A natural equation is an equation which specifies a curve independent of any choice of coordinates or parameterization. The study of natural equations began with the following problem: given two functions of one parameter, find the space curve for which the functions are the curvature and torsion.
Euler gave an integral solution for plane curves (which always have torsion 
). Call the angle
between the tangent line to the curve and the x-axis 
the tangential
angle, then
 |
(1)
|
where 
is the curvature. Then the equations
 |  |  |
(2)
|
 |  |  |
(3)
|
where 
is the torsion, are solved by the curve with parametric
equations
 |  |  |
(4)
|
 |  |  |
(5)
|
The equations 
and 
are called the natural (or intrinsic) equations of the space curve. An equation expressing
a plane curve in terms of 
and radius of curvature 
(or 
) is called a Cesàro
equation, and an equation expressing a plane curve in terms of 
and 
is called a Whewell equation.
The natural parametric equations of
a curve parametrize it in terms of arc length instead
of an arbitrary parameter such as 
.
Among the special planar cases which can be solved in terms of elementary functions are the circle, logarithmic spiral, circle involute, and epicycloid. Enneper showed that each of these is the projection of a helix on a conic surface of revolution along the axis of symmetry. The above cases correspond to the cylinder, cone, paraboloid, and sphere.
