info prev up next book cdrom email home

Hyperboloid Embedding

A 4-Hyperboloid has Negative Curvature, with

\begin{displaymath}
R^2 = x^2+y^2+z^2-w^2
\end{displaymath} (1)


\begin{displaymath}
2x{dx\over dw} + 2y{dy\over dw} + 2z{dz\over dw} - 2w = 0.
\end{displaymath} (2)

Since
\begin{displaymath}
{\bf r} \equiv x\hat{\bf x} + y\hat{\bf y} + z\hat{\bf z},
\end{displaymath} (3)


\begin{displaymath}
dw = {x\,dx + y\,dy + z\,dz\over w} = {{\bf r}\cdot d{\bf r}\over\sqrt{r^2-R^2}}.
\end{displaymath} (4)

To stay on the surface of the Hyperboloid,
$\displaystyle ds^2$ $\textstyle =$ $\displaystyle dx^2+dy^2+dz^2-dw^2$  
  $\textstyle =$ $\displaystyle dx^2+dy^2+dz^2 - {r^2\,dr^2\over r^2-R^2}$  
  $\textstyle =$ $\displaystyle dr^2+r^2d\Omega^2 + {dr^2\over 1 - {R^2\over r^2}}.$ (5)




© 1996-9 Eric W. Weisstein
1999-05-25