# General definition of angle/ rotation

It is well known that in the Euclidean plane a rotation about the origin can be computed with the formula

$$R_{\theta}(x,y) = \big(\cos(\theta)x-\sin(\theta)y, \sin(\theta)x+\cos(\theta)y\big)$$

It is somewhat well known that in the hyperbolic (Minkowski) plane a hyperbolic rotation (Lorentz boost) about the origin can be computed with the formula

$$HR_{\phi}(t,x) = \big(\cosh(\phi)t - \sinh(\phi)x, -\sinh(\phi)t+\cosh(\phi)x\big)$$

I'm well aware that given an inner product we can define the angle between two vectors $u,v$ by $\cos(\theta) = \dfrac{\langle u, v\rangle}{\|u\|\|v\|}$. But the hyperbolic plane (thought of as a vector space) isn't an inner product space. As far as I know the difference between the Euclidean plane and the hyperbolic plane is that they are equipped with different quadratic forms. In the Euclidean plane that quadratic form can be used to define an inner product, but not in the hyperbolic plane as it's not positive definite.

This leads me to think that there's a generalization of the formula for angles (or rotations as one seems to be expressible in terms of the other) for quadratic spaces. Does anyone know of a way of extending the concept of angle/ rotation to general quadratic spaces?