For a compact oriented
smooth surface immersed in Euclidean four-space (thought of as complex
two-space), the sum of the tangential and normal Euler numbers is equal
to the algebraic number of points where the tangent plane is a complex
line. This follows from the construction of an explicit homology between
the zero-chains of complex points and the zero-chains of singular points of
projections to lines and hyperplanes representing the tangential and normal Euler
classes.