  At each point in an immersed surface in $\mathbb R^4$ there is a curvature
ellipse in the normal plane which codifies all the local second order geometry
of the surface. More recently, at the singular point of a corank 1 singular
surface in $\mathbb R^3$, a curvature parabola in the normal plane which
codifies all the local second order geometry has been defined. When projecting
a regular surface in $\mathbb R^4$ to $\mathbb R^3$ in a tangent direction
corank 1 singularities appear generically. The projection has a cross-cap
singularity unless the direction of projection is asymptotic, where more
degenerate singularities can appear. In this paper we relate the geometry of an
immersed surface in $\mathbb R^4$ at a certain point to the geometry of the
projection of the surface to $\mathbb R^3$ at the singular point. In particular
we relate the curvature ellipse of the surface to the curvature parabola of its
singular projection.
