ABSTRACT
Differential geometry constitutes of a set of mathematical methods and operators which are useful in computing geometric quantities of discrete continuous structures using differential calculus as discussed by Koenderink [28],[29]. It can be described as a mathematical tool box for describing shape through derivatives. The assumption made in differential geometry is that the geometrical structures such as curves, surfaces, lattices etc. are everywhere differentiable and there are no sharp discontinuities such as corners or cuts. In differential geometry of shape measures, there is no global coordinate system. All measurements are made relative to the local tangent plane or normal. Although image is spatial data, there is no coordinate system associated with an image. In case of a geo-coded, geo-referenced satellite image, a global coordinate system such as UTM (Universal Transverse Mercator) is implicitly associated. Also for the sake of referencing, the positional value of each pixel in an image it is often associated with a coordinate system that defines the position of the origin. The intensity of the images is treated as perpendicular to the image plane or the ‘Z’ coordinate of the image making it a 3D surface. Further the application of pure geometry in GIS is discussed by Brannan et al. [6].
