Neste trabalho estudamos a geometria da evoluta afim e da curva normal afim associada à uma curva plana sem inflexões a partir do tipo de singularidade das funções suporte afim. O principal resultado estabelece que se \'\\gamma\' é uma curva plana sem inflexões, satisfazendo certas condições genéricas então dois casos podem ocorrer: 1. se p é um ponto da evoluta afim de \'\\gamma\' em \'s IND. 0\' então temos dois casos: se \'\\gamma\' (\'s IND. 0\') é um ponto sextático então, localmente em p, a evoluta afim é difeomorfa a uma cúspide em \'R POT. 2\' ; se não, localmente em p, a evoluta afim é difeomorfa à uma reta em \'R POT. 2\' , 2. se p = \'\\gamma\' (\'s IND. 0\') é um ponto da normal afim de \'\\gamma\' então temos dois casos: se \'\\gamma\'(\'s IND. 0\') é um ponto parabólico de \'\\gamma\' então, localmente em p, a curva normal afim é difeomorfa a uma cúspide em \'R POT. 2\' ; em outro caso, localmente em p, a curva normal afim é difeomorfa à uma reta em \'R POT. 2\' / In this work we study the geometry of the affine evolute and the affine normal curve associated with a plane curve without inflections from the type of singularity of affine support functions. The main result is setting if \'\\gamma\' is a flat curve without inflections, satisfying certain conditions generic then, if p is a point of the affine evolute of \'\\gamma\' at \'s IND. 0\' then two cases: if \'\\gamma\' (\'s IND. 0\') is a sextactic point then locally in p the affine evolute is diffeomorphic to a cusp at \'R POT. 2\', otherwise locally in p the affine evolute is diffeomorphic to a straight in \'R POT. 2\', and second if p = \'\\gamma\' (\'s IND. 0\') is a point of the affine normal curve then two cases: if \'\\gamma\'(\'s IND. 0\') is a parabolic point of \'\\gamma\' then locally in p the affine normal curve is diffeomorphic to a cusp at \'R POT. 2\' , in otherwise locally in p the affine normal curve is diffeomorphic to a line in \'R POT. 2\'

Neste trabalho estudamos a geometria da evoluta afim e da curva normal afim associada à uma curva plana sem inflexões a partir do tipo de singularidade das funções suporte afim. O principal resultado estabelece que se \'\\gamma\' é uma curva plana sem inflexões, satisfazendo certas condições genéricas então dois casos podem ocorrer: 1. se p é um ponto da evoluta afim de \'\\gamma\' em \'s IND. 0\' então temos dois casos: se \'\\gamma\' (\'s IND. 0\') é um ponto sextático então, localmente em p, a evoluta afim é difeomorfa a uma cúspide em \'R POT. 2\' ; se não, localmente em p, a evoluta afim é difeomorfa à uma reta em \'R POT. 2\' , 2. se p = \'\\gamma\' (\'s IND. 0\') é um ponto da normal afim de \'\\gamma\' então temos dois casos: se \'\\gamma\'(\'s IND. 0\') é um ponto parabólico de \'\\gamma\' então, localmente em p, a curva normal afim é difeomorfa a uma cúspide em \'R POT. 2\' ; em outro caso, localmente em p, a curva normal afim é difeomorfa à uma reta em \'R POT. 2\' / In this work we study the geometry of the affine evolute and the affine normal curve associated with a plane curve without inflections from the type of singularity of affine support functions. The main result is setting if \'\\gamma\' is a flat curve without inflections, satisfying certain conditions generic then, if p is a point of the affine evolute of \'\\gamma\' at \'s IND. 0\' then two cases: if \'\\gamma\' (\'s IND. 0\') is a sextactic point then locally in p the affine evolute is diffeomorphic to a cusp at \'R POT. 2\', otherwise locally in p the affine evolute is diffeomorphic to a straight in \'R POT. 2\', and second if p = \'\\gamma\' (\'s IND. 0\') is a point of the affine normal curve then two cases: if \'\\gamma\'(\'s IND. 0\') is a parabolic point of \'\\gamma\' then locally in p the affine normal curve is diffeomorphic to a cusp at \'R POT. 2\' , in otherwise locally in p the affine normal curve is diffeomorphic to a line in \'R POT. 2\'

This thesis is concerned with the utilization of object symmety as a cue for segmentation and object recognition. In particular it investigates the problem of detecting 3D bilaterally symmetric objects from affine views. The first part of the thesis investigates the problem of detecting 3D bilateral symmetry within a scene from known point correspondences across two or more affine views. We begin by extending the notion of skewed symmetry to three dimensions, and give a definition in terms of degenerate structure that applies equally to an affine 3D structure or to point correspondences across two or more affine views. We then consider the effects of measurement errors on symmetry detection, and derive an optimal statistical test of degenerate structure, and thereby of 3D-skewed symmetry. We then move on to the problem of searching for 3D skewed symmetric sets within a larger scene. We discuss two approaches to the problem, both of which we have implemented, and we demonstrate fully automatic detection of 3D skewed symmetry on images of uncluttered scenes. We conclude the first part by investing means of verifying the presence of bilateral rather than skewed symmetry in the Euclidean space, by enforcing mutual consistency between multiple skewed symmetric sets, and by drawing on partial knowledge about the camera calibration. The second part of the thesis is concerned with the problem of obtaining feature correspondences across multiple affine views, as required for the detection of symmetry. In particular we investigate the geometric matching constraints that exist between affine views. We start by specilizing the four projective multifocal tensors to the affine case, and use these to carry the bulk of all known projective multi-view matching relations to affine views, unearthing some new relations in the process. Having done that, we address the problem of estimating the affine tensors. We provide a minimal set of constraints on the affine trifocal tensor, and search for ways of estimating the affine tensors from point and line correspondences.