Active Shape Models (ASM) use an iterative algorithm to match statistically defined models of known but variable objects to instances in images. Each iteration of ASM search involves two steps: image data interrogation and shape approximation. Here we consider the shape approximation step in detail. We present a new method of shape approximation which uses directional constraints. We show how the error term for the shape approximation problem can be extended to cope with directional constraints, and present iterative solutions to the 2D and 3D problems. We also present an efficient algorithm for the 2D problem in which a modification of the error term permits a closed-form approximate solution which can be used to produce starting estimates for the iterative solution.