Suppose one has a string of length L, one end P0 fixed at -B/2, the other P1 fixed at B/2 (B < L). If it's more convenient, the ends of the string can be fixed at zero and B
For some FIXED point A on the string, what is the equation of the shape created if A is pushed to its extremes and rotated in all directions? i.e. What is the equation of the path of A, which bounds the possible locations of A?
NOTE: This is similar to — but NOT the same as — an ellipse, since the point A is fixed on the string, and I am interested in TWO parameterized curves.
MOTIVATION: I am exploring the joint distribution of distances on a string modeled as scaled jointly dirichlet random variables. If $X_{ij}$ is $Beta(\alpha_{ij},\beta_{ij})$, and $\ell_{ij}$ is the length of the string connecting point $i$ to point $j$, then $D_{ij}=\ell_{ij}X_{ij}$.
Put another way, if $\vec{x}$ ~ $Beta(\vec{\alpha},\vec{\beta})$ and $L$ is a diagonal matrix $\{\ell_{ij}\}_{i<j}$, then my random vector of distances are $D=L\vec{X}$.
Clearly, the set of possible distances of P0 to A and the distances of P1 to A depend heavily on the distance of P0 to P1, so I am interested in incorporating this covariance structure into my parameterization.