  Starting with the data of a curve of singularity types, we use the Legendre
transform to construct weak geodesic rays in the space of locally bounded
metrics on an ample line bundle L over a compact manifold. Using this we
associate weak geodesics to suitable filtrations of the algebra of sections of
L. In particular this works for the natural filtration coming from an algebraic
test configuration, and we show how in this case we recover the weak geodesic
ray of Phong-Sturm.
