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Projective Geometry: A Short Introduction

Projective geometry : A Short IntroductionLecture NotesEdmond BoyerMaster MOSIGI ntroduction to Projective GeometryContents1 Objective .. Historical Background .. Bibliography ..42 Projective Definitions .. Properties .. The hyperplane at infinity .. 123 The Projective Introduction .. Projective transformation ofP1.. The cross-ratio .. 144 The Projective Points and lines .. Line at infinity .. Homographies .. Conics .. Affine transformations .. Euclidean transformations .. Particular transformations .. Transformation hierarchy .. 25 Grenoble Universities1 Master MOSIGI ntroduction to Projective GeometryChapter ObjectiveThe objective of this course is to give basic notions and intuitions onprojectivegeometry. The interest of Projective geometry arises in several visual comput-ing domains, in particular computer vision modelling and computer provides a mathematical formalism to describe the geometry of cameras andthe associated transformations, hence enabling the design of computational ap-proaches that manipulates 2D projections of 3D objects.

Master MOSIG Introduction to Projective Geometry InÞnity non-parallel lines parallel lines Figure 1.2: Line intersections in a projective space 1.2 Historical Background The origins of geometry date back to Egypt and Babylon (2000 BC). It was rst designed to address problems of everyday life, such as area estimations and

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