Hello, Ideasman!

You're expected to know: .cosh²(x) - sinh²(x) .= .1

We will eliminate the parameters . . .Identify the following parametric surface.

. . [1] .x .= .sinh(v)

. . [2] .y .= .cosh(u)·cosh(v)

. . [3] .z .= .sinh(u)·cosh(v)

Square [2]: .y² .= .cosh²(u)·cosh²(v)

Square [3]: .z² .= .sinh²(u)·cosh²(v)

Subtract: .y² - z² .= .cos²(u)·cos²(v) - sinh²(u)·cosh²(v)

. . . . . . . . . . . . . = .[cosh²(u) - sin²(u)]·cosh²(v)

. . . . . . . . . . . . . = .cosh²(v)

We have: .y² - z² .= .cosh²(v) .= .sinh²(v) + 1 .[4]

Square [1]: .x² .= .sinh²(v)

Substitute into [4]: .y² - z² .= .x² + 1

We have: . y² - x² - z² .= .1 . . . a hyperboloid of two sheet.