Learning Linear Algebra for Computer Graphics?
July 22, 2018 8:45 AM Subscribe
Are there any good intro Linear Algebra books (or other media) that take an applied, computational approach, and specifically one based on graphics, image processing or computational geometry?
I was an engineering student, once upon a time, and so took calculus and linear algebra courses, with some success. After doing other, mostly non-technical things for 15 years, I find myself in a position where it is both personally fascinating and of immediate professional value to get something of handle on the maths involved in all things computer graphicsy, starting with linear algebra.
I know there is no royal road to geometry, but I would at least like to take a direct route that will begin being useful as soon as possible (even at the risk of obtaining a less deep or general grasp of basic principles).
I am a semi-professional Python programmer who already regularly uses tools like numpy and OpenCV without really understanding what's going on under the hood. I'm hoping for a text that could faciliate this, rather than emphasizing the old fashioned nitty-gritty of hand compuations.
My trig is still fluent, I retain a hand-wavy grasp of the principles of calculus, and I'm not afraid to do some serious research and head-scratching: resources that assume this level of preparation would be ideal.
I was an engineering student, once upon a time, and so took calculus and linear algebra courses, with some success. After doing other, mostly non-technical things for 15 years, I find myself in a position where it is both personally fascinating and of immediate professional value to get something of handle on the maths involved in all things computer graphicsy, starting with linear algebra.
I know there is no royal road to geometry, but I would at least like to take a direct route that will begin being useful as soon as possible (even at the risk of obtaining a less deep or general grasp of basic principles).
I am a semi-professional Python programmer who already regularly uses tools like numpy and OpenCV without really understanding what's going on under the hood. I'm hoping for a text that could faciliate this, rather than emphasizing the old fashioned nitty-gritty of hand compuations.
My trig is still fluent, I retain a hand-wavy grasp of the principles of calculus, and I'm not afraid to do some serious research and head-scratching: resources that assume this level of preparation would be ideal.
Ages ago (like the early '90s), I started writing a set of "computer graphics with high school algebra" tutorials. Then I got a job in the industry, and I think the critical "aha" moment for me when thinking about 3d transformations using 4x4 matrices is that the net effect of multiplying a point (1 in that last spot) or a vector (0 in the last spot) is to project each of the elements along the point along 4 different lines in 4d space
So creating a rotation matrix is just putting the 3 axes that you want to move the line into into the 4x4. Translation is using that last "1" to put movement into the first 3 (and a vector, then, doesn't translate).
I don't have a particular resource to help 'cause that was two and a half decades ago, and I'm kind of a non-traditional learner... And I was doing the hand calculations first. But when I've pointed this out to a few other people, some of whom are graphically adept, I've gotten "oh, duh, of course"...
posted by straw at 12:03 PM on July 22, 2018 [2 favorites]
So creating a rotation matrix is just putting the 3 axes that you want to move the line into into the 4x4. Translation is using that last "1" to put movement into the first 3 (and a vector, then, doesn't translate).
I don't have a particular resource to help 'cause that was two and a half decades ago, and I'm kind of a non-traditional learner... And I was doing the hand calculations first. But when I've pointed this out to a few other people, some of whom are graphically adept, I've gotten "oh, duh, of course"...
posted by straw at 12:03 PM on July 22, 2018 [2 favorites]
Response by poster: Thanks for the The Vulkan Tutorial recommendation; it is indeed fascinating, but possibly a little too much Applied and too little Linear Algebra for my present needs. The Learning Modern 3D Graphics Programming book they link to is much closer to what I was looking for (and I will definitely be reading it!), but I'm still hoping for something that emphasizes the linear algebra more explicitly...
posted by lordcorvid at 12:53 PM on July 22, 2018
posted by lordcorvid at 12:53 PM on July 22, 2018
Best answer: It sounds like you're looking for this course, so have a look at the materials it recommends.
posted by batter_my_heart at 4:05 PM on July 22, 2018 [1 favorite]
posted by batter_my_heart at 4:05 PM on July 22, 2018 [1 favorite]
Best answer: While I think you can find some really good applied linear algebra in image processing and computational geometry, that feels like jumping into the deep end. Basic rendering of 3D polygons, though, is pretty accessible, and, for better or for worse, depending on your purposes, uses just a tiny subset of linear algebra, exclusively concerning small (4x4 or smaller) matrices and vectors. For that, straw's advice is spot on.
Before diving head-first into the theory, I'd strongly recommend setting up a programming environment where you can experiment and visualize things. This seems like a pretty good tutorial on how to do that with Python and OpenGL: https://www.youtube.com/watch?v=R4n4NyDG2hI. (Also, I highly recommend starting with the old-school, vanilla OpenGL used in the video. Old-school GL is still the most straightforward way I know to get graphics displaying on the screen, without forcing you to abstract away much of the math. Vulkan, Metal, DirectX and modern OpenGL all have tons of boilerplate that you probably don't want to mess with, yet.)
Visualizing what straw said, a bit, if you have a point (Px, Py, Pz), you can transform it in 3D space by multiplying it by a special 4x4 matrix that looks like this (in column major format):
X, Y and Z are usually mutually orthogonal and normalized, and they define the rotation of the transformation (and when not orthogonal, or not normalized, they define the skew and scale, as well). For me, it's easiest to understand and visualize when you remember that multiplying by a matrix is transforming into a new basis (https://en.wikipedia.org/wiki/Basis_(linear_algebra)), and each column represents an axis of the new basis. The T column holds the translation. Taken all together, the matrix defines a new basis in 4D space, but I've found that impossible to visualize. If, instead, you think of transforming 2D points (using 3D transformation matrices), it becomes pretty possible to visualize what the transformation matrix means, geometrically. This website has a nice interface for exploring 3D transformations applied to 2D points: https://ncase.me/matrix/ (though, unfortunately, it doesn't visualize the 3D matrix as a basis, for you).
All this is just for moving points around in space, but actually projecting 3D points onto the 2D surface of your screen is another can of worms (and some of it technically no longer linear algebra, because it's non-linear). OpenGL can hide the projection process under the hood, initially, which is probably a good idea, until you feel really solid about basic transformation matrices. Once you want to explore deeper, the most clear and thorough explanation of projection I've encountered is Jim Blinn's article "The Homogeneous Perspective Transform," which, unfortunately, I haven't been able to find on the internet, but is collected in his book: "A Trip Down the Graphics Pipeline."
Diving deeper into rendering, "Learning Modern 3D Graphics Programming" is a fine resource, but if you're more interested in math than fancy shader effects, I'd recommend Christer Ericson's "Real-Time Collision Detection." With just a basic renderer in place, you could start implementing some of the collision detection algorithms in Christer's book, which is probably a faster path to developing linear algebra intuition than anything in the strict rendering world. Also, the book contains an excellent linear algebra primer, which is probably one of the reasons many graphics programmers I know have a well-loved and dog-eared copy on their shelves, even if they never have any reason to write collision detection code.
Concluding, I'd say graphics is a great way to learn a small part of linear algebra really really well (and also a lot of fun!). If your primary interest is truly general linear algebra, however, especially higher dimensional stuff, graphics may not be the most straightforward path.
posted by array at 4:32 PM on July 22, 2018 [5 favorites]
Before diving head-first into the theory, I'd strongly recommend setting up a programming environment where you can experiment and visualize things. This seems like a pretty good tutorial on how to do that with Python and OpenGL: https://www.youtube.com/watch?v=R4n4NyDG2hI. (Also, I highly recommend starting with the old-school, vanilla OpenGL used in the video. Old-school GL is still the most straightforward way I know to get graphics displaying on the screen, without forcing you to abstract away much of the math. Vulkan, Metal, DirectX and modern OpenGL all have tons of boilerplate that you probably don't want to mess with, yet.)
Visualizing what straw said, a bit, if you have a point (Px, Py, Pz), you can transform it in 3D space by multiplying it by a special 4x4 matrix that looks like this (in column major format):
| Xx Yx Zx Tx | | Px |
| Xy Yy Zy Ty | * | Py |
| Xz Yz Zz Tz | | Pz |
| 0 0 0 1 | | 1 |
X, Y and Z are usually mutually orthogonal and normalized, and they define the rotation of the transformation (and when not orthogonal, or not normalized, they define the skew and scale, as well). For me, it's easiest to understand and visualize when you remember that multiplying by a matrix is transforming into a new basis (https://en.wikipedia.org/wiki/Basis_(linear_algebra)), and each column represents an axis of the new basis. The T column holds the translation. Taken all together, the matrix defines a new basis in 4D space, but I've found that impossible to visualize. If, instead, you think of transforming 2D points (using 3D transformation matrices), it becomes pretty possible to visualize what the transformation matrix means, geometrically. This website has a nice interface for exploring 3D transformations applied to 2D points: https://ncase.me/matrix/ (though, unfortunately, it doesn't visualize the 3D matrix as a basis, for you).
All this is just for moving points around in space, but actually projecting 3D points onto the 2D surface of your screen is another can of worms (and some of it technically no longer linear algebra, because it's non-linear). OpenGL can hide the projection process under the hood, initially, which is probably a good idea, until you feel really solid about basic transformation matrices. Once you want to explore deeper, the most clear and thorough explanation of projection I've encountered is Jim Blinn's article "The Homogeneous Perspective Transform," which, unfortunately, I haven't been able to find on the internet, but is collected in his book: "A Trip Down the Graphics Pipeline."
Diving deeper into rendering, "Learning Modern 3D Graphics Programming" is a fine resource, but if you're more interested in math than fancy shader effects, I'd recommend Christer Ericson's "Real-Time Collision Detection." With just a basic renderer in place, you could start implementing some of the collision detection algorithms in Christer's book, which is probably a faster path to developing linear algebra intuition than anything in the strict rendering world. Also, the book contains an excellent linear algebra primer, which is probably one of the reasons many graphics programmers I know have a well-loved and dog-eared copy on their shelves, even if they never have any reason to write collision detection code.
Concluding, I'd say graphics is a great way to learn a small part of linear algebra really really well (and also a lot of fun!). If your primary interest is truly general linear algebra, however, especially higher dimensional stuff, graphics may not be the most straightforward path.
posted by array at 4:32 PM on July 22, 2018 [5 favorites]
Have you seen the YouTube Essence Of Linear Algebra series? It claims to be "aimed at animating the geometric intuitions underlying many of the topics taught in a standard linear algebra course." It does not look like it actually teaches the subject, but it makes a good complement (ha!) to a traditional course.
posted by Harvey Kilobit at 8:04 PM on July 22, 2018 [2 favorites]
posted by Harvey Kilobit at 8:04 PM on July 22, 2018 [2 favorites]
Best answer: Since it's in recent memory from another question.... 'Graphics Gems' (Google). Old school, bits and pieces of algebra of many sorts manipulating points down to scanning lines and putting it on the screen and up to reading mouse movements as a trackball. May be too low-level... it's all math of mostly the linear algebra tilt (really only the dimensions change). (modulo the non-linear visual perception sort of thing...)
That's only if you don't want to go into GL or some other Python libraries that do it all for you... but just have 2d output and want to do the rest of it yourself. How much do you want to do the math yourself?
posted by zengargoyle at 9:42 PM on July 22, 2018
That's only if you don't want to go into GL or some other Python libraries that do it all for you... but just have 2d output and want to do the rest of it yourself. How much do you want to do the math yourself?
posted by zengargoyle at 9:42 PM on July 22, 2018
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posted by Alterscape at 11:06 AM on July 22, 2018