The Angle Between Subspaces

[PatrickPowers]

It was clear early on that the notion of perpendicularity used in 3D Euclidean geometry wasn't adequate for higher dimensions. Everyone knows of perpendicular planes in 3D, but in 4D such planes are not perpendicular. Or are they? Eventually it became clear that there were two notions of perpendicularity. One we can call weak perpendicularity and the other strong. e₁₂ and e₂₃ are weakly perpendicular, e₁₂ and e₃₄ are strongly perpendicular. Once this distinction is made then angles may be calculated.

Given normalized subspaces A and B with grades a and b the weak angle is cos theta = ||AB<|a-b|>|| while the strong angle is sin tau = ||AB<a+b>||. In 3D they are usually the same, the exception being that the strong angle between two planes or volumes is always zero.

These notions may be extended to multivectors. Angles seem to me to have little meaning here so instead one may make use of the identity that if M and N are normalized multivectors with maximum grades m and n then MN<0>² + MN<1>² + MN<2>² ... MN<m+n>² = 1. The more weight have the lower order terms, the more the two multivectors have in common.

In 3D to find the subspace between two subspaces take the average. This doesn't necessarily work in higher dimensions. Suppose our two subspaces are e₁₂ and e₃₄. The normalized average of these will have strong and weak angles of pi/4 w.r.t. both e₁₂ and e₃₄ but is not a subspace. The subspaces that come closest to satisfying our requirement are e₁₃ and e₂₄. Either is weakly perpendicular to both e₁₂ and e₃₄.

[PatrickPowers]

Now note that the preceding definitions may not give the expected answer when applied to vectors. This is because when a vector is used to represent a subspace then it represents an infinite line. The angle between to intersecting infinite lines is between 0 and 90 degrees. If you are using that vector to represent a ray or a force or location or anything of that nature then you would expect the angle to be between 0 and 180 degrees. To get this behavior one may break the weak angle function into two cases. If a=b then instead use cos theta = AB^T<0> where B^T is the transversion of B. The transversion is the same multivector with the order of the vectors in each basis element reversed. e₁₂^T = e₂₁ and so forth. The purpose of this that if A=B then the weak angle will always be zero as it should be. With this definition the weak angle between vectors is as is usually expected. The new feature is that the angle between subspaces A and -A is always 180 degrees instead of zero. While e₁₂ and e₂₁ represent the same subspace their orientation is different so the angle between them is pi.

Can something like this be done with the strong angle? This seems possibly useful only in the case where a+b=N. Then we may use sin tau = AB<a+b>I^T. This will yield a strong angle that is between -90 and 90 degrees. The transversion is so that if AB=I then the strong angle between A and B is always pi/2.

The reason it is difficult to have a meaningful signed measure when a+b<N is that there is no apparent arbitrary standard such as I to compare with. The same holds true for the weak angle with subspaces of differing degrees.

[mr_e_man]

[...] the identity that if M and N are normalized multivectors with maximum grades m and n then MN<0>² + MN<1>² + MN<2>² ... MN<m+n>² = 1. [...]

Not true.

Take M = N = (1 + e₁)/√2, which are clearly normalized with maximum grade 1. Then

MN = (1 + e₁ + e₁ + e₁e₁)/2

= 1 + e₁,

so ||⟨MN⟩||² = ||⟨MN⟩₀||² + ||⟨MN⟩₁||² + ||⟨MN⟩₂||²

= 1 + 1 + 0

= 2 ≠ 1.

(Or take M = (1 + e₁)/√2 and N = (1 - e₁)/√2, so MN = 0.)

In fact I have been able to prove (not easily) that this is the maximum value, in 3D or lower: max{||MN||² / (||M||² ||N||²)} = 2. But then in 5D we have, for example,

M = N = (1 + e₁₂₃₄)(1 + e₅)/2

= (1 + e₁₂₃₄ + e₅ + e₁₂₃₄₅)/2,

so that ||M|| = 1, and

MN = (1 + e₁₂₃₄)²(1 + e₅)²/4

= (1 + 2e₁₂₃₄ + e₁₂₃₄e₁₂₃₄) (1 + 2e₅ + e₅e₅) /4

= (2 + 2e₁₂₃₄) (2 + 2e₅) /4

= (1 + e₁₂₃₄)(1 + e₅)/1

= 2M,

so ||MN||² = ||2M||² = 4||M||² = 4.

And then in 9D we can take M = N = (1 + e₁₂₃₄)/√2 (1 + e₅₆₇₈)/√2 (1 + e₉)/√2, which is normalized, and ||MN||² = 8.

And so on. The higher the dimension, the larger can be a product of unit multivectors.

[mr_e_man]

It doesn't work even if M and N have a single grade, say 2:

M = N = (e₁₂ + e₃₄)/√2

MN = (e₁₂e₁₂ + e₁₂e₃₄ + e₃₄e₁₂ + e₃₄₃₄) /2

= (-2 + 2e₁₂₃₄)/2

= -1 + e₁₂₃₄

||MN||² = 2

≠ ||M||² ||N||² = 1.

But it does work when M and N are blades, thus representing subspaces. Then indeed ||MN||² = ||⟨MN⟩₀||² + ||⟨MN⟩₁||² + ||⟨MN⟩₂||² + ... + ||M∧N||² = 1, so you can single out one grade of MN, which must have magnitude between 0 and 1, and call that cos θ or sin θ.

Proof of the identity: ||MN||² is the scalar part of

(MN) (MN)^T

= (MN) (N^TM^T)

= M (N N^T) M^T

= M ||N||² M^T

= ||N||² M M^T

= ||N||² ||M||².

This relies on the fact that a blade's reverse is ± the blade itself, and the square of a blade is a pure scalar (that is ± its squared magnitude).

..."Transversion"? I've heard "reversion" for multivectors, and "transpose" for matrices.

[mr_e_man]

This topic has been discussed before: viewtopic.php?f=3&t=2371

Personally, I prefer to revive old topics rather than start new ones, for the sake of organization.

[mr_e_man]

It doesn't work even if M and N have a single grade, say 2:

[...]

In that example, at least each particular grade of the product has magnitude ≤ 1. But here's an example where a single grade of the product has a larger magnitude:

M = N = (e₁₂ + e₃₄ + e₅₆)/√3

M∧N = ⟨MN⟩₄ = (2e₁₂₃₄ + 2e₁₂₅₆ + 2e₃₄₅₆)/3

||M∧N|| = 2√3/3 > 1.

[PatrickPowers]

This topic has been discussed before: viewtopic.php?f=3&t=2371

Personally, I prefer to revive old topics rather than start new ones, for the sake of organization.

That to me seems like the best definition of an angle between surtopes. Rotations, phase cancellations, and other physics type things don't come into that paradigm AFAIK, so wouldn't what I'm doing be off topic for them.

Multivector that aren't blades, seems to me that they aren't geometric objects at all. I'd be inclined to say they are operators and their geometric product is the composition of functions. But for all I know someone somewhere has some other use for them.

[mr_e_man]

This topic has been discussed before: viewtopic.php?f=3&t=2371

Personally, I prefer to revive old topics rather than start new ones, for the sake of organization.

That to me seems like the best definition of an angle between surtopes. Rotations, phase cancellations, and other physics type things don't come into that paradigm AFAIK, so wouldn't what I'm doing be off topic for them.

Rotations are relevant to polytopes. Phase cancellations (I guess like the 180° angle between e₁₂ and e₂₁) apply to multivectors, not to bare subspaces (no orientation/sign). Perhaps this topic should have been "The Angle Between Multivectors".

Anyway, the two topics are certainly related, but I won't insist that they be merged or renamed.