With the definition of discrete lines introduced by Réveillès [REV91], there has been a wide range of research in discrete geometry and more precisely on the study of discrete lines. By the use of the linear time segment recognition algorithm of Debled and Réveillès [DR94], Vialard [VIA96a] has proposed a O(l) algorithm for computing the tangent in one point of a discrete curve where l is the average length of the tangent. By applying her algorithm to n points of a discrete curve, the complexity becomes O(n.l). This paper proposes a new approach for computing the tangent. It is based on a precise study of the tangent evolution along a discrete curve. The resulting algorithm has a O(n) complexity and is thus optimal. Some applications in curvature computation and a tombstones contours study are also presented.

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With the aim of elaborating a mobile application, accessible to anyone and with educational purposes, we present a method for tree species identification that relies on dedicated algorithms and explicit botany-inspired descriptors. Focusing on the analysis of leaves, we developed a working process to help recognize species, starting from a picture of a leaf in a complex natural background. A two-step active contour segmentation algorithm based on a polygonal leaf model processes the image to retrieve the contour of the leaf. Features we use afterwards are high-level geometrical descriptors that make a semantic interpretation possible, and prove to achieve better performance than more generic and statistical shape descriptors alone. We present the results, both in terms of segmentation and classification, considering a database of 50 European broad-leaved tree species, and an implementation of the system is available in the iPhone application Folia 1 .