At each point of an ellipse one can attach a normal vector and a radial vector, the latter defined as the vector from the center of the ellipse. At the ends of the major and minor axes, the two vectors coincide, but at all other points they are separated by an angle [delta]. What is the maximum value that [delta] can attain, and where does it occur? This question leads to a surprisingly rich area of investigation, encompassing several different analytic solutions, geometric interpretations, an extension to "n" dimensions, and a geographical application.