I'm trying to find an equation that draws an ellipse in the same vein as drawing it with the pin and string method. Basically inputting the $2$ "pin" locations and the max distance between the $2$ pins aka the string length.
From my search, the pin locations would be called foci and the distance between them would be $2\cdot c$ where $$ c=\sqrt(a^2+b^2) $$
and the string length would be $$ \text{String Length} = 2b + 2c $$
The ellipse would not have to be locked to the $x$ or $y$ axis. Long story short, I have $2$ locations (points $A$ and $B$) on a map that I need to draw an ellipse around with the distance from $A$ to $E$ (point on the ellipse) $+$ distance from $B$ to $E = \text{max range}$.
Any help would be greatly appreciated.
Here is a way to organise the data:
Suppose the two fixed points are $A=(a, b)$ and $B= (c, d)$.
Let the distance $AB = l$.
Let the string have length $S=l+L$
Using the Euclidean distance between a point $P=(x,y)$ and the two points $A$ and $B$ you want $$\sqrt {(x-a)^2+(y-b)^2}+\sqrt {(x-c)^2+(y-d)^2}=L$$
It will take a little ingenuity to convert this to a more familiar form.