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