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Hyperboloid Embedding

A 4-hyperboloid has negative curvature, with

R^2=x^2+y^2+z^2-w^2
(1)
2x(dx)/(dw)+2y(dy)/(dw)+2z(dz)/(dw)-2w=0.
(2)

Since

r=xx^^+yy^^+zz^^,
(3)

it follows that

dw=(xdx+ydy+zdz)/w=(r·dr)/(sqrt(r^2-R^2)).
(4)

To stay on the surface of the hyperboloid, the line element is given by

ds^2=dx^2+dy^2+dz^2-dw^2
(5)
=dx^2+dy^2+dz^2-(r^2dr^2)/(r^2-R^2)
(6)
=dr^2+r^2dOmega^2+(dr^2)/(1-(R^2)/(r^2)).
(7)

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Cite this as:

Weisstein, Eric W. "Hyperboloid Embedding." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HyperboloidEmbedding.html

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