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People must get taught math terribly if they think "I don't need to worry about piles of abstract math to understand a rotation, all I have to do is think about

People must get taught math terribly if they think "I don't need to worry about piles of abstract math to understand a rotation, all I have to do is think about what happens to the XYZ axes under the matrix rotation". That is what you should learn in the math class! Anyone who has taken linear algebra should know that (1) a rotation is a linear operation, (2) the result of a linear operation is calculated with matrix multiplication, (3) the result of a matrix multiplication is determined by what it does to the standard basis vectors, the results of which form the columns of the matrix. This guy makes it sound like he had to come up with these concepts from scratch, and it's some sort of pure visual genius rather than math. But... it's just math.

vasco 1y ago
Math is just a standardized way to communicate those concepts though, it's a model of the world like any other. I get what you mean, but these intuitive or visualising approaches help many people with different thinking processes. Just imagine that everyone has equal math ability, except the model of math and representations of mathematical concepts and notation is more made for a certain type of brains than others. These kind of explanations allow bringing those people in as well.
omnicognate 1y ago
A lot of people who find themselves having to deal with matrices when programming have never taken that class or learned those things (or did so such a long time ago that they've completely forgotten). I assume this is aimed at such people, and he's just reassuring them that he's not going to talk about the abstract aspects of linear algebra, which certainly exist. I'd take issue with his "most programmers are visual thinkers", though. Maybe most graphics programmers are, but I doubt it's an overwhelming majority even there.
- foofoo12 1y ago
  > most programmers are visual thinkers I remember reading that there's a link between aphantasia (inability to visualize) and being on the spectrum. Being an armchair psychologist expert with decades of experience, I can say with absolute certainty that a lot of programmers are NOT visual thinkers.
  - voidUpdate 1y ago
    Do you have anything I can read about that? I'm definitely on the spectrum and have whatever the opposite of aphantasia is, I can see things very clearly in my head
    - foofoo12 1y ago
      "In Experiment 2 we have shown that people with aphantasia report higher AQ scores (more traits associated with autism than controls), and fall more often within the range suggestive of autism (≥32)." https://www.sciencedirect.com/science/article/abs/pii/S1053810021000131 https://www.sciencedirect.com/science/article/abs/pii/S10538...
      - voidUpdate 1y ago
        Interesting, I guess I'm an outlier then
  - treyd 1y ago
    This is interesting because, to me, programing is a deeply visual activity. It feels like wandering around in a world of forms until I find the structures I need and actually writing out the code is mostly a formality.
    - omnicognate 1y ago
      I would describe my experience of it similarly, but wouldn't call it "visual thinking" in the sense meant in the article, where one uses actual imagery and visual-spatial reasoning. Indeed, I almost completely lack the ability to conjure mental imagery (aphantasia) and I've speculated it might be because a part of my visual cortex is given over to the pseudo-visual activity that seems to take place when I program. I'm especially sure my sort of pseudo-visual thinking isn't what the article means by "visual thinking" because I also use it when working through "piles of abstract math", which I take to very kindly indeed. Is your "wandering" of this sort of pseudo-visual nature, or do you see actual visual images that could be drawn? Very intriguing if the latter, and I'd be curious to know what they look like.
zelphirkalt 1y ago
When I was studying and made the mistake of choosing 3D computer graphics as a lecture, I remember some 4x4 matrix that was used for rotation, with all kinds of weird terms in it, derived only once, in a way I was not able to understand and that didn't relate to any visual idea or imagination, which makes it extra hard for me to understand it, because I rely a lot on visualization of everything. So basically, there was a "magical formula" to rotate things and I didn't memorize it. Exam came and demanded having memorized this shitty rotation matrix. Failed the exam, changed lectures. High quality lecturing. Later in another lecture at another university, I had to rotate points around a center point again. This time found 3 3x3 matrices on wikipedia, one for each axis. Maybe making at least seemingly a little bit more sense, but I think I never got to the basis of that stuff. Never seen a good visual explanation of this stuff. I ended up implementing the 3 matrices multiplications and checked the 3D coordinates coming out of that in my head by visualizing and thinking hard about whether the coordinates could be correct. I think visualization is the least of my problems. Most math teaching sucks though, and sometimes it is just the wrong format or not visualized at all, which makes it very hard to understand.
- dgacmu 1y ago
  You can do rotation with a 3x3 matrix. The first lecture was using a 4x4 matrix because you can use it for a more general set of transformations, including affine transforms (think: translating an object by moving it in a particular direction). Since you can combine a series of matrix multiplications by just pre-multiplying the matrix, this sets you up for doing a very efficient "move, scale, rotate" of an object using a single matrix multiplication of that pre-calculated 4x4 matrix. If you just want to, e.g., scale and rotate the object, a 3x3 matrix suffices. Sounds like your first lecture jumped way too fast to the "here's the fully general version of this", which is much harder for building intuition for. Sorry you had a bad intro to this stuff. It's actually kinda cool when explained well. I think they probably should have started by showing how you can use a matrix for scaling: [[2, 0, 0], [0, 1.5, 0], [0, 0, 1]] for example, will grow an object by 2x in the x dimension, 1.5x in the y dimension, and keep it unchanged in the z dimension. (You'll note that it follows the pattern of the identity matrix). The derivation of the rotation matrix is probably best first derived for 2d; the wikipedia article has a decentish explanation: https://en.wikipedia.org/wiki/Rotation_matrix https://en.wikipedia.org/wiki/Rotation_matrix
whizzter 1y ago
Honestly, many math teachers are kinda bad at conveying all that. When everything clicked a few years down the line it all became so simple. Like you mention "linear operation", the word linear doesn't always make intuitive sense in terms of rotations or scaling if you have encountered simple 1 or 2 dimensional linear transformations when doing more basic graphics programming. As a teacher, I think the biggest lesson I had to learn was to always have at least 3 different ways of explaining everything to give different kinds of people different entrypoints into understanding concepts. For someone uninitiated a term like "basis vector" can be pure gibberish if it doesn't follow an example of a transform as a viewport change, and it needs to be repeated after your other explanations (of for example how vector components in the source view just are scalars upon the basis vectors when multiplied with a matrix instead of a heavy un-intuitive concept).
kccqzy 1y ago
I have taken several linear algebra courses, one from my high school and two from universities. The thing is, not all courses of linear algebra will discuss rotations the way you discuss it. One reason is that sometimes a high school linear algebra course cannot assume students have learned trigonometry. I've seen teachers teach it just to solve larger linear systems of equations. Another reason is that sometimes a course will focus just on properties of vector spaces without relating them to geometry; after all who can visualize things when the course routinely deals with 10-dimensional vectors or N-dimensional ones where N isn't a constant.
- ndriscoll 1y ago
  Usually you just draw a 2D or 3D picture and say "n" while pointing to it. e.g. I had a professor that drew a 2D picture on a board where he labeled one axis R^m and the other R^n and then drew a "graph" when discussing something like the implicit function theorem. One takeaway of a lot of linear algebra is that doing this is more-or-less correct (and then functional analysis tells you this is still kind-of correct-ish even in infinite dimensions). Actually SVD tells you in some sense that if you look at things in the right way, the "heart" of a linear map is just 1D multiplication acting independently along different axes, so you don't need to consider all n dimensions at once.
- yatopifo 1y ago
  I think teaching beginner linear algebra using matrices representing systems of equations is a pedagogical mistake. It gives the wrong impression that matrices are linear algebra and makes it difficult for students to think about it in an abstract way. A better way is to start by discussing abstract linear combinations and then illustrating what can be done with this using visualizations in various coordinate systems. Once the student understands this intuitively, systems of equations and matrices can be brought up as equivalent ways to represent linear transformations on paper. It’s important to emphasize that matrices are convenient but not the only way to write the language of linear algebra.
baruz 1y ago
This was part of Steve Baker’s (“Omniverous Hexapod”, sic) extensions to a long-standing Usenet FAQ about graphics programming, put out by “Carniverous Hexapod” (sic). It’s at least two decades old, and the FAQ from which he it on may be from the 1990s? I have the niggling recollection that the Carniverous name may have been based on Vernor Vinge’s _Fire upon the deep_ aliens. He did not invent it, but he probably had to deal with aspiring graphics programmers who were not very math-savvy.
ux266478 1y ago
> This guy makes it sound like he had to come up with these concepts from scratch, and it's some sort of pure visual genius rather than math. But... it's just math. The problem with this kind of thinking is that it encourages the exact kind of teaching you disparage. It's very easy to get lost in the sauce of arbitrary notational choices that the underlying concepts end up completely lost. Before you know it, matrices are opaque self-justified atoms in an ecosystem rather than an arbitrary tabular shorthand. Mathematics is not a singular immutable dogma that dropped out of the sky as natural law, and rediscovery is the most powerful tool for understanding.
Merad 1y ago
> Anyone who has taken linear algebra should know that [...] My university level linear algebra class didn't touch practical applications at all, which was frustrating to me because I knew that it could be very useful to some background doing hobbyist game dev. I still wish I had a better understanding of the use cases for things like eigenvectors/values.
- drdeca 1y ago
  Here are some applications of eigenvectors and eigenvalues: 1) If you have a set of states and a stochastic transition function, which gives for each starting state, the probability distribution over what the state will be at the next time step, you can describe this as a matrix. The long-term behavior of applying this can be described using the eigenvectors and eigenvalues of this matrix. Any stable distribution will be an eigenvector with eigenvalue 1. If there is periodicity to the behavior, where for some initial distributions, the distribution will change over time in a periodic way, where it ends up endlessly cycling through a finite set of distributions, then the matrix will have eigenvalues that are roots of unity (i.e. a complex number s such that s^n = 1 for some positive integer n). Eigenvalues with absolute value less than 1 correspond to transient contributions to the distribution which will decay (the closer to 0, the quicker the decay.). When there are finitely many states, there will always be at least one eigenvector with eigenvalue 1. 2) Related to (1), there is the PageRank algorithm, where one takes a graph where each node has links to other nodes, and one models a random walk on these nodes, and one uses the eigenvector one (approximately) finds in order to find the relative importance of the different nodes. 3) Rotations generally have eigenvalues that are complex numbers with length 1. As mentioned in (1), eigenvalues that are complex numbers with length 1 are associated with periodic/oscillating behavior. Well, I guess it sorta depends how you are using the matrix. If you have a matrix M with all of its eigenvalues purely imaginary, then exp(t M) (with t representing time) will describe oscillations with rates given by those eigenvalues. exp(t M) itself will have eigenvalues that are complex numbers of length 1. This is very relevant in solutions to higher order differential equations or differential equations where the quantity changing over time is a vector quantity. ____ But, for purposes of gamedev, I think probably the eigenvalues/eigenvectors are probably the less relevant things. Probably instead, at least for rendering and such, you want stuff like, "you can use homogeneous coordinates in order to incorporate translations and rotations into a single 4x4 matrix (and also for other things relating the 3d scene to the 2d screen)", and stuff about like... well, quaternions can be helpful. Of course, it all depends what you are trying to do...
recursive 1y ago
I took a linear algebra class, as well as many others. It didn't work. Most math classes I've taken granted me some kind of intuition for the subject material. Like I could understand the concept independent from the name of the thing. In linear algebra, it was all a series of arbitrary facts without reason for existing. I memorized them for the final exam, and probably forgot them all the next day, as they weren't attached to anything in my mind. "The inverse of the eigen-something is the determinant of the abelian". It was just a list of facts like this to memorize by rote. I passed the class with a decent grade I think. But I really understood nothing. At this point, I can't remember how to multiply matrices. Specifically do the rows go with the columns or do the columns go with the rows? I don't know if there's something about linear algebra or I just didn't connect with the instructor. But I've taken a lot of other math classes, and usually been able to understand the subject material readily. Maybe linear algebra is different. It was completely impenetrable for me.
- drdeca 1y ago
  To remind oneself how to multiply matrices together, it suffices to remember how to apply a matrix to a column vector, and that ((A B) v) = (A (B v)). For each 1-hot vector e_i (i.e. the row vector that has a 1 in the i-th position and 0s elsewhere), apply B e_i to get the i-th column of the matrix B. Then, apply the matrix A to the result, to obtain A (B e_i), which equals (A B) e_i . This is then the i-th column of the matrix A B. And, when applying the matrix A to some column vector v, for each entry/row of the resulting vector, it is obtained by combining the corresponding row of A, with the column vector v. So, to get the entry at the j-th row of the i-th column of (A B), one therefore combines the i-th column of B with the j-th row of A. Or, alternatively/equivalently, you can just compute the matrix (A B) column by column, by, for each e_i , computing that the i-th column of (A B) is (A (B e_i)) (which is how I usually think of it). To be clear, I don't have the process totally memorized; I actually use the above reasoning to remind myself of the computation process a fair portion of the time that I need to compute actual products of matrices, which is surprisingly often given that I don't have it totally memorized. When I took linear algebra, the professor emphasized the linear maps, and somewhat de-emphasized the matrices that are used to notate them. I think this made understanding what is going on easier, but made the computations less familiar. I very much enjoyed the class.
imtringued 1y ago
Rotations: SO(3) is the special orthogonal group for 3 dimensions, which is the set of rotation matrices. Rotation matrices have the determinant 1, which means that they keep the length of a vector the same (forcing it to change direction). Orthogonal group means that the rotation R times its transpose R^T yields the identity matrix aka R^T R=RR^T=I. This means that rotations are trivially reversible. You can obtain rotation matrices either by explicit use of cosine/sine along your desired axis or you can use the Rodrigues rotation formula to describe a rotation along an axis with an angle. so(3) is the Lie algebra to the Lie group SO(3). The capitalization here is critical. In particular, so(3) is the set of skew symmetric matrices. 3 by 3 skew-symmetric matrices essentially describe the cross product a x b for a constant a. Given a skew symmetric matrix and a rotation angle theta you can use the exponential function to map so(3) to SO(3) through e^(theta S). Note that the exponential is a general mapping from a Lie algebra to its Lie group. Transformations: SE(3) is the special euclidean group for 3 dimensions. Its structure consists of T = [[R, t], [0, 1]]. if R = I, then T is a pure translation by t. If R != I and t = 0, then it is a pure rotation. When you have both a rotation R and a translation t transformation then translating first and rotating second will give you T, but rotating first and translating second will give you T' = [[R, Rt], [0, 1]]. se(3) is the Lie algebra to the Lie group SE(3). One way to represent so(3) is through twists. Revolute joints are represented by a vector Xi = [ omega, q cross-product omega] where omega is a vector that represents the parameters of a skew symmetric matrix and q is a 3d point on the axis of rotation. Prismatic joints are represented by Xi = [0, v] where v is pointing in the direction of the movement. The transformation of a twist can be obtained as usual through the exponential function: exp(theta Xi)=T. https://en.wikipedia.org/wiki/3D_rotation_group https://en.wikipedia.org/wiki/3D_rotation_group https://en.wikipedia.org/wiki/Orthogonal_group https://en.wikipedia.org/wiki/Orthogonal_group https://en.wikipedia.org/wiki/Euclidean_group https://en.wikipedia.org/wiki/Euclidean_group https://en.wikipedia.org/wiki/Screw_theory https://en.wikipedia.org/wiki/Screw_theory

vasco 1y ago

Math is just a standardized way to communicate those concepts though, it's a model of the world like any other. I get what you mean, but these intuitive or visualising approaches help many people with different thinking processes. Just imagine that everyone has equal math ability, except the model of math and representations of mathematical concepts and notation is more made for a certain type of brains than others. These kind of explanations allow bringing those people in as well.

omnicognate 1y ago

A lot of people who find themselves having to deal with matrices when programming have never taken that class or learned those things (or did so such a long time ago that they've completely forgotten). I assume this is aimed at such people, and he's just reassuring them that he's not going to talk about the abstract aspects of linear algebra, which certainly exist. I'd take issue with his "most programmers are visual thinkers", though. Maybe most graphics programmers are, but I doubt it's an overwhelming majority even there.

foofoo12 1y ago
> most programmers are visual thinkers I remember reading that there's a link between aphantasia (inability to visualize) and being on the spectrum. Being an armchair psychologist expert with decades of experience, I can say with absolute certainty that a lot of programmers are NOT visual thinkers.
- voidUpdate 1y ago
  Do you have anything I can read about that? I'm definitely on the spectrum and have whatever the opposite of aphantasia is, I can see things very clearly in my head
  - foofoo12 1y ago
    "In Experiment 2 we have shown that people with aphantasia report higher AQ scores (more traits associated with autism than controls), and fall more often within the range suggestive of autism (≥32)." https://www.sciencedirect.com/science/article/abs/pii/S1053810021000131 https://www.sciencedirect.com/science/article/abs/pii/S10538...
    - voidUpdate 1y ago
      Interesting, I guess I'm an outlier then
- treyd 1y ago
  This is interesting because, to me, programing is a deeply visual activity. It feels like wandering around in a world of forms until I find the structures I need and actually writing out the code is mostly a formality.
  - omnicognate 1y ago
    I would describe my experience of it similarly, but wouldn't call it "visual thinking" in the sense meant in the article, where one uses actual imagery and visual-spatial reasoning. Indeed, I almost completely lack the ability to conjure mental imagery (aphantasia) and I've speculated it might be because a part of my visual cortex is given over to the pseudo-visual activity that seems to take place when I program. I'm especially sure my sort of pseudo-visual thinking isn't what the article means by "visual thinking" because I also use it when working through "piles of abstract math", which I take to very kindly indeed. Is your "wandering" of this sort of pseudo-visual nature, or do you see actual visual images that could be drawn? Very intriguing if the latter, and I'd be curious to know what they look like.

zelphirkalt 1y ago

When I was studying and made the mistake of choosing 3D computer graphics as a lecture, I remember some 4x4 matrix that was used for rotation, with all kinds of weird terms in it, derived only once, in a way I was not able to understand and that didn't relate to any visual idea or imagination, which makes it extra hard for me to understand it, because I rely a lot on visualization of everything. So basically, there was a "magical formula" to rotate things and I didn't memorize it. Exam came and demanded having memorized this shitty rotation matrix. Failed the exam, changed lectures. High quality lecturing. Later in another lecture at another university, I had to rotate points around a center point again. This time found 3 3x3 matrices on wikipedia, one for each axis. Maybe making at least seemingly a little bit more sense, but I think I never got to the basis of that stuff. Never seen a good visual explanation of this stuff. I ended up implementing the 3 matrices multiplications and checked the 3D coordinates coming out of that in my head by visualizing and thinking hard about whether the coordinates could be correct. I think visualization is the least of my problems. Most math teaching sucks though, and sometimes it is just the wrong format or not visualized at all, which makes it very hard to understand.

dgacmu 1y ago
You can do rotation with a 3x3 matrix. The first lecture was using a 4x4 matrix because you can use it for a more general set of transformations, including affine transforms (think: translating an object by moving it in a particular direction). Since you can combine a series of matrix multiplications by just pre-multiplying the matrix, this sets you up for doing a very efficient "move, scale, rotate" of an object using a single matrix multiplication of that pre-calculated 4x4 matrix. If you just want to, e.g., scale and rotate the object, a 3x3 matrix suffices. Sounds like your first lecture jumped way too fast to the "here's the fully general version of this", which is much harder for building intuition for. Sorry you had a bad intro to this stuff. It's actually kinda cool when explained well. I think they probably should have started by showing how you can use a matrix for scaling: [[2, 0, 0], [0, 1.5, 0], [0, 0, 1]] for example, will grow an object by 2x in the x dimension, 1.5x in the y dimension, and keep it unchanged in the z dimension. (You'll note that it follows the pattern of the identity matrix). The derivation of the rotation matrix is probably best first derived for 2d; the wikipedia article has a decentish explanation: https://en.wikipedia.org/wiki/Rotation_matrix https://en.wikipedia.org/wiki/Rotation_matrix

whizzter 1y ago

Honestly, many math teachers are kinda bad at conveying all that. When everything clicked a few years down the line it all became so simple. Like you mention "linear operation", the word linear doesn't always make intuitive sense in terms of rotations or scaling if you have encountered simple 1 or 2 dimensional linear transformations when doing more basic graphics programming. As a teacher, I think the biggest lesson I had to learn was to always have at least 3 different ways of explaining everything to give different kinds of people different entrypoints into understanding concepts. For someone uninitiated a term like "basis vector" can be pure gibberish if it doesn't follow an example of a transform as a viewport change, and it needs to be repeated after your other explanations (of for example how vector components in the source view just are scalars upon the basis vectors when multiplied with a matrix instead of a heavy un-intuitive concept).

kccqzy 1y ago

I have taken several linear algebra courses, one from my high school and two from universities. The thing is, not all courses of linear algebra will discuss rotations the way you discuss it. One reason is that sometimes a high school linear algebra course cannot assume students have learned trigonometry. I've seen teachers teach it just to solve larger linear systems of equations. Another reason is that sometimes a course will focus just on properties of vector spaces without relating them to geometry; after all who can visualize things when the course routinely deals with 10-dimensional vectors or N-dimensional ones where N isn't a constant.

ndriscoll 1y ago
Usually you just draw a 2D or 3D picture and say "n" while pointing to it. e.g. I had a professor that drew a 2D picture on a board where he labeled one axis R^m and the other R^n and then drew a "graph" when discussing something like the implicit function theorem. One takeaway of a lot of linear algebra is that doing this is more-or-less correct (and then functional analysis tells you this is still kind-of correct-ish even in infinite dimensions). Actually SVD tells you in some sense that if you look at things in the right way, the "heart" of a linear map is just 1D multiplication acting independently along different axes, so you don't need to consider all n dimensions at once.
yatopifo 1y ago
I think teaching beginner linear algebra using matrices representing systems of equations is a pedagogical mistake. It gives the wrong impression that matrices are linear algebra and makes it difficult for students to think about it in an abstract way. A better way is to start by discussing abstract linear combinations and then illustrating what can be done with this using visualizations in various coordinate systems. Once the student understands this intuitively, systems of equations and matrices can be brought up as equivalent ways to represent linear transformations on paper. It’s important to emphasize that matrices are convenient but not the only way to write the language of linear algebra.

baruz 1y ago

This was part of Steve Baker’s (“Omniverous Hexapod”, sic) extensions to a long-standing Usenet FAQ about graphics programming, put out by “Carniverous Hexapod” (sic). It’s at least two decades old, and the FAQ from which he it on may be from the 1990s? I have the niggling recollection that the Carniverous name may have been based on Vernor Vinge’s _Fire upon the deep_ aliens. He did not invent it, but he probably had to deal with aspiring graphics programmers who were not very math-savvy.

ux266478 1y ago

> This guy makes it sound like he had to come up with these concepts from scratch, and it's some sort of pure visual genius rather than math. But... it's just math. The problem with this kind of thinking is that it encourages the exact kind of teaching you disparage. It's very easy to get lost in the sauce of arbitrary notational choices that the underlying concepts end up completely lost. Before you know it, matrices are opaque self-justified atoms in an ecosystem rather than an arbitrary tabular shorthand. Mathematics is not a singular immutable dogma that dropped out of the sky as natural law, and rediscovery is the most powerful tool for understanding.

Merad 1y ago

> Anyone who has taken linear algebra should know that [...] My university level linear algebra class didn't touch practical applications at all, which was frustrating to me because I knew that it could be very useful to some background doing hobbyist game dev. I still wish I had a better understanding of the use cases for things like eigenvectors/values.

drdeca 1y ago
Here are some applications of eigenvectors and eigenvalues: 1) If you have a set of states and a stochastic transition function, which gives for each starting state, the probability distribution over what the state will be at the next time step, you can describe this as a matrix. The long-term behavior of applying this can be described using the eigenvectors and eigenvalues of this matrix. Any stable distribution will be an eigenvector with eigenvalue 1. If there is periodicity to the behavior, where for some initial distributions, the distribution will change over time in a periodic way, where it ends up endlessly cycling through a finite set of distributions, then the matrix will have eigenvalues that are roots of unity (i.e. a complex number s such that s^n = 1 for some positive integer n). Eigenvalues with absolute value less than 1 correspond to transient contributions to the distribution which will decay (the closer to 0, the quicker the decay.). When there are finitely many states, there will always be at least one eigenvector with eigenvalue 1. 2) Related to (1), there is the PageRank algorithm, where one takes a graph where each node has links to other nodes, and one models a random walk on these nodes, and one uses the eigenvector one (approximately) finds in order to find the relative importance of the different nodes. 3) Rotations generally have eigenvalues that are complex numbers with length 1. As mentioned in (1), eigenvalues that are complex numbers with length 1 are associated with periodic/oscillating behavior. Well, I guess it sorta depends how you are using the matrix. If you have a matrix M with all of its eigenvalues purely imaginary, then exp(t M) (with t representing time) will describe oscillations with rates given by those eigenvalues. exp(t M) itself will have eigenvalues that are complex numbers of length 1. This is very relevant in solutions to higher order differential equations or differential equations where the quantity changing over time is a vector quantity. ____ But, for purposes of gamedev, I think probably the eigenvalues/eigenvectors are probably the less relevant things. Probably instead, at least for rendering and such, you want stuff like, "you can use homogeneous coordinates in order to incorporate translations and rotations into a single 4x4 matrix (and also for other things relating the 3d scene to the 2d screen)", and stuff about like... well, quaternions can be helpful. Of course, it all depends what you are trying to do...

recursive 1y ago

I took a linear algebra class, as well as many others. It didn't work. Most math classes I've taken granted me some kind of intuition for the subject material. Like I could understand the concept independent from the name of the thing. In linear algebra, it was all a series of arbitrary facts without reason for existing. I memorized them for the final exam, and probably forgot them all the next day, as they weren't attached to anything in my mind. "The inverse of the eigen-something is the determinant of the abelian". It was just a list of facts like this to memorize by rote. I passed the class with a decent grade I think. But I really understood nothing. At this point, I can't remember how to multiply matrices. Specifically do the rows go with the columns or do the columns go with the rows? I don't know if there's something about linear algebra or I just didn't connect with the instructor. But I've taken a lot of other math classes, and usually been able to understand the subject material readily. Maybe linear algebra is different. It was completely impenetrable for me.

drdeca 1y ago
To remind oneself how to multiply matrices together, it suffices to remember how to apply a matrix to a column vector, and that ((A B) v) = (A (B v)). For each 1-hot vector e_i (i.e. the row vector that has a 1 in the i-th position and 0s elsewhere), apply B e_i to get the i-th column of the matrix B. Then, apply the matrix A to the result, to obtain A (B e_i), which equals (A B) e_i . This is then the i-th column of the matrix A B. And, when applying the matrix A to some column vector v, for each entry/row of the resulting vector, it is obtained by combining the corresponding row of A, with the column vector v. So, to get the entry at the j-th row of the i-th column of (A B), one therefore combines the i-th column of B with the j-th row of A. Or, alternatively/equivalently, you can just compute the matrix (A B) column by column, by, for each e_i , computing that the i-th column of (A B) is (A (B e_i)) (which is how I usually think of it). To be clear, I don't have the process totally memorized; I actually use the above reasoning to remind myself of the computation process a fair portion of the time that I need to compute actual products of matrices, which is surprisingly often given that I don't have it totally memorized. When I took linear algebra, the professor emphasized the linear maps, and somewhat de-emphasized the matrices that are used to notate them. I think this made understanding what is going on easier, but made the computations less familiar. I very much enjoyed the class.

imtringued 1y ago

Rotations: SO(3) is the special orthogonal group for 3 dimensions, which is the set of rotation matrices. Rotation matrices have the determinant 1, which means that they keep the length of a vector the same (forcing it to change direction). Orthogonal group means that the rotation R times its transpose R^T yields the identity matrix aka R^T R=RR^T=I. This means that rotations are trivially reversible. You can obtain rotation matrices either by explicit use of cosine/sine along your desired axis or you can use the Rodrigues rotation formula to describe a rotation along an axis with an angle. so(3) is the Lie algebra to the Lie group SO(3). The capitalization here is critical. In particular, so(3) is the set of skew symmetric matrices. 3 by 3 skew-symmetric matrices essentially describe the cross product a x b for a constant a. Given a skew symmetric matrix and a rotation angle theta you can use the exponential function to map so(3) to SO(3) through e^(theta S). Note that the exponential is a general mapping from a Lie algebra to its Lie group. Transformations: SE(3) is the special euclidean group for 3 dimensions. Its structure consists of T = [[R, t], [0, 1]]. if R = I, then T is a pure translation by t. If R != I and t = 0, then it is a pure rotation. When you have both a rotation R and a translation t transformation then translating first and rotating second will give you T, but rotating first and translating second will give you T' = [[R, Rt], [0, 1]]. se(3) is the Lie algebra to the Lie group SE(3). One way to represent so(3) is through twists. Revolute joints are represented by a vector Xi = [ omega, q cross-product omega] where omega is a vector that represents the parameters of a skew symmetric matrix and q is a 3d point on the axis of rotation. Prismatic joints are represented by Xi = [0, v] where v is pointing in the direction of the movement. The transformation of a twist can be obtained as usual through the exponential function: exp(theta Xi)=T. https://en.wikipedia.org/wiki/3D_rotation_group https://en.wikipedia.org/wiki/3D_rotation_group https://en.wikipedia.org/wiki/Orthogonal_group https://en.wikipedia.org/wiki/Orthogonal_group https://en.wikipedia.org/wiki/Euclidean_group https://en.wikipedia.org/wiki/Euclidean_group https://en.wikipedia.org/wiki/Screw_theory https://en.wikipedia.org/wiki/Screw_theory