In geometry, the antipode of a point 
is its antipodal point 
.
The term antipode is also used in plane geometry. Given a central conic section (or circle) and a point 
lying on it, draw a line
passing through 
and the center 
of the conic section. Then
the antipode 
of 
is the other point
lying on the conic section through which the line 
passes.
The antipode of a vertex 
in a graph is a vertex 
at greatest possible distance
from 
. An antipodal
graph is then defined as a connected graph
in which each vertex has exactly one antipode (Gorovoy
and Zmiaikou 2021).
In a Hopf algebra, the antipode is the linear transformation 
satisfying 
,
where 
, 
, 
,
and 
are the multiplication,
coproduct, unit, and counit. It generalizes group
inversion and is part of the structure of a quantum
group.
