||In this thesis, the problem of the deformation of a planar curve with constraints on its length is studied. Curve length is an integral property, which is typically computed by numerical methods; this makes the deformation problem a complex one. Recently some researchers have attempted to solve such problems for multi-resolution representations of curves. However, we take a differential geometric approach. The modification problem is formulated as a constrained optimization problem, which is subsequently converted to an unconstrained minimization problem. Two techniques are explored to solve the resulting system: the Uzawa method and a Newtonian method. In the latter approach, we attempt to take advantage of the finding that the NURBS length function is convex with respect to the coordinates of the control points. Both approaches are implemented in MATLAB™, and some examples are presented.