This article addresses the question of describing the differential properties of shapes which are invariant to the action of a group. The shapes of interest are differentiable manifolds such as curves and surfaces but can also be differentiable sets of lines such as complexes or congruences. The groups of interest in computer vision are the euclidean, affine (or unimodular affine) and projective groups. Among the methods that can be used to obtain such descriptions there is one that clearly emerges because of its simplicity, elegance, generality, and because it is quite amenable to computer implementation. This method is known as the Cartan's moving frame method and has been developed in the first decades of this century by Elie Cartan and his students [2, 3]. The method is widely used in mathematics and physics but has not yet attracted many researchers in computer vision with the notable exception of ter Haar Romeny and his coworkers [17]. In section 2 of this article we give a detailed description of the moving frame method which is completely general and can be used (and automated) in all practical cases. This description uses the tools of the modern exterior differential calculus which were being invented at the time Cartan was developing his moving frame method and is an extended version of what can be found in [2]. We then attempt to help the reader develop some intuition about how the method actually works by using it on three simple and useful examples: plane curves subject to the action of the euclidean, affine, and projective groups. To help even further the intuition we present geometric interpretations of the affine and projective arc lengths. We also relate projective and affine invariants to the more familiar euclidean ones. We found these relations quite...

This paper examines projectively invariant local properties of smooth curves and surfaces. Oriented projective differential geometry is proposed as a theoretical framework for establishing such invariants and describing the local shape of surfaces and their outlines. This framework is applied to two problems: a projective proof of Koenderink's famous characterization of convexities, concavities, and inflections of apparent contours; and the determination of the relative orientation of rim tangents at frontier points.