Summary of Arc Length and Curvature

The arc-length function for a vector-valued function is calculated using the integral formula $s(t)=\displaystyle\int_{a}^{t} \parallel{\bf{r}}^{\prime}(u)\parallel{du}$ . This formula is valid in both two and three dimensions.
The curvature of a curve at a point in either two or three dimensions is defined to be the curvature of the inscribed circle at that point. The arc-length parameterization is used in the definition of curvature.
There are several different formulas for curvature. The curvature of a circle is equal to the reciprocal of its radius.
The principal unit normal vector at $t$ is defined to be ${\bf{N}}(t)=\dfrac{{\bf{T}}^{\prime}(t)}{\parallel{\bf{T}}^{\prime}(t)\parallel}$ .
The binormal vector at $t$ is defined as ${\bf{B}}(t)={\bf{T}}(t)\times{\bf{N}}(t)$ , where ${\bf{T}}(t)$ is the unit tangent vector.
The Frenet frame of reference is formed by the unit tangent vector, the principal unit normal vector, and the binormal vector.
The osculating circle is tangent to a curve at a point and has the same curvature as the tangent curve at that point.

Key Equations

Arc length of space curve
$s={\displaystyle\int_{a}^{b}} \sqrt{\left[f^{\prime}(t)\right]^{2}+\left[g^{\prime}(t)\right]^{2}+\left[h^{\prime}(t)\right]^{2}} dt =\displaystyle\int_{a}^{b} \parallel{\bf{r}}^{\prime}(t)\parallel{dt}$
Arc-length function
$s(t)={\displaystyle\int_{a}^{t}} \sqrt{\left(f^{\prime}(u)\right)^{2}+\left(g^{\prime}(u)\right)^{2}+\left(h^{\prime}(u)\right)^{2}} du$ or $s(t)=\displaystyle\int_{a}^{t} \parallel{\bf{r}}^{\prime}(u)\parallel{du}$
Curvature
$\kappa=\dfrac{\parallel{\bf{T}}^{\prime}(t)\parallel}{\parallel{\bf{r}}^{\prime}(t)\parallel}$ or $\kappa=\dfrac{\parallel{\bf{r}}^{\prime}(t)\times{\bf{r}}^{\prime\prime}(t)\parallel}{\parallel{\bf{r}}^{\prime}(t)\parallel^{3}}$ or $\kappa=\dfrac{|y^{\sigma}|}{[1+(y^{\prime})^{2}]^{3{/}2}}$
Principal unit normal vector
${\bf{N}}(t)=\dfrac{{\bf{T}}^{\prime}(t)}{\parallel{\bf{T}}^{\prime}(t)\parallel}$

arc-length function: a function $s(t)$ that describes the arc length of curve $C$ as a function of $t$

arc-length parameterization: a reparameterization of a vector-valued function in which the parameter is equal to the arc length

binormal vector: a unit vector orthogonal to the unit tangent vector and the unit normal vector

curvature: the derivative of the unit tangent vector with respect to the arc-length parameter

Frenet frame of reference: (TNB frame) a frame of reference in three-dimensional space formed by the unit tangent vector, the unit normal vector, and the binormal vector

normal plane: a plane that is perpendicular to a curve at any point on the curve

osculating circle: a circle that is tangent to a curve $C$ at a point $P$ and that shares the same curvature

osculating plane: the plane determined by the unit tangent and the unit normal vector

principal unit normal vector: a vector orthogonal to the unit tangent vector, given by the formula $\frac{{\bf{T}}^{\prime}(t)}{\parallel{\bf{T}}^{\prime}(t)\parallel}$

smooth: curves where the vector-valued function ${\bf{r}}(t)$ is differentiable with a non-zero derivative