Intrinsic coordinates describe a plane curve using its arc length 
and tangential angle 
, rather than coordinates measured
from fixed axes. When 
, the signed curvature
is 
(Yates 1952).
Given an initial point 
, the Cartesian
coordinates can be recovered from
 |  |  |
(1)
|
 |  |  |
(2)
|
Thus the intrinsic description determines the plane curve up to a translation and rotation. Rotating the coordinate axes changes 
by a constant while leaving 
unchanged. A relation between arc
length and tangential angle, or equivalently
between arc length and curvature, is called a natural
equation, or intrinsic equation, of a plane curve. For a space curve, the natural
equations give curvature and torsion as functions of
arc length.
