# 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?

## 1 Answer

While the indefinite bilinear form for the hyperbolic plane technically doesn't count as an inner product, it is still a symmetric bilinear form, and that is really the most appropriate generalization of the two.

Things like orthogonality and rotation can be and are defined in the same way that they are in real inner product spaces. Orthogonal transformations are those preserving the bilinear form, and rotations are the subset of those with determinant 1, reflections are those with determinant -1 etc.

Of course, the bilinear form corresponds to the quadratic form you mentioned. Perhaps you didn't trust that the analogy could be stretched to indefinite forms. It is still definitely useful. The quadratic form no longer suggests length ( no pun intended) but I know that while interpreting real spaces with indefinite metrics for relativity, they sometimes call the quantity an 'interval' in spacetime, rather than the length. Of course the form can sort out the timeline, lightlike, and spacelike vectors, too.

The theory of bilinear forms is very rich and approachable. I recommend Kaplansky's book Linear algebra and geometry for this, but enjoyment of the exposition may vary based on your temperament. Reading this, I found out why spaces with indefinite forms are as useful, if not more useful, compared to definite spaces.

• Oh, I forgot about the fact that rotations are defined as elements of the special orthogonal group. That way of generalization makes a lot of sense. Thanks. :) – user300497 Dec 23 '15 at 4:40