Intrinsic equation

In geometry, an intrinsic equation of a curve is an equation that defines the curve using a relation between the curve's intrinsic properties, that is, properties that do not depend on the location and possibly the orientation of the curve. Therefore an intrinsic equation defines the shape of the curve without specifying its position relative to an arbitrarily defined coordinate system. 
The intrinsic quantities used most often are arc length formula_1, tangential angle formula_2, curvature formula_3 or radius of curvature, and, for 3-dimensional curves, torsion formula_4. Specifically:
The equation of a circle (including a line) for example is given by the equation formula_5 where formula_6 is the arc length, formula_3 the curvature and formula_8 the radius of the circle.
These coordinates greatly simplify some physical problem. For elastic rods for example, the potential energy is given by 
where formula_10 is the bending modulus formula_11. Moreover, as formula_12, elasticity of rods can be given a simple variational form.