All Comments
TopTalkedBooks posted at August 19, 2017
I'd say alternative is an unlucky choice of words. I'd rather say geometric algebra (GA) is an extension of linear algebra (LA). In order to really understand GA you need first to firmly understand LA. Then it becomes clear that all that GA does is to turn a Hilbert space into an algebra called a Clifford algebra, and to examine the geometric semantics of the various operations that pop up in the process.

Here are three great sources that helped me to understand GA:



3. , pages 749-752

The first source gives great motivation and intuition for GA and its various products. Its mostly coordinate free approach is very refreshing and makes the subject feel exciting and magical. This is also the problem of the book, it's easy to end up confused and disoriented after working through it for a while. The second source is great because it grounds GA firmly on LA, and makes everything very clear and precise. The third source gives a short and concise definition of what a Clifford algebra is.

TopTalkedBooks posted at August 20, 2017

Here is a short code snippet which works whenever the cross product makes sense: the 3D version returns a vector and the 2D version returns a scalar. If you just want simple code that gives the right answer without pulling in an external library, this is all you need.

# Compute the vector cross product between x and y, and return the components
# indexed by i.
CrossProduct3D <- function(x, y, i=1:3) {
  # Project inputs into 3D, since the cross product only makes sense in 3D.
  To3D <- function(x) head(c(x, rep(0, 3)), 3)
  x <- To3D(x)
  y <- To3D(y)

  # Indices should be treated cyclically (i.e., index 4 is "really" index 1, and
  # so on).  Index3D() lets us do that using R's convention of 1-based (rather
  # than 0-based) arrays.
  Index3D <- function(i) (i - 1) %% 3 + 1

  # The i'th component of the cross product is:
  # (x[i + 1] * y[i + 2]) - (x[i + 2] * y[i + 1])
  # as long as we treat the indices cyclically.
  return (x[Index3D(i + 1)] * y[Index3D(i + 2)] -
          x[Index3D(i + 2)] * y[Index3D(i + 1)])

CrossProduct2D <- function(x, y) CrossProduct3D(x, y, i=3)

Does it work?

Let's check a random example I found online:

> CrossProduct3D(c(3, -3, 1), c(4, 9, 2)) == c(-15, -2, 39)

Looks pretty good!

Why is this better than previous answers?

  • It's 3D (Carl's was 2D-only).
  • It's simple and idiomatic.
  • Nicely commented and formatted; hence, easy to understand

The downside is that the number '3' is hardcoded several times. Actually, this isn't such a bad thing, since it highlights the fact that the vector cross product is purely a 3D construct. Personally, I'd recommend ditching cross products entirely and learning Geometric Algebra instead. :)

Top Books
We collected top books from hacker news, stack overflow, Reddit, which are recommended by amazing people.