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MONKEY SADDLE
Cylindrical equation: .
Cartesian parametrization: . Cartesian equation: . Cubic surface. 
Surface shaped like a saddle that allows to place a monkey's legs but also its tail (the monkey is not the mount, but the rider!).
By the point O (which is a planar
point of the surface) there pass 3 real lines of the surface, forming
between one another angles of 120° (in red above): the point O
is an Eckardt point of the surface;
by the point at infinity of Oz, which is a singular point of the
surface, there pass the 3 other lines of the surface, that are also real,
but at infinity.
O is also an umbilic.
Generalization: surface with a saddle where end n
valleys and n mountains:
(hence a surface with
rotational symmetry).
The case n = 2 gives the horse saddle or hyperbolic paraboloid and the case n = 3 the monkey saddle. Proposition of name by Paul Micaelli: octopus saddle... 

Other surface with 3 valleys, 3 mountains: ; but the saddle is not comfortable, there is no tangent plane at O (it is a cone, cf. Cartan's umbrella).  
Surface with 4 mountains, 4 valleys, flat this time : ; named "crossed trough". 

Compare to the Enneper
surface.
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© Robert FERRÉOL 2017