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At each point in an immersed surface in $\mathbb R^4$ there is a curvatureellipse in the normal plane which codifies all the local second order geometryof the surface. More recently, at the singular point of a corank 1 singularsurface in $\mathbb R^3$, a curvature parabola in the normal plane whichcodifies all the local second order geometry has been defined. When projectinga regular surface in $\mathbb R^4$ to $\mathbb R^3$ in a tangent directioncorank 1 singularities appear generically. The projection has a cross-capsingularity unless the direction of projection is asymptotic, where moredegenerate singularities can appear. In this paper we relate the geometry of animmersed surface in $\mathbb R^4$ at a certain point to the geometry of theprojection of the surface to $\mathbb R^3$ at the singular point. In particularwe relate the curvature ellipse of the surface to the curvature parabola of itssingular projection.

A relation between the curvature ellipse and the curvature parabola