Computer Graphics (CMU 15-462/662)
CMU 15-462/662, Fall 2020
Tue/Thu 1:30-2:50pm
Instructor: Keenan Crane
Course Description

This course provides a comprehensive introduction to computer graphics. It focuses on fundamental concepts and techniques, and their cross-cutting relationship to multiple problem domains in graphics (rendering, animation, geometry, imaging). Topics include: sampling, aliasing, interpolation, rasterization, geometric transformations, parameterization, visibility, compositing, filtering, convolution, curves & surfaces, geometric data structures, subdivision, meshing, spatial hierarchies, ray tracing, radiometry, reflectance, light fields, geometric optics, Monte Carlo rendering, importance sampling, camera models, high-performance ray tracing, differential equations, time integration, numerical differentiation, physically-based animation, optimization, numerical linear algebra, inverse kinematics, Fourier methods, data fitting, example-based synthesis.

[keenanc at andrew]
Your fun and helpful TAs:
Maxwell Slater
[mjslater at andrew]
Office hours:
Monday 1-3pm
Location: Graphics Monday Zoom (see Piazza for link)
Alice Lai
[alicel at andrew]
Office hours:
Tuesday 5:30-7:30pm
Location: Graphics Tuesday Zoom (see Piazza for link)
Vidya Narayanan
[vidyan at andrew]
Office hours:
Wednesday 10am-noon
Location: Graphics Wednesday Zoom (see Piazza for link)
Oscar Dadfar
[odadfar at andrew]
Office hours:
Thursday 5-7pm
Location: Graphics Thursday Zoom (see Piazza for link)
Hesper Yin
[hyin2 at andrew]
Office hours:
Friday 7-9pm
Location: Graphics Friday Zoom (see Piazza for link)
As a policy, we ask that you please do not email TAs directly for help with assignments, grading, etc. TAs can be contacted either through Piazza or during office hours.

Course prerequisites are (15-213, 21-259, and 21-240) or (15-213, 21-259, and 21-241) or (18-213 and 18-202). Basic vector calculus and linear algebra will be an important component of this course. Previous exposure to basic programming in C/C++ or similar languages is very helpful as course programming assignments will involve significant implementation effort.

Remote Lectures

Lecture videos will be pre-recorded and uploaded to YouTube. During the lecture period, you are free to watch the video at your own pace. During this time, the instructor and/or TAs will be available to answer questions via Zoom (see Piazza for Zoom links). So, if you have a question, you can just pause, switch to Zoom, and ask your question. If for some reason you are unable to watch the videos during the lecture period, you are free to ask questions (i) on the lecture slides themselves, via this webpage, (ii) via Piazza, and/or (iii) during office hours.

Some tips for making the most out of videos:

  • Some students have found it helpful to watch videos at 1.25x or 1.5x speed.
  • A few of the videos are broken down into sections via the time index (see the video comments). These are especially helpful for studying, or going back to an earlier concept.
  • Some people have trouble staying focused on a video for a complete 80-minute lecture. If that's true for you, try splitting it up into two (or more) watching sessions. E.g., one 40-minute viewing during the lecture period, and another the next day. The video sections can help you to find a logical break.

Some notes on the Zoom Q&A sessions:

  • Feel free to ask questions either via voice or in the chat. If you ask by voice, I will respond by voice; if you ask via chat, I will respond via chat!
  • You are welcome to come and go as you please, but if you'd like to keep a persistent Zoom running while you watch the lecture video, note that you can disable the Zoom audio temporarily by selecting the “Leave Computer Audio” option from the menu attached to the Mute/Unmute button.
  • Please make sure to mute yourself while you're not asking questions—and please don't be offended if I mute you! ;-)
  • Finally, people seem to like asking questions about all sorts of graphics-related stuff purely out of curiosity. Please don't worry if you don't understand everything that's being discussed. If someone asks me about something that's not directly related to the class/lecture, I'm very glad to answer—but the answer may involve concepts and jargon that you are not expected to know for the class itself. And of course if you do want to know more, just ask! There are no “dumb” questions.


The TAs will hold several recitations throughout the semester, to help supplement lecture material and provide key information about assignments. Recitations will also be used to review solutions to the mini-homework. Please refer to Piazza for the recitation schedule and associated Zoom links.

Discussion Boards

There are two different mechanisms for discussing course content and asking questions—you will need to sign up for both!.

  • Course Web Page — Our course web page has a nice setup that lets you directly ask questions / make comments on course slides. These slides will appear before each lecture. Part of your course participation grade will be to leave one question/comment per lecture (see below for more info).
  • Piazza — We will be using Piazza for general class announcements, as well as homework questions/help and general discussion. The 15-462/662 Piazza page is located here. The link to sign up for the class on Piazza is here.


(15%) Written Homework. Following most lectures, you will be assigned a short “mini homework” assignment (just a few questions) that reinforces the most essential concepts. At the beginning of the semester, you will also complete some written exercises reviewing linear algebra and vector calculus. All written homework can be submitted via GradeScope; mini homeworks must be submitted before the beginning of the next lecture period. Solutions will be reviewed in recitation. To mitigate potential absences (sick days, etc.), students can omit up to four mini homeworks without penalty. Students are encouraged to discuss concepts with their peers, on Piazza, and/or in office hours, but final homework answers must be written independently and individually.

(60%) Programming Assignments. The bulk of the coursework is centered around four major programming assignments; each assignment is worth 25% of the programming component of the course, or 15% of the overall course grade. Like the written homework, programming assignments should be completed independently by each student.

(20%) Midterm / Final. There will be a midterm and a final, each worth 10% of the overall course grade. Both exams will cover the cumulative material seen in the course so far.

(5%) Class Participation. To earn the first 3% of your participation grade, you are required to make one question/comment on each set of lecture slides, via the 462 webpage. In order to receive credit, this comment must be made prior to the beginning of the next lecture. The remaining 2% of the participation grade will be based on in-class comments, Piazza discussions, participation in office hours, and other contributions to the class (at the discretion of the instructors).

Late Policy

All programming assignments are due at 11:59:59pm Eastern time on the due date, as reported by Each student is allotted a total of five late-day points for the semester. Late-day points are for use on the four programming assignments only (A1–A4). Late-day points work as follows:

  • A student can extend a programming assignment deadline by one day using one point.
  • If a student does not have remaining late day points, late hand-ins will incur a 10% penalty per day (10% of max points on the assignment).
  • Late-day points get used up for each checkpoint. So for instance, even if you already used one late point for A2.0, you would still have to use another late point if you wanted to turn in A2.1 a day late.
  • No assignments will be accepted more than three days after the deadline. This is true whether or not the student has late-day points remaining.

The dates on the course schedule are the official due dates. If you find conflicting information elsewhere (e.g., on Piazza, in-class, talking to a TA...), you should always assume that the assignment is actually due on the date stated on the front page of the course web site. (But please let us know if something seems totally wrong! ;-))

Collaboration Policy

Students in 15-462 are absolutely encouraged to talk to each other, to the TAs, to the instructors, or to anyone else about course assignments. Any assistance, though, must be limited to discussion of the problems and sketching general approaches to a solution. Each student should write their own code and produce their own writeup. Consulting another student's solution is prohibited and submitted solutions may not be copied from any source. These and any other form of collaboration on assignments constitute cheating. If you have any question about whether some activity would constitute cheating, just be cautious and ask the instructors before proceeding!

If you are caught cheating, you will get a zero for the entire course (not just the assignment). Also, if two identical assignments are handed in, both students will be accountable for cheating (no questions asked). So please be careful to ensure that nobody is copying your work!

You may not supply code, assignment writeups, or exams you complete during 15-462/662 to other students in future instances of this course or make these items available (e.g., on the web) for use in future instances of this course (just as you may not use work completed by students who've taken the course previously). Make sure to make repositories private if you use public source control hosts like GitHub.



There is no required textbook for 15-462, though a variety of books may provide good supplementary material:

Pete Shirley and Steve Marschner with Michael Ashikhmin, Michael Gleicher, Naty Hoffman, Garrett Johnson, Tamara Munzner, Erik Reinhard, Kelvin Sung, William B. Thompson, Peter Willemsen, and Bryan Wyvill
Fundamentals of Computer Graphics. A K Peters, 2009
[ On Amazon ]

John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley
Computer Graphics: Principles and Practice
[ On Amazon ]

Matt Pharr and Greg Humphreys
Physically Based Rendering: From Theory to Implementation
[ On Amazon ]
This book (PBRT) is the book for learning about modern ray tracing techniques. It has a great website with full source code online for an advanced physically-based ray tracer. It even won an oscar for its impact on the film industry!