Minimal number of points to define an elliptic arc?

Question

What is the minimal number of points to uniquely define an elliptic arc (portion of an ellipse) ?

The points are not restricted to be part of the path of the ellipse, each of them can have a different meaning/semantics (i.e. they can be "the center", "a focus", or any other reference).

So far my idea is:

• Point #1: one endpoint of the elliptic arc
• Point #2: the other endpoint of the elliptic arc
• Point #3: the center of the ellipse

But, how many concentric ellipses that pass through the same two points are there? Only 2? ... or many?

Answer 1

In general something else is needed. If the two endpoints of the arc happen to be on the line through the center, and the ellipse is not rotated (so has its major/minor axes parallel to the $x,y$ axes), then there will be an infinite family of such elliptic arcs, not to mention the choice of whether to use the top or bottom or left or right arc.

Answer 2

For an ellipse, you need five independent pieces of information, so for an arc, which has two more degrees of freedom, you would need seven. Note that this isn't always enough to uniquely determine the arc, but it is enough to ensure that only finitely many possible arcs exist.

A point on the arc is a single piece. An end point of the arc is two pieces, and the length of the arc is one piece. A focal point of the ellipse is two pieces, as is the center. The line through the focal points is two pieces, although if you already know the center or one of the focal points, it is only a single piece of information.

Answer 3

Three points and a real are enough.

You can describe the ellipse by a focus, excentricity and directrix.

• focus: one point or two reals
• directrix: an arbitrary line not passing through the focus, so two reals (say, distance from focus and orientation). That can be a point too: the point of the directrix nearest to the ellipse.
• eccentricity: one real

You then need two reals to describe the two endpoints (say, distance from the point nearest to the directrix, clockwise).

That's 7 reals, which you can "encode" in 3 points and a real.

Of course, you could theoretically encode the seven reals within only one real, since there is a bijection between $\Bbb R^7$ and $\Bbb R$, but that's cheating.