Tractricoid Pseudosphere

the term used refer surface called tractricoid: result of revolving tractrix asymptote. example, (half) pseudosphere (with radius 1) surface of revolution of tractrix parametrized by

t
↦

(
t
−
tanh
⁡

t

,
sech


t

)

,


0
≤
t
<
∞
.


{\displaystyle t\mapsto \left(t-\tanh {t},\operatorname {sech} \,{t}\right),\quad \quad 0\leq t<\infty .}

it singular space (the equator singularity), away singularities, has constant negative gaussian curvature , therefore locally isometric hyperbolic plane.

the name pseudosphere comes because two-dimensional surface of constant negative curvature sphere positive gauss curvature. sphere has @ every point positively curved geometry of dome whole pseudosphere has @ every point negatively curved geometry of saddle.

as 1693 christiaan huygens found volume , surface area of pseudosphere finite, despite infinite extent of shape along axis of rotation. given edge radius r, area 4πr sphere, while volume 2/3πr , therefore half of sphere of radius.