Radius of curvature (mathematics)
If the curve is given in Cartesian coordinates as y(x), then the radius of curvature is (assuming the curve is differentiable up to order 2):
and | z | denotes the absolute value of z.
If the curve is given parametrically by functions x(t) and y(t), then the radius of curvature is
Heuristically, this result can be interpreted as
Semicircles and circles
For a semi-circle of radius a in the upper half-plane
For a semi-circle of radius a in the lower half-plane
The circle of radius a has a radius of curvature equal to a.
In an ellipse with major axis 2a and minor axis 2b, the vertices on the major axis have the smallest radius of curvature of any points , and the vertices on the minor axis have the largest radius of curvature of any points .
- do Carmo, Manfredo (1976). Differential Geometry of Curves and Surfaces. ISBN 0-13-212589-7.