  A point P on a smooth hypersurface X of degree d in an N-dimensional
projective space is called a star point if and only if the intersection of X
with the embedded tangent space T_P(X) is a cone with vertex P. This notion is
a generalization of total inflection points on plane curves and Eckardt points
on smooth cubic surfaces in three-dimensional projective space. We generalize
results on the configuration space of total inflection points on plane curves
to star points. We give a detailed description of the configuration space for
hypersurfaces with two or three star points. We investigate collinear star
points and we prove that the number of star points on a smooth hypersurface is
finite.
