Math Art
Trihyperboloid

Trihyperboloid

Trihyperboloid is a surface, given by a parametrisation, immersed.

Open Trihyperboloid in the interactive viewer →

aperiodic def-parametric immersed implemented misc

Formula

x(u, v)
sin⁡(u)⁢cos⁡(v)max⁡(max⁡(sin⁡(u)2−cos⁡(u)2,sin⁡(u)2⁢sin⁡(v)2+cos⁡(u)2−sin⁡(u)2⁢cos⁡(v)2),cos⁡(u)2+sin⁡(u)2⁢cos⁡(v)2−sin⁡(u)2⁢sin⁡(v)2)\frac{\sin\left(u\right) \cos\left(v\right)}{\sqrt{\max\left(\max\left(\sin\left(u\right)^{2} - \cos\left(u\right)^{2}, \sin\left(u\right)^{2} \sin\left(v\right)^{2} + \cos\left(u\right)^{2} - \sin\left(u\right)^{2} \cos\left(v\right)^{2}\right), \cos\left(u\right)^{2} + \sin\left(u\right)^{2} \cos\left(v\right)^{2} - \sin\left(u\right)^{2} \sin\left(v\right)^{2}\right)}}
y(u, v)
sin⁡(u)⁢sin⁡(v)max⁡(max⁡(sin⁡(u)2−cos⁡(u)2,sin⁡(u)2⁢sin⁡(v)2+cos⁡(u)2−sin⁡(u)2⁢cos⁡(v)2),cos⁡(u)2+sin⁡(u)2⁢cos⁡(v)2−sin⁡(u)2⁢sin⁡(v)2)\frac{\sin\left(u\right) \sin\left(v\right)}{\sqrt{\max\left(\max\left(\sin\left(u\right)^{2} - \cos\left(u\right)^{2}, \sin\left(u\right)^{2} \sin\left(v\right)^{2} + \cos\left(u\right)^{2} - \sin\left(u\right)^{2} \cos\left(v\right)^{2}\right), \cos\left(u\right)^{2} + \sin\left(u\right)^{2} \cos\left(v\right)^{2} - \sin\left(u\right)^{2} \sin\left(v\right)^{2}\right)}}
z(u, v)
cos⁡(u)max⁡(max⁡(sin⁡(u)2−cos⁡(u)2,sin⁡(u)2⁢sin⁡(v)2+cos⁡(u)2−sin⁡(u)2⁢cos⁡(v)2),cos⁡(u)2+sin⁡(u)2⁢cos⁡(v)2−sin⁡(u)2⁢sin⁡(v)2)\frac{\cos\left(u\right)}{\sqrt{\max\left(\max\left(\sin\left(u\right)^{2} - \cos\left(u\right)^{2}, \sin\left(u\right)^{2} \sin\left(v\right)^{2} + \cos\left(u\right)^{2} - \sin\left(u\right)^{2} \cos\left(v\right)^{2}\right), \cos\left(u\right)^{2} + \sin\left(u\right)^{2} \cos\left(v\right)^{2} - \sin\left(u\right)^{2} \sin\left(v\right)^{2}\right)}}

Properties

Family
misc
Given by
a parametrisation
Curvature
no condition imposed
Periodicity
not periodic
Ends
0
Embedding
immersed
Fidelity
exact
Exactness
elementary

Definition

the radial graph r = 1/sqrt(max of the three hyperboloid forms) along the unit direction (sin u cos v, sin u sin v, cos u); the max of the three is >= 1/3, so 1 <= r <= sqrt 3 everywhere

Sources

Chart reproduced numerically against the shipped implementation: fwd max 0.30 of tol, 0.0% of oracle beyond tol (tol floor 2.5 x 0.05605).