Is there a geometric construction to determine an ellipse tangent to an arbitrary curve?

Answer

Dear Theophile,

The answer to your question is generally no. This is because your boundary conditions generally do not allow this. Let me clarify what I mean, because the answer is YES if the arbitrary curve, exactly 1 focus of the ellipse and the tangent point on the arbitrary curve are given.

What goes wrong if two focal points are given and not the tangent point on the curve? The problem is simple. Since the curve cannot be described analytically, you must be able to construct the tangent point geometrically. This tangent point is an intersection with the arbitrary curve which can be a very ‘dirty’ function, so this intersection need not be a so-called ‘constructable point’.

The situation is even much worse. If you do have an analytical prescription for the curve, then you must analytically determine an intersection with the curve, namely the tangent point on the curve. This function will have to be sufficiently good (or simple) to be able to do this analytically. If this function is not simple enough, you will either have to approximate the curve or use a numerical algorithm (such as Newton-Gauss or Levenberg-Marquardt).

Let’s change your boundary conditions. Suppose we know where we want to hit the curve, and know a focal point. Then there will be an infinite number of ellipses tangent to the curve in your chosen tangent point with your chosen focal point. Below is a method that constructs one, namely the easiest.

In the figures you can see the different steps that you have to go through. The first figure is the starting position. Then you draw the tangent line through the tangent point C (Fig. 2). Next, mirror the focal point F1 from the tangent to the point F1′ (Fig. 3). In the next step, draw the horizontal through the focal point F1 (Fig.4). Then you draw the line through the points C and F1′ and the intersection of these two lines is the focal point F2 (Fig. 5). (Actually any point, on the same side of the curve as the focal point F1, on the line CF1′ is a good second tangent point. This just ensures that there are infinitely many tangent ellipses)

Now we’re going to create the minor axis of the ellipse. The midpoint between F1 and F2 is the center of the ellipse. Draw the vertical through the center point. Measure the sum of the distances from F1 to C and from C to F2. Measure the point on the vertical axis where the distance to F1 is just half the distance you just calculated. That is the point B and along the bottom the point B’ (Fig. 6). The long axis is easier, it’s half the distance you calculated. This gives you points A and A’. Then you have everything to draw the ellipse. (Fig.7)

Hopefully I have offered you a nice alternative, because in general mathematics will not provide you with an answer with your conditions.

A very nice book about such problems is: Eagles TH, Constructive geometry of plane curves. With numerous examples, Ann Arbor, Michigan: University of Michigan Library, 2005.

Regards,

Kurt