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The sphere in real analysis which is known as the [[Riemann sphere|Riemann sphere]] in the theory of functions of a complex variable.

Let  $  \Sigma : X  ^ {2} + Y  ^ {2} + Z  ^ {2} = 1 $
be the unit sphere in the Euclidean  $  (X, Y, Z) $-
space, and let  $  N(0, 0, 1) $
and  $  S(0, 0, -1) $
be its north and south pole, respectively; let  $  \nu $
and  $  \sigma $
be planes tangent to  $  \Sigma $
at the points  $  N $
and  $  S $
respectively; let  $  xSy $
and  $  uNv $
be coordinate systems in  $  \sigma $
and  $  \nu $
with axes parallel to the corresponding axes of the system  $  XOY $
in the plane  $  Z = 0 $
and pointing in the same directions; let  $  \Pi $
be the [[Stereographic projection|stereographic projection]] of  $  \Sigma $
onto  $  \sigma $
from the centre  $  N $, 
and let  $  \Pi ^ { \prime } $
be the stereographic projection of  $  \Sigma $
onto  $  \nu $
from the centre  $  S $. 
Then  $  \Sigma $
is the Bendixson sphere with respect to any one of the planes  $  \sigma $, 
$  \nu $. 
It generates the bijection  $  \phi = \Pi ^ { \prime } \Pi  ^ {-1} $
of the plane  $  \sigma $(
punctured at the point  $  S $) 
onto the plane  $  \nu $, 
which is punctured at the point  $  N $. 
This bijection is employed in the study of the behaviour of the trajectories of an autonomous system of real algebraic ordinary differential equations of the second order in a neighbourhood of infinity in the phase plane (the right-hand sides of the equations are polynomials of the unknown functions). This bijection reduces the problem to a similar problem in a neighbourhood of the point  $  (0, 0) $. 
Named after I. Bendixson.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.A. Andronov,   E.A. Leontovich,   I.I. Gordon,   A.G. Maier,   "Qualitative theory of second-order dynamic systems" , Wiley  (1973)  (Translated from Russian)</TD></TR></table>