Definition:Ellipse/Equidistance

Definition

 * EllipseEquidistance.png

Let $F_1$ and $F_2$ be two points in the plane.

Let $d$ be a length greater than the distance between $F_1$ and $F_2$.

Let $K$ be the locus of points $P$ which are subject to the condition:
 * $d_1 + d_2 = d$

where:
 * $d_1$ is the distance from $P$ to $F_1$
 * $d_2$ is the distance from $P$ to $F_2$.

Then $K$ is an ellipse.

This property is known as the equidistance property.

The points $F_1$ and $F_2$ are the foci of $K$.

Proof
Let $K$ be an ellipse aligned in a cartesian plane in reduced form.

Thus its foci are at $\tuple {\mathop \pm c, 0}$.

It will be shown that $d_1 + d_2 = d$ as above, being equivalent to pin-and-string construction (with the foci being pins, and $d$ being the length of a string), will produce an ellipse.

This is the form of a unit-radius circle $x^2 + y^2 = 1$ but with scaling factors applied to $x$ ($\dfrac d 2$) and $y$ ($\sqrt {\dfrac {d^2} 4 - c^2}$).

This is the definition of an ellipse aligned in a cartesian plane in reduced form.

Also see

 * Equivalence of Definitions of Ellipse
 * Equidistance of Ellipse equals Major Axis