Curvature

The characterization of the curvature in terms of the derivative of the unit tangent vector is probably less intuitive than the definition in terms of the osculating circle, but formulas for computing the curvature are easier to deduce. Therefore, and also because of its use in kinematics, this characterization is often given as a definition of the curvature.

This definition is difficult to manipulate and to express in formulas. Therefore, other equivalent definitions have been introduced.

and the center of curvature is on the normal to the curve, the center of curvature is the point

As the first and second derivatives of x are 1 and 0, previous formulas simplify to

In the general case of a curve, the sign of the signed curvature is somewhat arbitrary, as it depends on the orientation of the curve. In the case of the graph of a function, there is a natural orientation by increasing values of x. This makes significant the sign of the signed curvature.

This results from the formula for general parametrizations, by considering the parametrization

Above formula for the curvature can be derived from the expression of the curvature of the graph of a function by using the implicit function theorem and the fact that, on such a curve, one has

It can be useful to verify on simple examples that the different formulas given in the preceding sections give the same result.

It follows, as expected, that the radius of curvature is the radius of the circle, and that the center of curvature is the center of the circle.

The circle is a rare case where the arc-length parametrization is easy to compute, as it is

As planar curves have zero torsion, the second Frenet–Serret formula provides the relation

where the prime denotes differentiation with respect to the parameter t. This can be expressed independently of the coordinate system by means of the formula

where × denotes the vector cross product. This last formula is valid for the curvature of curves in a Euclidean space of any dimension:

The curvature of curves drawn on a surface is the main tool for the defining and studying the curvature of the surface.

Because (Gaussian) curvature can be defined without reference to an embedding space, it is not necessary that a surface be embedded in a higher-dimensional space in order to be curved. Such an intrinsically curved two-dimensional surface is a simple example of a Riemannian manifold.

Thus the second fundamental form encodes both the intrinsic and extrinsic curvatures.