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So to make this familiar, you're probably used to traditional coordinate vectors in geometry. For example a 3D vector [x, y, z]. This seems sane enough, but is
by orlp 5y ago
So to make this familiar, you're probably used to traditional coordinate vectors in geometry. For example a 3D vector [x, y, z]. This seems sane enough, but is actually somewhat ambiguous. Which cardinal direction map to x, y, z respectively (e.g. is z for up/down or forwards/backwards)? What direction do the axes point, e.g. is +z away or towards us? To make it unambiguous, we can define three oriented basis vectors. That is, each basis vector defines what the unit is in that direction, and which way is "positive". So I could write [3, 2, 9] as 3*xunit + 2*yunit + 9*zunit.
Now in geometric algebra we also have oriented basis vectors. So they determine the unit, and the direction of the embedding space (note that this not necessarily map directly to points in your world space, which becomes especially clear when you start doing projective geometry, e.g. in 3D PGA you represent lines in your world as planes in the embedding space). Rather than xunit, yunit, etc, they are simply called e1, e2, ..., en.
The key difference is that geometric algebra has the exterior product, notated ^. For example, e1 ^ e2 is the exterior product of two oriented basis vectors. You can interpret this as being an oriented basis for the space spanned by e1, e2. And similarly for e1 ^ e2 ^ e3 being an oriented basis of the volume spanned by e1, e2, e3, etc. These are called basis blades.
The most general closed numerical representation of an object in geometric algebra is the multivector. Just like a traditional vector is just a bag of K coefficients for K basis vectors, a multivector is a bag of 2^K coefficients for the 2^K basis blades, one for each possible combination of the exterior product of the K basis vectors.
- faller 5y ago> So to make this familiar, you're probably used to traditional coordinate vectors in geometry. For example a 3D vector [x, y, z]. This seems sane enough, but is actually somewhat ambiguous. The concept of a coordinate system is not ambiguous. You have dimensions, and each can be represented as a vector that complies with specific properties, such as linear independence. > Which cardinal direction map to x, y, z respectively (e.g. is z for up/down or forwards/backwards)? That's a function of whatever coordinate transformation you wish to apply. Nevertheless, I vaguely recall from school the concept of an oriented vector space and direct coordinate system, whose definition was something like the cross product of consecutive director vectors resulted in a positive vector (right-hand rule) i.e., the direction of z is determined unambiguously by the direction of x and y. > Now in geometric algebra we also have oriented basis vectors. If I'm not mistaken, oriented referential systems are covered in intro to euclidean geometry classes. > The key difference is that geometric algebra has the exterior product, notated ^. For example, e1 ^ e2 is the exterior product of two oriented basis vectors. You can interpret this as being an oriented basis for the space spanned by e1, e2. And similarly for e1 ^ e2 ^ e3 being an oriented basis of the volume spanned by e1, e2, e3, etc. These are called basis blades. Sounds like a convoluted way to refer to basic concepts like direction vectors of a direct coordinate system. Quite bluntly, this all sounds like an attempt to reinvent euclidean geometry following a convoluted way. I mean, what does all this buy you that applying a subset of affine transformations (scaling, translation, rotation) to an orthogonal coordinate system doesn't give you already?
- chombier 5y ago> Quite bluntly, this all sounds like an attempt to reinvent euclidean geometry following a convoluted way I think that's the idea. > I mean, what does all this buy you that applying a subset of affine transformations (scaling, translation, rotation) to an orthogonal coordinate system doesn't give you already? Well, algebra: with the geometric product one can solve geometric equations in a nice, unified way.
- orlp 5y ago> Quite bluntly, this all sounds like an attempt to reinvent euclidean geometry following a convoluted way. I mean, what does all this buy you that applying a subset of affine transformations (scaling, translation, rotation) to an orthogonal coordinate system doesn't give you already? It is attempting to reinvent Euclidean geometry, yes. I don't think it's convoluted though. To give a prime example, take the article we're commenting on: interpolating rotations. Or more generally: interpolating transformations. Just doing this with rotations without suffering gimbal locks already brings you to quaternions. Are quaternions 'convoluted'? The fact that all objects are native to the algebra means they're composable. Take for example this slide of the formula of a 4D torus in coordinates, and in 4D PGA: https://i.imgur.com/T4hofL2.png https://i.imgur.com/T4hofL2.png The talk in general has a bunch of example applications: https://youtu.be/tX4H_ctggYo?t=4232 https://youtu.be/tX4H_ctggYo?t=4232 Questions such as "the intersection of this line and this plane", "the line through two points" "the circle where these two spheres intersect", "the point at the intersection of three planes", "the projection of this line on this plane" and such are trivial, native (the resulting object is part of the algebra) and exception-free in geometric algebra. E.g. two planes always intersect, it just happens that the intersection is a line at infinity if they're parallel. The exact same code used to translate and rotate a point around the origin can be used to translate and rotate a line, or a plane around the origin. Also note that most of computer graphics already realizes that embedding our geometric space into a larger space is useful. Projective geometry (embedding 3D into 4D) is already everywhere, because it unifies translations and rotations into a single concept (matrix multiplication). Geometric algebra simply goes a step further.