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Matrices can be your friends (2002)

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Re: Matrices can be your friends (2002)

#41

Post by z0r »

kevindamm wrote: post #22
dgacmu wrote: post #17

Earlier quoted context omitted.

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 usi…

The first time I learned it was from a book by LaMothe in the 90s and it starts with your demonstration of 3D matrix transforms, then goes "ha! gimbal lock" then shows 4D transforms and the extension to projection transforms, and from there you just have an abstraction of your world coordinate transform and your camera transform(s) and most everything else becomes vectors. I think it's probably the best way to teach…

Tricks of the * Game Programming Gurus :)

Re: Matrices can be your friends (2002)

#43

Post by owlbite »

Viliam1234 wrote: post #5

> Mathematicians like to see their matrices laid out on paper this way (with the array indices increasing down the columns instead of across the rows as a programmer would usually write them). Could a mathematician please confirm of disconfirm this? I think that different branches of mathematics have different rules about this, which is why careful writers make it explicit.

What I suspect he really means is that FORTRAN lays out its arrays column-major, whilst C choose row-major. Historically most math software was written in the former, including the de facto standard BLAS and LAPACK APIs used for most linear algebra. Mix-and-matching memory layouts is a recipe for confusion and bugs, so "mathematicians" (which I'll read as people writing a lot of non-ML matrix-related code) tend to prefer to stick with column major.

Of course things have moved on since then and a lot of software these days is written in languages that inherited their array ordering from C, leading to much fun and confusion.

The other gotcha with a lot of these APIs is of course 0 vs 1-based array numbering.

Re: Matrices can be your friends (2002)

#44

Post by neffy »

notmyjob wrote: post #35
foofoo12 wrote: post #19

Earlier quoted context omitted.

> 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.

Math achievement correlates strongly with visuospatial reasoning. Programmers may not be as proficient in math as economists, but they are better at it than biologists or lawyers.

And since the economist's main skill at math is fitting a very short ruler to a very large curve... i wouldn't put them ahead of lawyers...

Re: Matrices can be your friends (2002)

#45

Post by Ono-Sendai »

nyrikki wrote: post #38
dgacmu wrote: post #17

Earlier quoted context omitted.

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 usi…

> You can do rotation with a 3x3 matrix. You can do a rotation or some rotations but SO(3) is not simply connected. It mostly works for rigid bodies centered on the origin, but gimbal lock or Dirac's Plate Trick are good counter example lenses. Twirling a baton or a lasso will show that 720 degrees is the invariant rotation in SO(3) The point at infinity with a 4x4 matrix is one solution, SU(3), quaternions, or recen…

I think you are confused about what 'simply connected' means. A 3x3 matrix can represent any rotation. Also from a given rotation there is a path through the space of rotations to any other rotation. It's just that some paths can't be smoothly mapped to some other paths.

Re: Matrices can be your friends (2002)

#46

Post by PhilipRoman »

vintermann wrote: post #14
account42 wrote: post #9

Earlier quoted context omitted.

Not a mathematician, but programmers definitely don't agree on whether matrices should be row-major or column-major.

I'm surprised we even agree that they should be top-down.

At this point might as well make them match the x/y convention, with first index increasing to the right, and second index increasing from bottom to top.

Re: Matrices can be your friends (2002)

#47

Post by kccqzy »

getnormality wrote: post #6

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…

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.

Re: Matrices can be your friends (2002)

#48

Post by Figs »

zelphirkalt wrote: post #15
getnormality wrote: post #6

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…

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 w…

In computer graphics, 4x4 matrices let you do a rotation and a translation together (among other things). There's the 3x3 rotation block you found later as well as a translation vector embedded in it. Multiplying a sequence of 4x4 matrices together accumulates the rotations and translations appropriately as if they were just a bunch of function applications. i.e. rotate(translate(point)) is just rotation_matrix * translation_matrix * point_vector if you construct your matrices properly. Multiplying a 4x4 matrix with another 4x4 matrix yields a 4x4 matrix result, which means that you can store an arbitrary chain of rotations and translations accumulated together into a single matrix...

Re: Matrices can be your friends (2002)

#49

Post by simpaticoder »

zkmon wrote: post #12

There are a lot more ways to look at and understand these mysterious beasts called matrices. They seem to represent a more fundamental primordial truth. I'm not sure what it is. Determinant of a matrix indicate the area of or volume spanned by its component vectors. Complex matrices used in Fourier transform are beautiful. Quantum mechanics and AI seem to be built on matrices. There is hardly any area of mathematics…

>[Matrices] seem to represent a more fundamental primordial truth.

No, matrices (or more specifically matrix multiplication) are a useful result picked out of a huge search space defined as "all the ways to combine piles of numbers with arithmetic operators". The utility of the discovery is determined by humans looking for compact ways to represent ideas (abstraction). One of the most interesting anecdotes in the history of linear algebra was how Hamilton finally "discovered" a way to multiply them. "...he was out walking along the Royal Canal in Dublin with his wife when the solution in the form of the equation i2 = j2 = k2 = ijk = −1 occurred to him; Hamilton then carved this equation using his penknife into the side of the nearby Broom Bridge" [0]

The "primordial truth" is found in the selection criteria of the human minds performing the search.

0 - https://en.wikipedia.org/wiki/William_Rowan_Hamilton

Re: Matrices can be your friends (2002)

#50

Post by 1980phipsi »

notmyjob wrote: post #35
foofoo12 wrote: post #19

Earlier quoted context omitted.

> 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.

Math achievement correlates strongly with visuospatial reasoning. Programmers may not be as proficient in math as economists, but they are better at it than biologists or lawyers.

I would distinguish between visual imagination and visuospatial reasoning.

For people like myself with aphantasia, there are often problems solving strategies that can help you when you can’t visualize. Like draw a picture.

And lots of problems don’t really require as much visual imagination as you would think. I’m pretty good at math, programming, and economics. Not top tier, but pretty good.

If there are problems out there that you struggle with compared to others, then that’s the universe telling you that you don’t have a comparative advantage in it. Do something else and hire the people who can more easily solve them if you need it.

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