Curvature Center

The point on the Positive Ray of the Normal Vector at a distance $\rho(s)$, where $\rho$ is the Radius of Curvature. It is given by

$${\bf z}={\bf x}+\rho {\bf N} = {\bf x}+\rho^2 {{\bf T}\over ds},$$ (1)

where ${\bf N}$ is the Normal Vector and ${\bf T}$ is the Tangent Vector. It can be written in terms of ${\bf x}$ explicitly as

$${\bf z}={\bf x}+{{\bf x}''({\bf x}'\cdot{\bf x}')^2-{\bf x}'... ...\bf x}')({\bf x}''\cdot{\bf x}'')-({\bf x}'\cdot{\bf x}'')^2}.$$ (2)

For a Curve represented parametrically by $(f(t), g(t))$,

$$\alpha = f-{(f'^2-g'^2)g'\over f'g''-f''g'}$$ (3)

$$\beta = g+{(f'^2-g'^2)f'\over f'g''-f''g'}.$$ (4)

References

Gray, A. Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, 1993.