We investigate the relations between the structure of the image and events in the geometry of the underlying surface. We introduce some elementary differential geometry and use it to define a coordinate system on the object based on the lines of curvature. Using this coordinate system we can prove results connecting the extrema, ridges and zero-crossings in the image to geometrical features of the object. We show that extrema of the image typically correspond to points on the surface with zero Gaussian curvature and that parabolic lines often give rise to ridges, or valleys, in the image intensity. We show that directional zero-crossings of the image along the lines of curvature generally correspond to extrema of curvature along such lines.
