Walking Paths with Complex Numbers(dannythecoder.wordpress.com)
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Walking Paths with Complex Numbers
https://dannythecoder.wordpress.com/2016/12/02/walking-paths-with-complex-numbers/
22 comments
I'd say if it depends on how you think about complex numbers. If your understanding of the imaginary unit is limited to "that thing that when you square it you get -1", I agree with you. But if you think of the imaginary unit as the 2-blade in the geometric algebra of R^2, I would argue it is one of the best ways to think about 2D geometry.
i is a rotation operator in R3 as well - it's just a 3-vector instead of a 1-vector, and textbooks usually call these "quaternions".
If you do want to think of geometry in terms of linear algebra structures, you probably want to be studying Geometric Algebra.
If you do want to think of geometry in terms of linear algebra structures, you probably want to be studying Geometric Algebra.
ah, I'm aware of those things and have studied them extensively --
My point is, I think it's harmful to say "hey, you - you're trying to solve a problem in 2d space? Have you considered using complex numbers?" When the correct thing to do is "have you considered using the algebra of N-d spaces?" (that is, Clifford/geometric algebra), and leave all the magical complex numbers out of it.
My point is, I think it's harmful to say "hey, you - you're trying to solve a problem in 2d space? Have you considered using complex numbers?" When the correct thing to do is "have you considered using the algebra of N-d spaces?" (that is, Clifford/geometric algebra), and leave all the magical complex numbers out of it.
What would that look like, code-wise? Is there a good library for this stuff?
It's easy to reach for complex numbers when they're built-in. Do you have to use Maple/Mathematica for this stuff, or can you do it easily in Python/JS/&c ?
It's easy to reach for complex numbers when they're built-in. Do you have to use Maple/Mathematica for this stuff, or can you do it easily in Python/JS/&c ?
If you are interested in this stuff, I have heard that the book Geometric Algebra for Computer Science is good.
You can find the first chapter as a pdf download here: http://geometricalgebra.org/downloads/ga4cs_chapter1.pdf. It contains some code examples, and the complete source code is here: http://geometricalgebra.org/sandbox.html
You can find the first chapter as a pdf download here: http://geometricalgebra.org/downloads/ga4cs_chapter1.pdf. It contains some code examples, and the complete source code is here: http://geometricalgebra.org/sandbox.html
I made a list recently: https://news.ycombinator.com/item?id=12941658
The GACS book is indeed great. The “conformal model” defined there is very pleasant to work with. If you want to learn about it without buying a textbook, there’s also this PhD thesis: http://www2.eng.cam.ac.uk/~rjw57/pdf/r_wareham_pdh_thesis.pd...
The GACS book is indeed great. The “conformal model” defined there is very pleasant to work with. If you want to learn about it without buying a textbook, there’s also this PhD thesis: http://www2.eng.cam.ac.uk/~rjw57/pdf/r_wareham_pdh_thesis.pd...
Well, implementing rotation matrices in code isn't hard, and they should be supported in every mathematical library already.
Even more lightweight than matrices is just implementing the transformation on vector objects yourself: a 90-degree rotation left is R(x,y) = (-y, x), and (-R)(x,y) = -(R(x,y)). In higher dimensions this remains true for any 2-dimension subplane, so R_zx(x,y,z) = (z,y,-x).
Rotation matrices / other formulations become necessary when you're dealing with being able to rotate through any angle.
Though, doing rotations of the matrix 'y = Rx' form suffers mathematical problems like numerical instability and bad behavior. Quaternion rotations or Geometric Algebra's rotors (which are equivalent for N=3, and both take the form y = RxR^-1) work a lot better, and also eliminate a lot of the redundancy in the (skew-symmetric) rotation matrices formulation (4 variables instead of 9, for 3d).
Even more lightweight than matrices is just implementing the transformation on vector objects yourself: a 90-degree rotation left is R(x,y) = (-y, x), and (-R)(x,y) = -(R(x,y)). In higher dimensions this remains true for any 2-dimension subplane, so R_zx(x,y,z) = (z,y,-x).
Rotation matrices / other formulations become necessary when you're dealing with being able to rotate through any angle.
Though, doing rotations of the matrix 'y = Rx' form suffers mathematical problems like numerical instability and bad behavior. Quaternion rotations or Geometric Algebra's rotors (which are equivalent for N=3, and both take the form y = RxR^-1) work a lot better, and also eliminate a lot of the redundancy in the (skew-symmetric) rotation matrices formulation (4 variables instead of 9, for 3d).
I think it does extend to higher dimension. I have to read up on hypercomplex numbers, Clifford algebra and geometric algebra. Quaternions are known to facilitate arbitrary rotations in 3D. I suppose, complex numbers are more powerful in analysis rather than discrete mathematics, if there is a difference at all.
Quaternions model 3d rotations, and Clifford algebras in general model rotations in arbitrary higher dimensions, but it's definitely non-trivial to go from "complex numbers model 2d" to all of that machinery.
What I mean is that the intuition around complex numbers doesn't extend. Certainly not the "i is the square root of -1"-level understanding of complex numbers.
Clifford algebras are better thought of as structures built over the higher-dimension vectors spaces, with little relationship to algebras/fields. They have much more in common with the matrix representation than the algebraic one. The algebraic perspective only supports complex, quaternion, and octonion numbers before breaking down.
What I mean is that the intuition around complex numbers doesn't extend. Certainly not the "i is the square root of -1"-level understanding of complex numbers.
Clifford algebras are better thought of as structures built over the higher-dimension vectors spaces, with little relationship to algebras/fields. They have much more in common with the matrix representation than the algebraic one. The algebraic perspective only supports complex, quaternion, and octonion numbers before breaking down.
I still don't get the significance of complex numbers, to me they seem like vectors with certain operations and I still suppose there is no deeper revelation, that sqrt(-1) represents, and every example could have a geometric interpretation, even if this isn't motivated by the example. Likewise, the only explanation for e^(pi*i)-1=0 I know is the conversion to polar coordinates.
>Since you need to keep track of your current orientation this slightly complicates things a bit
>If at each step we keep the imaginary multiplication term from the previous step we can effectively encode the rotation in our expression
Why, you could just keep track of the rotation as an angle, which would likewise be a result of the previous step. In binary there wouldn't even be much of a difference: Look at a 2-bit value either as a modulo 4 ring of type pi/2, or as a modulo 2 vector of the two complex components. That's my first thought, at least. Although, the real and imaginary parts would always have opposite values, so only one bit is needed. I wonder if complexity analysis would show a difference.
>Since you need to keep track of your current orientation this slightly complicates things a bit
>If at each step we keep the imaginary multiplication term from the previous step we can effectively encode the rotation in our expression
Why, you could just keep track of the rotation as an angle, which would likewise be a result of the previous step. In binary there wouldn't even be much of a difference: Look at a 2-bit value either as a modulo 4 ring of type pi/2, or as a modulo 2 vector of the two complex components. That's my first thought, at least. Although, the real and imaginary parts would always have opposite values, so only one bit is needed. I wonder if complexity analysis would show a difference.
The best non-geometric interpretation of complex numbers is algebraic closure, IMO. Basically, mathematicians invent new kinds of numbers when they want to do things and have the result of the operation always be a number. Adding and multiplying natural numbers gets you a natural number. Subtraction on integers yields integers. Division for rational numbers. Infinite series for real numbers. And factoring all polynomials is the raison d'etre for complex numbers.
A complex number is a “complex” consisting of a scalar part (“real” or “parallel” number) and a bivector part (“imaginary” or “perpendicular” number) complected together into one single object.
IMO the best way to think of it is as the quotient or ratio between two arbitrary vectors T = v/u in a Euclidean vector space. Or equivalently, as a type of mathematical operator which you can apply (by multiplication on the left) to one vector u to produce another vector v, i.e. Tu = (v/u)u = v. This operator T will scale and rotate any other vector in the plane spanned by your two vectors u and v in the same way.
http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf http://geocalc.clas.asu.edu/pdf/GrassmannsVision.pdf
IMO the best way to think of it is as the quotient or ratio between two arbitrary vectors T = v/u in a Euclidean vector space. Or equivalently, as a type of mathematical operator which you can apply (by multiplication on the left) to one vector u to produce another vector v, i.e. Tu = (v/u)u = v. This operator T will scale and rotate any other vector in the plane spanned by your two vectors u and v in the same way.
http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf http://geocalc.clas.asu.edu/pdf/GrassmannsVision.pdf
(first, I second ThrustVectoring's comment about algebraic closure. That's a key part of understanding complex numbers)
I recommend reading this[1] article. I found it very enlightening about why complex numbers are useful.
[1] https://acko.net/blog/how-to-fold-a-julia-fractal/
edit: Warning - [1] requires WebGL.
I recommend reading this[1] article. I found it very enlightening about why complex numbers are useful.
[1] https://acko.net/blog/how-to-fold-a-julia-fractal/
edit: Warning - [1] requires WebGL.
[deleted]
>e^(pi*i)+1=0
fixed, otherwise e^pi would be trivially wrong
fixed, otherwise e^pi would be trivially wrong
Good stuff! Even though this may not be the best application for complex numbers, it is a very simple example for a practical use. For some person out there, this may even serve as a "gateway drug" to complex analysis.
Here is the pseudo-code for matrix rotation for comparison:
def rotate(vector, angle):
rotation_matrix = [
[cos(angle), -sin(angle)],
[sin(angle), cos(angle)]
]
return rotation_matrix * vector
position = (0, 0)
velocity = (0, 1)
for instruction in [x.strip() for x in input.split(",")]:
direction = instruction[0]
distance = int(instruction[1:])
angle = 90 if direction == 'L' else -90
velocity = rotate(velocity, angle)
position += velocity * distance
print positionSince you only work with 90 degrees, you don't even need the full machinery. In my solution I used (a,b) -> (-b,a) and its dual.
(edit: added reference to geometric algebras, which are cleaner than rotation matrices for intuitively modeling this.)