Curvature and torsion ($\kappa ,\tau$)

For unit speed, $\kappa =|T^{\prime }|$ and $\tau =-B^{\prime }\cdot N$. Generally $\kappa =|r^{\prime }\times r^{\prime \prime }|/|r^{\prime }{|}^3$ and $\tau =\det (r^{\prime },r^{\prime \prime },r^{\prime \prime \prime })/|r^{\prime }\times r^{\prime \prime }{|}^2$.

For a unit-speed curve $\gamma$, the unit tangent is $T={\gamma }^{\prime }$ and the curvature is $\kappa =|T^{\prime }|$. It measures the rate at which the tangent direction turns.

For a regular plane curve $(x(u),y(u))$,For a graph $(u,f(u))$, this reduces to $|f^{\prime \prime }(u)|/(1+f^{\prime }(u{)}^2{)}^{3/2}$.

At a regular planar level set of a twice differentiable function, its oriented curvature with normal $\nabla u/|\nabla u|$ is $\kappa =(\Delta u-\nabla u^T(D^{2}u)\nabla u/|\nabla u{|}^2)/|\nabla u|$. Zero curvature at a point gives second-order contact with the tangent line, not a whole straight segment. For example the regular zero level line of $u(x,y)=y+x^3$ has zero curvature at the origin but is curved nearby.

Given a smooth function $\kappa (s)$, setThen $\gamma$ is unit speed and has signed curvature $\kappa$. This gives the existence part of the fundamental theorem of plane curves.

The total curvature of a regular curve isIt equals the length traced by the unit tangent on the unit sphere, so lifting a planar circle into a helix can reduce total curvature even when both curves make one revolution around the same axis.

For a unit-speed curve with nonzero curvature, torsion is $\tau =-B^{\prime }\cdot N$, measuring rotation of the osculating plane.

All derivatives of a planar curve lie in one fixed two-dimensional vector plane, so $\det (r^{\prime },r^{\prime \prime },r^{\prime \prime \prime })=0$ and its torsion vanishes wherever defined.