0

It is known that a point $(x,y)$ on the circle $x^2+y^2=1$ can be expressed as $(\sin(\theta), \cos(\theta))$, where $\theta$ is the angle between the points $(1,0)$, $(0,0)$ and $(x,y)$. It is also equal to the area of the circular sector and the length of the arc inscribed by the angle.

Analogously, a point $(x,y)$ on the hyperbola $x^2-y^2=1$ can be expressed as $(\cosh(\alpha), \sinh(\alpha))$.

Can someone explain geometrically what is $\alpha$ in terms of angles, areas and arclengthes, and in general the relation between these three in hyperbolas, similar to how it is with circles?

Rócherz
  • 4,241
math guy
  • 44
  • 1
    See, for instance, my answer to "Alternative definition of hyperbolic cosine without relying on exponential function". – Blue Oct 06 '22 at 17:52
  • Ok now I understand how it is related to areas, but I still can not seem to find a clear relation to angles (just like circular trig is usually defined in terms of angles and not areas) – math guy Oct 06 '22 at 18:00
  • The key is to think of circular radians as twice the area of the unit-circular sector corresponding to an angle. (Eg, the full circle has area $\pi$. A right-angle sector fills a quarter of it, for an area of $\pi/4$; doubling that gives $\pi/2$ as the radian measure of the angle.) This matches how hyperbolic radians are twice the area of the unit-hyperbolic sector corresponding to an angle. – Blue Oct 06 '22 at 18:07
  • I think I understand what you mean, but what are hyperbolic radians exactly? (in terms of degrees/circular radians) – math guy Oct 06 '22 at 18:19
  • Hyperbolic radians are just ... whatever the corresponding double-areas say they are. A $0^\circ$ angle has measure $0$ in both circular and hyperbolic radians; a $45^\circ$ angle has measure $\pi/4$ in circular radians and measure $\infty$ in hyperbolic radians; in-between angles are a bit messy. The Gudermannian function converts between circular and hyperbolic radians; the linked Wikipedia article provides context that may be helpful. – Blue Oct 06 '22 at 19:26

0 Answers0