,
where u is a variable parameter (``time''), the derivative (``velocity'')
is a tangent vector to the curve at the point 
.
The arc length s along the curve is defined by the equation 
and the unit tangent vector is
By definition, 
.
Differentiating the equation 
, we get 
, hence the vector
is normal to the curve. By definition, the curvature at the point is the length of this normal vector,
where R is the radius of curvature. Note that since 
is a unit tangent vector, 
is simply the angle by which the direction of the curve changes over the infinitesimal distance 
.
Example 1.
Let x,y,z
be Cartesian coordinates, 
let u=x, and introduce the notation
Then
In the special case of a plane curve with 
we get
Example 2. For a charged particle in a magnetic field the radius of curvature of the track is proportional to the momentum component perpendicular to the field.