbf{r}) = \sum_{j=1}^{n} |\mathbf{r}-\mathbf{p}_j|

The set of points {\mathbf{r}} for which {d(\mathbf{r}) = k}, the level set, is called an {n}-ellipse. Thus, a 1-ellipse is a circle and a 2-ellipse is a standard common-or-garden ellipse.

Drawing a 3-Ellipse

Suppose we have three fixed points, the foci of a 3-ellipse. Once the distance sum to the three foci is fixed, the curve is fully determined. However, it has to be computed numerically. Here, we consider a rough-and-ready way of approximating the 3-ellipse.

We imagine a loop of rope around three stakes. If it is drawn tight by a stick that is moved around, a curve is traced out, as in the gardener’s method. However, the result will not be a single mathematical curve. The rope can be taut around two stakes or three. As the stick moves around, different sets of two or three stakes act to constrain the rope.

The result is a closed curve with six different segments joined up smoothly but not analytically. Each segment is the arc of a different ellipse. Let the three foci be represented by the three vertices of a triangle. For each of the three pairs, there is a family of ellipses. Two members of each family are shown here.

Three-Ellipse-Pairs

Three foci. Each pair of foci determines a family of ellipses. Two ellipses from each family are shown.

We can select elliptical arcs from the three families that, together, form a smooth curve (whose tangent varies smoothly with arc-length). Such a selection is shown here.

Six-Ell-Arc

Left: Ellipses from the three families. Right: six elliptical arcs, smoothly joined.

Finally, we compare the curve constructed piece-wise with the true 3-ellipse. The elliptical arcs are shown in blue, green and magenta, while the analytical curve is in black. The agreement between the two curves is reasonably close.

SixArcVnellipse

Colour: Six elliptical arcs, smoothly joined. Black: Exact 3-ellipse.

Of course, the gardener’s method will not work unless all the foci are within the n-ellipse. Can it be modified to cater for the case of one or more foci outside the curve?

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