3 ms·

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

by arh68 10y ago

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 ?

andars 10y ago
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 http://geometricalgebra.org/downloads/ga4cs_chapter1.pdf. It contains some code examples, and the complete source code is here: http://geometricalgebra.org/sandbox.html http://geometricalgebra.org/sandbox.html
- jacobolus 10y ago
  I made a list recently: https://news.ycombinator.com/item?id=12941658 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.pdf http://www2.eng.cam.ac.uk/~rjw57/pdf/r_wareham_pdh_thesis.pd...
ajkjk 10y ago
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).

by arh68 10y ago

andars 10y ago

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 http://geometricalgebra.org/downloads/ga4cs_chapter1.pdf. It contains some code examples, and the complete source code is here: http://geometricalgebra.org/sandbox.html http://geometricalgebra.org/sandbox.html

jacobolus 10y ago
I made a list recently: https://news.ycombinator.com/item?id=12941658 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.pdf http://www2.eng.cam.ac.uk/~rjw57/pdf/r_wareham_pdh_thesis.pd...

ajkjk 10y ago

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).