antisphere | tractricoid | tractrisoid | tractroid

x(u, v) = a sech(u) cos(v) y(u, v) = a sech(u) sin(v) z(u, v) = a (u - tanh(u))

z^2 = (a sech^(-1)(sqrt(x^2 + y^2)/a) - sqrt(a^2 - x^2 - y^2))^2

S = 4 π a^2

ds^2 = a^2 tanh^2(u) du^2 + a^2 sech^2(u) dv^2

dA = a^2 tanh(u) sech(u) du dv

x^_ = (0, 0, 0)

V = (2 π a^3)/3

K(u, v) = -1/a^2

g_(uu) = a^2 tanh^2(u) g_(vv) = a^2 sech^2(u)

Γ | u | | | uu = csch(u) sech(u) Γ | u | | | vv = csch(u) sech(u) Γ | v | | | uv = -tanh(u) Γ | v | | | vu = -tanh(u)

E(u, v) = a^2 tanh^2(u) F(u, v) = 0 G(u, v) = a^2 sech^2(u)

e(u, v) = a tanh(u) (-sech(u)) f(u, v) = 0 g(u, v) = a tanh(u) sech(u)

left double bracketing bar x(u, v) right double bracketing bar = a sqrt(u^2 - 2 u tanh(u) + 1)

N^^(u, v) = (abs(tanh(u)) cos(v), abs(tanh(u)) sin(v), sech(u) sgn(tanh(u)))

constant (Gaussian) curvature surfaces | surfaces of revolution