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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},
\end{displaymath} (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}.
\end{displaymath} (2)

For a Curve represented parametrically by $(f(t), g(t))$,
$\displaystyle \alpha$ $\textstyle =$ $\displaystyle f-{(f'^2-g'^2)g'\over f'g''-f''g'}$ (3)
$\displaystyle \beta$ $\textstyle =$ $\displaystyle g+{(f'^2-g'^2)f'\over f'g''-f''g'}.$ (4)


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

© 1996-9 Eric W. Weisstein