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Geometry Of Vision

Overview

ABOUT THE COURSE:This course will explore how one of our senses -- our sense of sight -- relies on our brain's seemingly hardwired understanding of a fascinating geometric space. What's more, this space -- known as the two dimensional Projective Plane -- is surprisingly different from the familiar Euclidean plane.By examining some basic questions related to perspective drawing, like why parallel lines appear to converge in our vision, we can identify and analyze the defining properties of this space — properties which are well known to our visual sense, but defy our logical intuition.As the course progresses, we'll situate perspective drawing in the framework of Projective Geometry, and get familiar with the algebra of homogeneous coordinates, which will allow us to translate our sensory intuition into precise information we can easily communicate. We'll also get to know the topological space known as the real projective plane, as well the matrix group that governs its transformations.Being a four-week long elective, this course will nicely complement standard courses on geometry and topology, by giving hands-on experience with several important mathematical objects you’ll meet in those classes. Some basic linear algebra will be required, especially in the final week of the course (in particular, we will assume familiarity with vector calculus and with matrices as linear transformations).As part of the series 'Our Mathematical Senses', we plan to offer follow-up courses in subsequent semesters, exploring our sense of spatial orientation (the Topology of Movement) and our sense of hearing (the Calculus of Sound).INTENDED AUDIENCE :Any Interested LearnersPRE-REQUISITES :Linear Algebra is the only prerequisite

Syllabus

WEEK 1: Perspective Drawing from First Principles Lecture 1: Why do families of parallel lines appear to converge in our vision? Lecture 2: Can we determine the position of the observer in a perspective drawing?   WEEK 2: An Invitation to Projective Geometry Lecture 3: A geometry without parallel lines? Lecture 4: Does projective geometry tell us anything about nonlinear curves?   WEEK 3: The Cross Ratio: A Projective Invariant Lecture 5: What is preserved under perspective transformations? Lecture 6: What can the cross ratio tell us?   WEEK 4: Automorphisms of the Projective Plane Lecture 7: Homogeneous coordinates and the topology of the real projective plane Lecture 8: Automorphisms of the real projective plane

Taught by

Prof. Vijay Ravikumar

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