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Section 1.5 A Compendium of Curve Formula
In the following \(\vr(t)=\big(x(t)\,,\,y(t)\,,\,z(t)\big)\) is a parametrization of some curve. The vectors \(\hat\vT(t),\ \hat\vN(t),\ \) and \(\ \hat\vB(t)\ \) are the unit tangent, normal and binormal vectors, respectively, at \(\vr(t)\text{.}\) The tangent vector points in the direction of travel (i.e. direction of increasing \(t\) ) and the normal vector points toward the centre of curvature. The arc length from time \(0\) to time \(t\) is denoted \(s(t)\text{.}\) The binormal \(\ \hat\vB(t)=\hat\vT(t)\times \hat\vN(t)\ \) is perpendicular to the plane that fits the curve best at \(\vr(t)\text{.}\) Some formulae use an arc length parametrization, which is denoted \(\vr(s)\text{.}\)
the velocity
\(\displaystyle \vv(t)=\diff{\vr}{t}(t)=\diff{s}{t}(t)\,\hat\vT(t)\)
the unit tangent vector
\(\hat\vT(t)=\frac{\vv(t)}{|\vv(t)|}\) (general parametrization)
\(\hat\vT(s)=\diff{\vr}{s}(s)\) (arc length parametrization)
the acceleration
\(\displaystyle \va(t)=\difftwo{\vr}{t}(t)=\difftwo{s}{t}(t)\,\hat\vT(t)
+\ka(t)\big(\diff{s}{t}(t)\big)^2\hat\vN(t)\)
the speed
\(\displaystyle \diff{s}{t}(t) = |\vv(t)| = \big|\diff{\vr}{t}(t)\big|\)
the arc length
\(\displaystyle s(T) = \int_0^T\! \diff{s}{t}(t)\,\dee{t}
= \int_0^T\! \sqrt{x'(t)^2\!+\!y'(t)^2\!+\!z'(t)^2}\,\dee{t}\)
the curvature
\(\ka(t)
= \big|\diff{\hat\vT}{t}(t)\big|/\diff{s}{t}(t)
=\displaystyle{ \frac{|\vv(t)\times\va(t)|}{(\diff{s}{t}(t))^3} }\)
\(\ka(s)
= \big|\diff{\phi}{s}(s)\big|
= \big|\diff{\hat\vT}{s}(s)\big|\)
the unit normal vector
\(\displaystyle \hat\vN(t) = \diff{\hat\vT}{t}(t)/\big|\diff{\hat\vT}{t}(t)\big|
\qquad
\hat\vN(s) = \diff{\hat\vT}{s}(s)/\ka(s)\)
the radius of curvature
\(\displaystyle \rho(t)=\frac{1}{\ka(t)}\)
the centre of curvature
\(\displaystyle \vr(t)+\rho(t)\hat\vN(t)\)
the torsion
\(\displaystyle \displaystyle
\tau(t)=\frac{\big(\vv(t)\times\va(t)\big) \cdot \diff{\va}{t}(t)}
{|\vv(t)\times\va(t)|^2}\)
the binormal
\(\displaystyle \displaystyle
\hat\vB(t)=\hat\vT(t)\times \hat\vN(t)=\frac{\vv(t)\times\va(t)}{|\vv(t)\times\va(t)|}\)
Under arclength parametrization (i.e. if \(t=s\) ) we have \(\hat\vT(s)=\frac{d\vr}{ds}(s)\) and the Frenet-Serret formulae
\begin{align*}
\diff{\hat\vT}{s}(s)&=\phantom{-}\ka(s)\ \hat\vN(s)\cr
\diff{\hat\vN}{s}(s)&=\phantom{-}\tau(s)\ \hat\vB(s)-\ka(s)\ \hat\vT(s)\cr
\diff{\hat\vB}{s}(s)&=-\tau(s)\ \hat\vN(s)\cr
\end{align*}
which in matrix form is
\begin{align*}
\diff{}{s}
\left[ \begin{matrix}\hat\vT(s) \\ \hat\vN(s)\\ \hat\vB(s)\end{matrix} \right]
=\left[\begin{matrix} 0 & \ka(s) & 0 \\
-\ka(s) & 0 &\tau(s) \\
0 &-\tau(s) & 0 \end{matrix}\right]
\left[\begin{matrix}\hat\vT(s) \\ \hat\vN(s)\\ \hat\vB(s)\end{matrix}\right]
\end{align*}
When the curve lies entirely in the \(xy\) -plane the curvature is given by
\begin{gather*}
\ka(t)
=\frac{\big|
\diff{x}{t}(t)\ \difftwo{y}{t}(t)-\diff{y}{t}(t)\ \difftwo{x}{t}(t)
\big|}{\Big[\big(\diff{x}{t}(t)\big)^2
+\big(\diff{y}{t}(t)\big)^2\Big]^{3/2}}
\end{gather*}
When the curve lies entirely in the \(xy\) -plane and the \(y\) -coordinate is given as a function, \(y(x)\text{,}\) of the \(x\) -coordinate, the curvature is
\begin{gather*}
\ka(x)
=\frac{\big|\difftwo{y}{x}(x)\big|}
{\Big[1+\big(\diff{y}{x}(x)\big)^2\Big]^{3/2}}
\end{gather*}
Notice that this follows from the previous formula since \(\diff{x}{x}=1\) and \(\difftwo{x}{x}=0\text{.}\)
