# '3-point' curve

If you have a loop of string, a fixed point and a pencil, and stretch the string as much as possible, you draw a circle. With 2 fixed points you draw an ellipse. What do you draw with 3 fixed points?

• What exactly is the setup you're imagining? Is it a string whose ends are fixed and has another fixed point in the middle, or are you imagining a loop with three fixed points? – Hayden Jan 31 '15 at 12:41
• I have a loop of string with n fixed points inside. Similarly therefore to constructing a circle (with 1 fixed point) – JMP Jan 31 '15 at 12:57
• You mean 3 fixed points? – Narasimham Jan 31 '15 at 15:41
• In this question yes. My previous comment is a further generalization and only relevant as such. You are free to attempt to answer it if you want. – JMP Jan 31 '15 at 15:51

Assume that three non-collinear points $A$, $B$, $C$ in the plane are given, and that you have a loop of string of length $\ell>|AB|+|BC|+|CA|$. Slinging this string around the three points and a pencil you can draw a loop $\gamma$ around $\triangle(ABC)$ in the obvious way, keeping the string tight at all times. This loop will be a continuous curve. In order to describe $\gamma$ more precisely we draw the lines $A\vee B$, $B\vee C$, $C\vee A$ in full. In this way the plane is divided into $7$ compartments, one of them the triangle $\triangle(ABC)$. The loop $\gamma$ traverses the $6$ unbounded compartments, and at each intersection with one of the above lines it has a corner. Within a given compartment $\gamma$ is an arc of an ellipse, whereby two of the three vertices $A$, $B$, $C$ act as foci. The points $A$, $B$ are foci for two such arcs. One of these arcs belongs to an ellipse with major axis $\ell-|AB|$ and the other to an ellipse with major axis $\ell-|BC|-|CA|$.

Let length of sting be > max(AB,BC,CA). If the given three points are A,B and C are not in a straight line, then

there are three possibilities to trace three ellipses and the locus is composed of the these three discontinuous ellipses.

Center point B can be ignored when inter-focal distance is AC,

Center point C can be ignored when inter-focal distance is BA, and

Center point A can be ignored when inter-focal distance is BC.

So in effect each is only a 2-point curve.

• Draw a figure and check what is really happening. By the way: What is a "discontinuous ellipse"? – Christian Blatter Jan 31 '15 at 12:57
• Since there are 3 foci taken two at a time the locus cannot be a single ellipse without 2nd order curvature discontinuities. Can upload a picture later. – Narasimham Feb 1 '15 at 11:41