What is the minimum information required to define an equation for ellipse?

Question

What is the minimum information ie. amount of points in 2-dimensional plane in order to define the equation for an ellipse?

I know that unique ellipse cannot be defined when only one of the foci is known in addition to two points in the ellipses path.

I also know, that we can define unique ellipse when both foci are known and the distance from both of them onto point in ellipse (this is as far as I know the least information required).

Answer 1

I am interpreting the question as asking for a minimum number of points that determine a unique ellipse passing through them.

Four points is not enough. The two ellipses $$ x^2+2y^2=3\qquad\text{and}\qquad 2x^2+y^2=3 $$ both pass through the points $(x,y)=(\pm1,\pm1)$ (all four sign combinations).

Five points $P_j=(x_j,y_j), j=1,2,3,4,5,$ (in general) does suffice. But we get more than we bargained for. Namely five points determine a conic. The general quadratic equation $$ a_1x^2+a_2xy+a_3y^2+a_4x+a_5y+a_6=0\qquad(*) $$ has six unknown coefficients. By plugging in the coordinates of the points $P_j$ we get a homogeneous system of five equations in the six unknowns. Linear algebra then tells us that the system has a non-trivial solution. Usually the solution is determined up to a scalar multiple, but scaling all the coefficients in $(*)$ won't change the curve, so we can ignore that.

Two caveats:

• The resulting conic may be a hyperbola, parabola, or even two intersecting lines - not necessarily an ellipse. All depending.
• The system that we get may have a 2-dimensional solution space (in which case the curve would not be determined uniquely). But this happens only, when some subset of four points are collinear, so we will not encounter this problem when looking at ellipses. Though well known, this fact is not immediately obvious, and needs an argument. In Joe Harris' book this is an exercise, but does follow from a more general theorem (Theorem 1.18) stating that through any $d+3$ points in general position in $\Bbb{P}^d$ there passes a unique rational normal curve.

Answer 2

I believe that the following equation defines an ellipse:

$\sqrt{(x-x_1)^2 + (y-y_1)^2} + \sqrt{(x-x_2)^2 + (y-y_2)^2} = 2a$

The foci are the two points: $(x_1,y_1) , (x_2,y_2)$

Since there are 5 unknown, there is a need for 5 points.