Definition:Ellipse/Equidistance

Definition

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 $K$.

Also see

• Equivalence of Definitions of Ellipse
• Equidistance of Ellipse equals Major Axis
• Equation of Ellipse in Reduced Form

Sources

• 1933: D.M.Y. Sommerville: Analytical Conics (3rd ed.) ... (previous) ... (next): Chapter $\text {IV}$. The Ellipse: $1 \text a$. Focal properties
• 1937: Eric Temple Bell: Men of Mathematics ... (previous) ... (next): Chapter $\text{VI}$: On the Seashore
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): ellipse
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): ellipse
• 2008: Ian Stewart: Taming the Infinite ... (previous) ... (next): Chapter $2$: The Logic of Shape: Problems for the Greeks
• 2008: Ian Stewart: Taming the Infinite ... (previous) ... (next): Chapter $6$: Curves and Coordinates: Fermat

• Weisstein, Eric W. "Ellipse." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Ellipse.html