pseudosphere - Wolfram|Alpha : Cartesian equation, plot, ...

pseudosphere: Cartesian equation, plot, algebraic degree, classes, nodes, ...Input: pseudosphere

pseudosphere (surface)

Example plot

Alternate names

antisphere | tractricoid | tractrisoid | tractroid

Properties

parametric equations | x(u, v) = a sech(u) cos(v)
y(u, v) = a sech(u) sin(v)
z(u, v) = a (u-tanh(u))
Cartesian equation | z^2 = (a sech^(-1)(sqrt(x^2+y^2)/a)-sqrt(a^2-x^2-y^2))^2
classes | constant (Gaussian) curvature surface | surface of revolution
surface area | 4 a^2 pi
centroid of solid | (0, 0, 0)
volume of solid | (2 a^3 pi)/3
Gaussian curvature | -1/a^2
mean curvature | (sinh(u)-csch(u))/(2 a)
principal curvatures | ((sinh(u))/a, -(csch(u))/a)

Metric properties

metric tensor | g_(11) = a^2 tanh^2(u)
g_(12) = 0
g_(21) = 0
g_(22) = a^2 sech^2(u)
Christoffel symbol of the second kind | Gamma | 1 |
| | 22 = 1/a
Gamma | 2 |
| | 12 = -1/a
coefficients of the first fundamental form | E(u, v) = a^2 tanh^2(u)
F(u, v) = 0
G(u, v) = a^2 sech^2(u)
coefficients of the second fundamental form | e(u, v) = -a tanh(u) sech(u)
f(u, v) = 0
g(u, v) = a tanh(u) sech(u)