# Our Mathematical Senses Part 1: Geometry of Vision

By Prof. Vijay Ravikumar   |   CMI
Learners enrolled: 1337
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 Learners
PRE-REQUISITES   : Linear Algebra is the only prerequisite
Summary
 Course Status : Completed Course Type : Elective Duration : 4 weeks Start Date : 23 Aug 2021 End Date : 17 Sep 2021 Exam Date : 24 Oct 2021 IST Enrollment Ends : 30 Aug 2021 Category : Mathematics Credit Points : 1 Level : Undergraduate

### Course layout

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

### Books and references

• The Four Pillars of Geometry by John Stillwell
• Foundations of Projective Geometry by Robin Hartshorne
• Perspective and Projective Geometry by Annalisa Crannell, Marc Frantz, and Fumiko Futamura
• Projective Geometry by HM Coxeter

### Prof. Vijay Ravikumar

CMI
Prof. Vijay Ravikumar is a mathematician and artist based in Chennai, with an interest in math  outreach. He was an Assistant Professor at the Chennai Mathematical Institute from 2016 to 2020.  He received his PhD from Rutgers University in 2013.

### Course certificate

The course is free to enroll and learn from. But if you want a certificate, you have to register and write the proctored exam conducted by us in person at any of the designated exam centres.
The exam is optional for a fee of Rs 1000/- (Rupees one thousand only).
Date and Time of Exams: 24 October 2021 Morning session 9am to 12 noon; Afternoon Session 2pm to 5pm.
Registration url: Announcements will be made when the registration form is open for registrations.
The online registration form has to be filled and the certification exam fee needs to be paid. More details will be made available when the exam registration form is published. If there are any changes, it will be mentioned then.
Please check the form for more details on the cities where the exams will be held, the conditions you agree to when you fill the form etc.

CRITERIA TO GET A CERTIFICATE

Average assignment score = 25% of average of best 3 assignments out of the total 4 assignments given in the course.
Exam score = 75% of the proctored certification exam score out of 100

Final score = Average assignment score + Exam score

YOU WILL BE ELIGIBLE FOR A CERTIFICATE ONLY IF AVERAGE ASSIGNMENT SCORE >=10/25 AND EXAM SCORE >= 30/75. If one of the 2 criteria is not met, you will not get the certificate even if the Final score >= 40/100.

Certificate will have your name, photograph and the score in the final exam with the breakup.It will have the logos of NPTEL and IIT Madras .It will be e-verifiable at nptel.ac.in/noc.

Only the e-certificate will be made available. Hard copies will not be dispatched.

Once again, thanks for your interest in our online courses and certification. Happy learning.

- NPTEL team