The total absolute curvature of a two-dimensional surface 
is the integral
 |
where 
is its Gaussian curvature at a point 
and 
is its area element (White 2016, p. 15). It
is nonnegative and may be infinite.
The signed total curvature is instead 
. When 
is absolutely integrable, the triangle
inequality for integrals gives 
. For a minimal
surface in three-dimensional Euclidean space,
, so its total absolute curvature is the negative
of its signed total curvature. This identity includes
infinite total absolute curvature and signed total
curvature 
.
For a regular, twice continuously differentiable space curve 
parametrized by arc length 
, the total absolute curvature, often simply called
total curvature, is
 |
where 
is the nonnegative curvature (Milnor 1953). For an
oriented plane curve with signed curvature 
, the same quantity is 
. This integral differs from the third-curvature
convention involving 
,
where 
is the torsion and these classical quantities are defined.
