Geometry of Vision

By Prof. Vijay Ravikumar   |   CMI
Learners enrolled: 977

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. Finally, we'll  get to know the topological space known as the real projective plane, as well as 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.  But much of the course will also be interesting for artists, designers, or anyone else looking to understand deeper connections between math and art that humankind has been exploring for thousands of years.

The course is best experienced through the companion page: https://this-vijay.github.io/vision-math/
Here you can easily navigate between videos, lectures, and chapters, understand the context of each video, and access interactive exercises that transition from one video to the next.

INTENDED AUDIENCE : Any Interested Learners.
PRE-REQUISITES   : None, but some experience with mathematical proofs will be useful. Linear Algebra will be useful in the fourth week, but not essential.
Course Status : Completed
Course Type : Elective
Duration : 4 weeks
Category :
  • Mathematics
Credit Points : 1
Level : Undergraduate
Start Date : 22 Aug 2022
End Date : 16 Sep 2022
Enrollment Ends : 22 Aug 2022
Exam Date : 29 Oct 2022 IST

Note: This exam date is subjected to change based on seat availability. You can check final exam date on your hall ticket.

Page Visits

Course layout

WEEK 1: The Power of Vanishing Points
Lecture 1: Why do images of parallel lines converge?
Lecture 2: Solving puzzles with vanishing points
Lecture 3: Using vanishing points to draw in perspective

WEEK 2: A Geometry of Coincidence
Lecture 4: The extended Euclidean plane
Lecture 5: Three coincidences in projective geometry
Lecture 6: Extending space to prove Desargues's Theorem

WEEK 3: The Shadow of a Square
Lecture 7: A closer look at perspectivities
Lecture 8: Projectivities and the Three-Fixed-Points Theorem
Lecture 9: The Fundamental Theorem of Projective Geometry

WEEK 4: The Analytic Framework
Lecture 10: Examining the cross ratio
Lecture 11: The Real Projective Plane
Lecture 12: Transformations of the Real Projective Plane

Books and references

  1. The Four Pillars of Geometry by John Stillwell
  2. Foundations of Projective Geometry by Robin Hartshorne
  3. Perspective and Projective Geometry by Annalisa Crannell, Marc Frantz, and Fumiko Futamura
  4. Projective Geometry by HM Coxeter

Instructor bio

Prof. Vijay Ravikumar

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: 29 October 2022 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.


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

MHRD logo Swayam logo


Goto google play store