The signed curvature formula for a planar parametric curve is given in
(2.25). Due to the regularity condition, a
necessary condition to determine inflection points is
(8.8)
An inflection point on a planar curve is shown in Fig.
8.7, marked by
.
The curvature
of a space curve is given in
(2.26). The formula can be expressed as
(8.9)
Since we are assuming a regular curve, a condition to determine a
point on a curve, where the curvature
vanishes, is [57]
(8.10)
or
(8.11)
Curvature vanishing points on a space curve
are shown in Fig.
8.8, marked by
.
.
of a space curve is given in
(2.26). The formula can be expressed as
are shown in Fig.
8.8, marked by ![$\displaystyle \dot{x}(t)\ddot{y}(t) - \dot{y}(t)\ddot{x}(t) = 0, \quad \quad t\in[t_1,t_2]\;.$](img2482.gif){kind=small}
![$\displaystyle K_{0}(t) {\bf\equiv} (\dot{x}\ddot{y} - \dot{y}\ddot{x})^...
...{y})^2 + (\dot{z}\ddot{x} - \dot{x}\ddot{z})^2
=0, \quad \quad t\in[t_1,t_2]\;,$](img2485.gif){kind=small}
![$\displaystyle \dot{x}\ddot{y} - \dot{y}\ddot{x}=\dot{y}\ddot{z} - \dot{z}\ddot{y}= \dot{z}\ddot{x} - \dot{x}\ddot{z}=0, \quad \quad t\in[t_1,t_2]\;.$](img2486.gif){kind=small}