This thesis realises the need for describing computer curves and surfaces in terms of intrinsic quantities and certain properties relative to the Euclidean space in which they are embedded. Chapter 1 introduces some of the ideas and problems involved in what can be termed computational differential geometry. Chapter 2 presents some analysis of the major types of computer curves in terms of a number of shape control parameters. Chapter 3 gives a similar analysis of computer surfaces. Chapter 4 considers the' calculus of variations in connection with the minimal immersion and a particular invariantly defined functional analogous to energy. Chapter 5 applies the energy functional to a class of computer curves. Chapter 6 looks at a number of surfaces in relation to surface mappings and distortion. Some mappings are also derived. This generally involves the solution of non linear differential equations the linearisation of which will almost certainly remove the salient features of the theory. A bibliography and a number of figures are provided following chapter 6.