Parametric representations Hyperboloid

-


animation of hyperboloid of revolution


cartesian coordinates hyperboloids can defined, similar spherical coordinates, keeping azimuth angle θ ∈ [0, 2π), changing inclination v hyperbolic trigonometric functions:


one-surface hyperboloid: v ∈ (−∞, ∞)











x



=
a
cosh
⁡
v
cos
⁡
θ




y



=
b
cosh
⁡
v
sin
⁡
θ




z



=
c
sinh
⁡
v






{\displaystyle {\begin{aligned}x&=a\cosh v\cos \theta \\y&=b\cosh v\sin \theta \\z&=c\sinh v\end{aligned}}}



two-surface hyperboloid: v ∈ [0, ∞)











x



=
a
sinh
⁡
v
cos
⁡
θ




y



=
b
sinh
⁡
v
sin
⁡
θ




z



=
±
c
cosh
⁡
v






{\displaystyle {\begin{aligned}x&=a\sinh v\cos \theta \\y&=b\sinh v\sin \theta \\z&=\pm c\cosh v\end{aligned}}}




hyperboloid of 1 sheet: generation rotating hyperbola (top) , line (bottom: red or blue)



hyperboloid of 1 sheet: plane sections







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