Given a real matrix $X$ with $n$ rows and 2 columns, can the matrix be transformed to a real matrix $Y$ such that all the points formed by the rows of $Y$ lie on a circle (2d) and their inter-point curved/geodesic distances on the circle approximate or are exactly equal to the the inter-point Euclidean distances (straight line distances) between the rows of $X$.
Question: Can all the distances be preserved with no error for all real matrices $X$ given that the points in the rows of $X$ are 2 dimensional (X has two columns) and the circle is a 2d curve (not a n-sphere in 3 or greater dimensional).
I believe the spherical distance between $d_s(Y_{i.},Y_{j.})=arccos(x_{i.},x_{j.}^T)$ for rows $i,j$?