Unlearn rotation matrices as rotations

https://kodkodgames.gitlab.io/rotation_matrices/

by dosshell • 6 years ago

258 124 6 years ago

kodkodgames.gitlab.io

mdtusz• 6 years ago

The first column of the rotation matrix is the new x-axis expressed in the old coordinate system, the second column is the y-axis and so on. An identity matrix would yield in no rotation since all unit vectors would be the same as the previous coordinate system.

It's a shame that this isn't made more clear in most tutorials and classes. The idea of axis application order really does make things more confusing than is needed.

lallysingh• 6 years ago

I've spent the last year correcting for the poor linear algebra taught in my undergrad. The gaps and inconsistencies left by those classes are a lot harder to repair when you've got a full-time job and kids.

martincmartin• 6 years ago

What do you suggest as a good text for learning it right the first time?

ziotom78• 6 years ago

I remember that in my freshman's year, my linear algebra teacher explained that every matrix (i.e., not only rotation matrices) has this property: each column is the result of the operation on each basis element in the old coordinate system. This is true even for rectangular matrices.

rocqua• 6 years ago

Once you get this, you can start building matrices for any (finite dimensional) 'vector space' not just the standard 'vectors as a list of numbers'.

All you need to do is to pick a basis for your vector space, and you can start to represent linear transformations as matrices w.r.t. that basis. Even cooler is when you let go of specific basis choices, and start talking about properties of linear transformations that do not depend on a choice of basis. Things like a determinant, eigenvalues, etc.

layoutIfNeeded2• 6 years ago

I’ve encountered this misunderstanding many times, usually in the context of creating a rotation matrix for a given direction and “up” vector. People look baffled when I replace their elaborate Euler-rotation algorithm with three lines of code:

    z = direction
    x = up.cross(z).normalized()
    y = z.cross(x)
    M = [x, y, z]

andi999• 6 years ago

Could you explain this is more detail?

kaetemi• 6 years ago

Creates a matrix with one axis in a given direction. Then a second axis at a straight angle to that, based on a reference up axis. The third axis is simply perpendicular to both. Position goes into the last vector of the matrix.

bl0b• 6 years ago

It is usually used to create a matrix for the camera orientation given a) the camera position in XYZ coordinates, b) a point the camera is looking at in XYZ coordinates, and c) the direction of 'up' in the world (otherwise the camera would be able to spin freely along it's Z axis while still satisfying the 'look at point A from point B' requirement). You build up the 3 axes of the camera orientation step-by-step. The 3 axes can then be plugged directly into the transformation matrix, and voila.

When I was learning 3D graphics, I found this site an excellent resource. Here is the page with a more in-depth explanation of the 'look at' function:

https://learnopengl.com/Getting-started/Camera

tripletao2• 6 years ago

A matrix-vector multiplication (or any other linear operation) can be understood as a change of basis, with the matrix rows or columns as the new basis vectors. A rotation[1] changes to a new set of basis vectors that are (a) all unit length and (b) all perpendicular to each other.

In the code, direction is assumed to be a unit vector, and that's your new z. x = up.cross(direction) is perpendicular to both up and direction, but may have arbitrary length since up and direction aren't generally perpendicular to each other; so x must also be normalized to length of one. y = z.cross(x) is perpendicular to both z and x, and doesn't need normalizing since they were already perpendicular (though extra normalizing doesn't hurt and sometimes--probably not here for most purposes--helps to clean up rounding error).

It's almost never a good idea to use Euler angles unless you're modeling a system where they have physical meaning (e.g., robot arm, gimbal, etc.). The math is almost always simpler when you work in terms of the rotation matrix directly (or quaternions, or axis-angle rotation, or pretty much anything else).

1. Or mirroring, which is the same as a rotation but with any two of the basis vectors swapped or any one negated.

beefield3• 6 years ago

Funny. Only some months ago I needed to figure out how to represent unit vectors of (a rotated) coordinate system in another coordinate system. After some days of drawing on the paper I came to the conclusion that actually the rotation vector columns are just that. But found no way to verify that claim in the internet. Thank you for this.

leereeves2• 6 years ago

That's not only true for rotation matrices. The columns of any matrix are where the unit vectors in the original coordinate system are mapped to when multiplied by the matrix.

MiroF10• 6 years ago

What are colleges teaching these days if not this??

im3w1l• 6 years ago

A really neat thing I learned in physics was Bra-Ket (get it bracket?) notation.

Ket's are vectors and written like |u>. Bra's are objects that compute dot products with vectors. They are written like <u|. So <u| |v> = u · v.

However if you have |u> <v| then nothing happens. This is a new object. It will first take dot product with v and then multiply with |u> vector.

Letting e1, e2.. denote the coordinate basis vectors, we can write a matrix A with coefficients a_ij as the sum Σa_ij |ej> <ei|.

This can be rearranged as

Σ_i |Σ_j a_ij ej> <ei|

Written like this, it is clear that the column is where the unit vector ends up.

dosshell• 6 years ago

I'm glad you appreciated the text.

bawana2• 6 years ago

I thought I knew what rotation matrices were until I read this article. Now I am confused. I used to think I needed a 3x3 matrix to rotate a point (with x,y,z coordinates) in 3 D space. But the article refers to Rx,Ry and Rz EACH of which are a 3x3 matrix.Why are there 3 matrices? To move an x coordinate to a new coordinate system, I only need to displace it among the the existing three axes. So each coordinate only needs a 3x1 vector to displace it . That a makes a single 3x3 matrix.

leereeves2• 6 years ago

Rx rotates about the x-axis (in the yz plane), Ry about the y-axis (in the xz plane), and Rz about the z-axis (in the xy plane).

We need all three because none of them can be made from the other two, but any other rotation can be made from a combination of these three.

dahart• 6 years ago

Those 3 mats Rx Ry Rz are just 3 separate examples from Wikipedia of how to construct a rotation matrix where the rotation axis is in the x, y, or z direction. They are separate rotations that don’t have to be combined.

diffeomorphism• 6 years ago

I only need to displace it among the the existing three axes.

That is exactly what these three matrices do. R_x depends on a single parameter and tells you how much you rotate around the x-axis. R_y tells you how much you rotate around the y-axis and R_z for the z-axis. Any rotation can be written as a product of these three (which then is a single 3x3 matrix depending on three parameters).

aesthesia• 6 years ago

Those three matrices describe three different rotations. Rx is a rotation about the x axis, Ry about the y axis, Rz about the z axis. They're parameterized by an angle theta that describes how much to rotate.

brummm• 6 years ago

R_x, R_y and R_z are the basis to generate all rotations. Think of them as the equivalent of the basis vectors in 3d euclidian space: e_x = (1, 0, 0), e_y=(0, 1, 0) and e_z = (0, 0, 1)

kergonath• 6 years ago

Any number of successive rotations can be represented as a single matrix. Both are right at the same time: an arbitrary rotation (represented by a single matrix) is a combination of 3 rotations about the axes of the reference frame (represented by 3 matrices).

bawana2• 6 years ago

Thank you to all who replied. My clue should have been the use of the same theta instead of alpha, beta and gamma.

IshKebab• 6 years ago

This is a great clarification. It's a shame that Wikipedia is such a bad reference for learning maths. I think it is a result of most maths articles being edited by people who just learnt something and therefore don't understand it enough to explain it well, and also want to show off their knowledge rather than actually explain things.

Mathworld is much much better in general.

(Somebody's probably going to reply "Wikipedia is an encyclopaedia not a tutorial".)

nl• 6 years ago

This is absolutely correct.

Wikipdia math pages (and for some sciences too) are correct in the details and yet provide no insight.

It's almost backwards what you want from an encyclopedia, which should be an overview with references to more details.

ducttapecrown• 6 years ago

Well, the insight each person needs varies.

edflsafoiewq• 6 years ago

The matrix article is really unusually bad.

One of the problems, I think, is that there are competing "views" of linear algebra. Mathematics students are taught a different class than everyone else where vector spaces and linear functions are brought to the fore. Everyone else seems to get a more computation-focused course. So there's a tension there where if you write a math-style article about matrices where this fact about the columns would probably be the definition of a matrix, everyone else might not even recognize what's going on.

There's also the issue that the current article is so sprawling that refactoring it to subject it to a unifying plan would require a large rewrite, and that's hard in an environment like Wikipedia.

jeffreyrogers• 6 years ago

Wikipedia is great when you already know a math topic and just need to look something up, but is terrible at teaching math. Generally the level of abstraction is too high and too general for a first introduction to a topic.

swimfar• 6 years ago

I find that sometimes even when I'm familiar with a topic, it ends up being a bad reference. This is especially true when they start introducing variables and constants that haven't even been defined.

benrbray2• 6 years ago

For what it's worth, I find that Wikipedia is a great place to learn about more advanced topics in abstract algebra / analysis / category theory. For example, the representation theory [1] page gives some pretty good context for how it relates to other fields.

It's not the same as a textbook, but generally, after reading the Wikipedia page, I know enough new terminology to intelligently Google for more complete resources like lecture notes, papers, etc..

[1] https://en.wikipedia.org/wiki/Representation_theory

dazzawazza• 6 years ago

I was eight years in to professional videogame dev. When I decided to debug matrices used to generate racing track surfaces by drawing the columns in the game world when I realised they were X/Y/Z axis. I felt so stupid. I jokingly told the other coders around me how stupid I was... turns out we were all stupid.

One old timer laughed in the background and told us how he discovered this on the Amiga a few years before.

vivekseth• 6 years ago

I learned this concept by watching 3blue1brown's series on Linear Algebra: https://www.3blue1brown.com/essence-of-linear-algebra-page

Would highly recommend. I feel like Grant's videos gave me a better understanding of Linear Algebra than the course I took in college.

mmmBacon• 6 years ago

Blue Brown is really a great resource. His visualization library looks really nice too. He clearly spends a lot of time thinking about how to explain things. A lot of colleges could learn a lot about how to teach from people like him.

aidos• 6 years ago

This series taught me all sorts of things I never groked from years of university education.

calebkaiser2• 6 years ago

I would heavily second this. I would also recommend, for those who are interested in computational linear algebra, going through fast.ai's course when you finish 3blue1brown: https://github.com/fastai/numerical-linear-algebra/blob/mast...

edge172• 6 years ago

Could you expand a bit on why this is a good course? Lots of great linear algebra stuff out there, why this one?

pengaru• 6 years ago

The 3b1b videos are really high quality, but I don't find them particularly useful when learning new material. The fancy animations distract and mesmerize, I slip into a mode of being entertained, becoming mostly a waste of time.

For subjects I've already learned however they can be useful for gaining new visual perspectives.

On the topic of learning rotation matrices / linear algebra in video games, I'd strongly recommend the Handmade Hero youtube channel. Casey explains these subjects at length in multiple videos, using plain chalkboard-style drawings I personally find far less distracting.

bananaface• 6 years ago

Second on Handmade Hero. It's great.

AlchemistCamp• 6 years ago

They give the "feeling of learning", which is worth only a little. They also give the motivation to learn, which is invaluable.

atoav• 6 years ago

Also recommended: https://natureofcode.com/book/

MiroF10• 6 years ago

"Don’t think of them as rotations, think of them as a unit vectors of a new coordinate systems."

This makes it clear to me that many people have been very poorly served by their linear algebra courses if that is not obvious.

criddell• 6 years ago

I'm one of those people. I really struggled with my undergrad linear algebra courses. It was taught in a very abstract way with no suggestion ever that any of this had useful applications. I passed (barely) and was glad to be free of it.

Years later when I got into graphics programming I found myself dealing with linear algebra concepts but this time it made much more sense to me. Being able to apply the ideas to real problems with visual solutions made all the difference.

0xfaded• 6 years ago

I went through Stephen Boyd's "Introduction to Linear Dynamical Systems", and at some point he said something like "Eigen vectors are just the directions linear systems grow in", and bam!, many years of linear algebra came rushing back and everything made sense.

Teknoman117• 6 years ago

At my university they rolled differential equations and linear algebra into the same course. It ... wasn't very good.

MiroF10• 6 years ago

I was taught linear algebra largely through various physics courses I took through undergrad. Not necessarily the most traditional way, but I think that it should be taught that way more frequently - with physics, there is something concrete to grasp which can be helpful to a lot of people.

recursive• 6 years ago

I technically passed a linear algebra course, but developed zero intuition for any of the facts I memorized. As a result, I no longer remember any of it other than something-something eigen-matrix.

failuser2• 6 years ago

They probably skipped classes to build their new app’s prototype. I wouldn’t even say it’s worse, most coder won’t need linear algebra.

MiroF10• 6 years ago

If you ever want to really do math at some point in your life, I think you need linear algebra or something equivalent. Better to build the intuitions earlier in your life.

arcticbull• 6 years ago

Having a breadth of knowledge is one of those things you never realize the value of, if you don't have it. That's something I realized over the course of my career.

For instance, I have a broad math background, nothing too deep, but enough to have an intuition on how complex things worked. One of the defining projects of my career to date was an embedded signal processing project given to me because "it seems like you know math" -- and I figured it out on the job by intuiting my way through. I figured out Fourier analysis and windows and FIR filters by intuition, and was able to use the pieces I had available as API to do something truly impactful.

Most people won't need most things from school, but if you're the one who knows that one thing that's needed that one time, it can be disproportionately rewarding.

mytailorisrich• 6 years ago

Recently I worked on a project involving rotation matrices. Engineers were worried about matrix multiplications. I told them it's just a dot product with each of the 3 unit vectors... Sighs of relief all round.

jheriko2• 6 years ago

many people did not learn about them from a linear algebra course

ranger207• 6 years ago

I've never really needed to use linear algebra, but I did take a class on it recently, so...

Why does it matter that it's unit vectors in a new coordinate system? I don't really care about the new coordinate system, I care about rotating something to this position in my old coordinate system. Saying "oh yeah it's <1,1,1> in this coordinate system" doesn't really help because I still need to translate <1,1,1> in the old coordinate system to <1,1,1> in the new coordinate system... and now I've got the same problem again.

I mean, I get that a rotation matrix, or any matrix for that matter, represents unit vectors in a new coordinate system. If I got nothing else from that course I at least learned that a matrix represents like 50 different concepts like systems of equations, coordinate spaces, etc. A big part of my course was applying rules for one representation of a problem to another equivalent representation to a problem. What I never understood is why it matters how you're representing the problem. Identifying that a rotation matrix is just unit vectors in a new coordinate space doesn't help me solve my problem of wanting to rotate something, yet it felt like 75% of my course was about converting problems from one representation to another.

Twisol• 6 years ago

I think the idea is to recontextualize the coordinate system of your object as not being the global coordinate system in the first place. The matrix defines the orientation of your object relative to the global coordinate system.

The question of "how to rotate something" rests on operational semantics. The question of "how is this related to that" rests on denotational semantics, and is much more general. The beauty of linear algebra is how closely the two are related: every linear transformation (the relationship) gives a matrix given a choice of basis, and matrices have a direct algorithm for transforming coordinate vectors.

overgard• 6 years ago

In this case, the idea is to make it easier to debug by looking at it. If you work in 3D graphics, usually looking at euler angles (ie, rotation around x,y,z) is pretty intuitive, but it has a lot of issues with gimbal lock, order the rotations happen in, etc. Quaternion's fix that, but they're very unintuitive to look at (well, unless there's another tricky like this I'm unaware off -- I'm sure there is). On the other hand, this trick with the rotation matrix lets me represent an arbitrary rotation, but it's also pretty easy to take that representation back to something I can understand.

In 3d graphics (or physics or what have you) you're constantly moving between coordinate systems/frames of reference because certain things are very easy to represent in a local coordinate system but are difficult to work with in a global one

fifticon• 6 years ago

But that is actually very useful problem solving. Identifying a problem in your hand as identical to a different problem you know solutions for, allows you to solve it effortlessly like one of those impossible smith puzzles with knots and strings.

white-flame• 6 years ago

Even if you didn't learn this from linear algebra, doing any measure of graphics programming at the matrix level very directly reveals these sorts of associations as you see how the numbers accumulate, and get everything debugged. Children learned these concepts, too, as they hobbied around with programming rotating visuals. It doesn't require advanced math, it's just a very practical perspective.

SNES programming also directly introduces people into the notion of 2d vectors advancing over an original axis, which ends up in a rotation in mode 7 stuff.

taneq• 6 years ago

I remember this being one of the huge "aha!" moments in my linear algebra course. I'd been doing 3D graphics programming for a while at that point and suddenly all the mechanics behind the calculations snapped into place.

SagelyGuru• 6 years ago

I would emphasise one more thing in connection with this: scalar products being projections onto the new axis and the matrix multiplication (so called rotation) of a vector is just calculating those scalar products. Now you can forget all those confusing angles altogether.

rumanator3• 6 years ago

"Don’t think of them as rotations, think of them as a unit vectors of a new coordinate systems."

That's exactly what all square non-singular matrices with unit column spaces are. In fact, all square non-singular matrices are coordinate transformations. That's pretty much the central point of linear algebra.

Framing rotation matrices as changes to the coordinate system doesn't add much info like framing polynomials as continuous functions. Technically it's true, but it doesn't add much to clarify.

undershirt• 6 years ago

It’s clearer in 2D. Rotating by angle `a` tilts the x-axis to (cos(a), sin(a)), because that’s the very definition of cosine and sine (i.e. the x and y components of angles on the unit circle). The y-axis just flips the components and negates one, because that’s what 90° rotations do.

adenadel2• 6 years ago

I want to share my favorite linear algebra textbook (Linear Algebra Done Right by Axler)

http://linear.axler.net/

It's meant for a second course in linear algebra and the focus is on abstract vector spaces and linear transformations (rather than a table of numbers perspective). It also doesn't use determinants until the last chapter.

sannee2• 6 years ago

LADR is probably the best linear algebra book around. It's a bit of a shame that the "marketing" mostly involves the fact that it does not use determinants.

The last edition also contains chapters on stuff like dual spaces and quotient spaces, which are important yet very often missing in other textbooks.

That being said, his reliance on C/R in about half the book could be a problem for people with CS background. Like, what if I want to compute determinants of linear maps which are over Z_2? There are also no chapters on number wrangling (like the gaussian elimination and friends), which is a big plus in my view but possibly a problem for others.

adenadel2• 6 years ago

I think these are great points. I justify the lack of some of the fundamental concepts like Gaussian elimination to be due to the fact that the book is meant for a second course in linear algebra. My copy has dual and quotient spaces and I think these are important topics to get an introduction to if you're looking to learn functional analysis, topology, etc. Some discussion of finite fields could be interesting. This has prompted me to look up linear algebra over finite fields.

jpeloquin2• 6 years ago

If we relax the restriction that the columns of the 3x3 transformation matrix must be unit vectors, we also get the ability to apply scaling and shear transformations. In solid mechanics, this is called the F tensor, and is used to describe deformation of materials [1].

There is also an augmented 4x4 form if translation is needed as well:

  | Q ...  Δx |         | vx |
  | .      Δy |         | vy |
  | .      Δz |   dot   | vz |   = new vector
  | 0 0 0  1  |         | 1  |

where "Q" is the 3 × 3 transformation matrix (e.g., rotation matrix) that is the subject of the OP, and [vx, vy, vz, 1] is the augmented form of a vector "v" = [vx, vy, vz] that is being transformed.

The augmented form is especially useful for transforming voxel indices in an 3D image array to spatial coordinates, such as for MRI or CT image data [2]. It is helpful if the index coordinate system has its origin at zero (zero-indexed arrays), like a normal coordinate system.

Finally, if we have many [x, y, z, 1] data points that each have an old and a new position, we can compute the overall best-fit transformation from the old to new positions with the least-squares solution to

  A x = b

where "A" is the (unknown) augmented matrix representing the transformation, "x" is a 4 × n array storing the points' old augmented-form positions, and "b" is a 4 × n array storing the points' new augmented-form positions. If we allow the points to have arbitrary dimension, without any particular spatial interpretation, we get least squares curve fitting and the foundation of machine learning.

[1] http://homepages.engineering.auckland.ac.nz/~pkel015/SolidMe...

[2] https://nifti.nimh.nih.gov/nifti-1/documentation/nifti1field...

shezi• 6 years ago

What you describe is very standard in 3D graphics, and it's alluded to in the article by the question at the end. You can even use the last row of the matrix as scaling factors (you'll have to re-normalize the output vector).

From a mathematical standpoint, this is now a projective vector space.

Some references:

https://en.wikipedia.org/wiki/Transformation_matrix

https://www.quora.com/Why-does-OpenGL-use-4D-matrices-for-ev...

https://guide.handmadehero.org/code/day362/

sannee2• 6 years ago

Note that projective spaces are not vector spaces. There is no meaningful way of adding vectors nor multiplying by a scalar.

Meaning we have to be very careful about treating our projective transformations as linear maps, as they are, in fact, nothing like that.

Of course people doing graphics often pick a section of the projective space ("last component is 1"), then map that onto R^3 and off they go, sort of ignoring the "planes at infinity" and praying for the best. This is messy but I do not see a better way.

jpeloquin2• 6 years ago

Your wikipedia link reminds me that I should have listed rotation, scaling, shear, and reflection transformations. Oops.

In your second link, I like the trick of storing positions as [x, y, z, 1] and directions as [vx, vy, vz, 0].

Regarding, "Hey, Markus! How come this matrix is 4x4?", I'm not sure if Markus was referring to an affine transformation, part of a perspective projection, a matrix representation of a quaternion, or something else. The mere fact a matrix is 4x4 (or some other size) unfortunately doesn't say much about what it's meant to do. Which probably contributes to Markus getting so many questions.

activatedgeek3• 6 years ago

Rotation matrices just describe the unit vectors of a new coordinate system

More generally, _every_ matrix describes how a change of coordinate system should happen. For any m x n (m rows, n columns) matrix, each of the the n column vectors represent how the current coordinate system (which may or may not be unit vectors) should be represented in the new coordinate system (for a left multiplied matrix).

Whenever a matrix's determinant is zero, it means that you squashed some dimensions. As you can imagine, when m is not equal to n, there will always be dimension squashing. Even when they are equal, that can happen. If you take a 3 x 3 matrix and it transforms all the 3-D vectors into only planes (which are 2-D objects), the determinant will be zero. This would be stated as having a rank of 2. More simply, you'd say the "volume" of the transform to be 0 (because planes have zero volume).

EDIT: Make sure to read important clarifications by JadeNB below.

JadeNB2• 6 years ago

Whenever a matrix's determinant is zero, it means that you squashed some dimensions. As you can imagine, when m is not equal to n, there will always be dimension squashing.

Although these two sentences are correct individually, it may be worth it to emphasise that they should not be read together: the determinant is defined only for square matrices. (One can do something like computing the determinant of det(A^{adjoint}A) if you want a general numerical invariant that behaves as you like—and then we're getting in the neighbourhood of the SVD).

If you take a 3 x 3 matrix and it transforms all the 3-D vectors into only planes (which are 2-D objects), the determinant will be zero.

Although your meaning seems clear, I think you may have misspoken slightly. Of course 3D vectors are transformed by a 3 x 3 matrix into vectors, not planes. To me, what I think you mean would read better as "If you take a 3 x 3 matrix and there is a plane such that all 3D vectors are transformed into that plane, then …", or, perhaps even better, "… such that the transforms of all 3D vectors lie in that plane …".

activatedgeek3• 6 years ago

Those are very important points. I've edited the parent to point out this clarifying comment.

smlckz• 6 years ago

I remember asking in freenode irc channel ##math that why matrix multiplication is defined like that, why not in any other way. I was told that:

> cuz the composition of linear maps are defined like that.

It took a long time to understand what they meant. Still I see how it is taught to work out mechanically. Brave friends of mine went to the Wikipedia page to come back with more confusion. Wikipedia is a great reference when you know the subject well enough.

https://en.wikipedia.org/wiki/Wikipedia:WikiProject_Mathemat...

exmadscientist• 6 years ago

Wikipedia is a great reference when you know the subject well enough.

Some math pages on Wikipedia are great. Some are... horrid. A while back (2017?) I couldn't remember the terms in the Taylor expansion of sqrt(x), so I went to Wikipedia. The article was one of the most jumbled mishmashes I'd ever tried to read. But, even though I'm qualified to fix it, I didn't dare wade into Wikipedia politics. The rewrite would have taken a while (it's a long article), but I can't even imagine how long shoving it through would have taken!

a_zaydak2• 6 years ago

This is so important! I spend my life writing EKFs for navigation systems however did not have a good linear algebra education. It was a hard struggle! Now, every time I have a new hire or intern, the first thing that I go over is basic linear algebra and rotations including DCM, Euler angles, quaternions, and so on. I also make sure they understand it graphically such as how you can scale and project vectors using a matrix.

tomo_kallang• 6 years ago

Hi, do you mind if I reach out to you? I am working on similiar projects and somehow every filters I tried have some issues.

a_zaydak2• 6 years ago

Sure, just send me an email.

failuser2• 6 years ago

I'd say quatenions are a nicer way to represent rotations, but 4x4 matrix can describe any projective transformation in 3d.

jheriko2• 6 years ago

i have also encountered this problem. i learned the nature of rotation from first principles in order to solve these problems without documentation... which has turned out to be a massive boon.

tutorials are the devil. they do not teach so much as demonstrate a lot of the time.

its a sad place to be

uzbit• 6 years ago

Speaking of unlearning rotations, has anyone looked at https://bivector.net ? Some other hacker posted it a while back on another thread and I found it to be really intriguing.

rory_h_r• 6 years ago

There's nothing wrong with understanding rotation matrices as rotations, it's just that in their application it's not the right mental model. Ideally you'll understand both representations and switch between them as needed.

dmch-1• 6 years ago

For some time I was surprised that anyone would represent rotations by anything else but rotation matrices. I sometimes do find wikipedia useful for quick answers for maths, however agree its quite disorganised.

saeranv• 6 years ago

The way I think about it: any matrix multiplication consists of the linear combination of the column vectors of your transformation matrix, with each column vector 'weighted' by the the input vector's x, y, z, ... n values.

FabHK• 6 years ago

Two remarks:

1. > Right handed, z forward through the nose and x through the left ear.

Interesting, in aviation engineering the convention is different: the body frame has x forward (positive rotation around x is right roll), y to the right (... pitch up), z down (... right yaw).

2. The best reference (reference, not explanation/textbook) for this whole spiel (rotation matrices, Euler angles, quaternions) I've seen is a paper by Diebel, Representing attitude: Euler angles, unit quaternions, and rotation vectors

https://www.astro.rug.nl/software/kapteyn-beta/_downloads/at...

(Or maybe I like it because "Diebels Alt" is my favourite local beer... https://en.wikipedia.org/wiki/Altbier)

seventytwo• 6 years ago

Is there a good web app where you can play around in a visual 3D space with transforms like this and other ones like quaternions?

DreamScatter• 6 years ago

Yes, check out https://bivector.net

https://github.com/enkimute/ganja.js

and my own geometric algebra also:

https://github.com/chakravala/Grassmann.jl

syntaxing• 6 years ago

I think the biggest struggle with rotation matrix isn't the theory but the nomenclature. Is it R(ZYX)?! Is it rotation axis rather than Euler angles?! All the convention differences can make signficant mistake very quickly.

Unlearn rotation matrices as rotations

Related Stories