3 ms·

Would you mind expanding on the idea that an angle measure is a logarithmic quantity and that sin/cos/tan are exponentials (or providing a phrase I could google

Would you mind expanding on the idea that an angle measure is a logarithmic quantity and that sin/cos/tan are exponentials (or providing a phrase I could google)? I've never come across a formulation/characterization like that, and it sounds useful...

jacobolus 8y ago
You can break the exponential function into the part acting on a scalar and the part acting on a bivector, and then furthermore break those into even and odd parts: exp(ρ + θi) = (cosh ρ + sinh ρ)(cos θ + i sin θ) The angle measure in the above is the θ part. The “linear space” expression of the orientation/rotation is a 2-dimensional quanitity cos θ + i sin θ. Or if you start with the 2-coordinate version x + iy, where x^2 + y^2 = 1, you can find the angle measure by taking the natural logarithm, imag(log(x + yi)). Rotations are naturally a multiplicative concept. If you have a rotation a + bi and a rotation c + di, then you can compose them like (a + bi)(c + di) = (ac - bd) + (ad + bc)i (you might recognize this as the the “angle sum” identities from high school trigonometry class). The idea of the logarithm is that you can convert a multiplicative structure into an additive one. So if you have rotations Q and R, you could normally compose them like QR, but as an alternative you can use a log-space representation (angle measure): QR = exp(log(Q) + log(R)). A rotation inherently comes with an orientation. The i (“imaginary unit”) is in my opinion properly thought of as a bivector, oriented like a plane. When we express something as a scalar-valued angle measure, we have stripped the orientation out.
- twic 8y ago
  Thanks for this - i have studied very little maths, so even this perhaps quite basic stuff is like magic for me! I love the revelation that log/exp shifts between Cartesian and (log-)polar meanings of complex numbers. That's enormously satisfying for some reason. I note that the pattern in multiplying rotations also comes up if you do it with matrices: a -b c -d ac-bd -(ad+bc) b a . d c = ad+bc ac-bd Does something similar happen with quaternions / geometric products in 3D?
  - jacobolus 8y ago
    Taking 3D rotations into log space gets tricky. 3D rotation is not commutative, so if you have two quaternions Q and R, you can’t in general say that QR = exp(log(Q))exp(log(R)) = exp(log(Q) + log(R)). That identity only holds if the bivector (“imaginary”) parts of Q and R are oriented the same way in space, i.e. are scalar multiples of each-other. Otherwise in general exp(log(Q))exp(log(R)) ≠ exp(log(R))exp(log(Q)). The log space version is still useful though if you want to e.g. take the nth root of a spatial rotation.

jacobolus 8y ago

You can break the exponential function into the part acting on a scalar and the part acting on a bivector, and then furthermore break those into even and odd parts: exp(ρ + θi) = (cosh ρ + sinh ρ)(cos θ + i sin θ) The angle measure in the above is the θ part. The “linear space” expression of the orientation/rotation is a 2-dimensional quanitity cos θ + i sin θ. Or if you start with the 2-coordinate version x + iy, where x^2 + y^2 = 1, you can find the angle measure by taking the natural logarithm, imag(log(x + yi)). Rotations are naturally a multiplicative concept. If you have a rotation a + bi and a rotation c + di, then you can compose them like (a + bi)(c + di) = (ac - bd) + (ad + bc)i (you might recognize this as the the “angle sum” identities from high school trigonometry class). The idea of the logarithm is that you can convert a multiplicative structure into an additive one. So if you have rotations Q and R, you could normally compose them like QR, but as an alternative you can use a log-space representation (angle measure): QR = exp(log(Q) + log(R)). A rotation inherently comes with an orientation. The i (“imaginary unit”) is in my opinion properly thought of as a bivector, oriented like a plane. When we express something as a scalar-valued angle measure, we have stripped the orientation out.

twic 8y ago
Thanks for this - i have studied very little maths, so even this perhaps quite basic stuff is like magic for me! I love the revelation that log/exp shifts between Cartesian and (log-)polar meanings of complex numbers. That's enormously satisfying for some reason. I note that the pattern in multiplying rotations also comes up if you do it with matrices: a -b c -d ac-bd -(ad+bc) b a . d c = ad+bc ac-bd Does something similar happen with quaternions / geometric products in 3D?
- jacobolus 8y ago
  Taking 3D rotations into log space gets tricky. 3D rotation is not commutative, so if you have two quaternions Q and R, you can’t in general say that QR = exp(log(Q))exp(log(R)) = exp(log(Q) + log(R)). That identity only holds if the bivector (“imaginary”) parts of Q and R are oriented the same way in space, i.e. are scalar multiples of each-other. Otherwise in general exp(log(Q))exp(log(R)) ≠ exp(log(R))exp(log(Q)). The log space version is still useful though if you want to e.g. take the nth root of a spatial rotation.