Essential Concepts
- The arc-length function for a vector-valued function is calculated using the integral formula [latex]s(t)=\displaystyle\int_{a}^{t} \parallel{\bf{r}}^{\prime}(u)\parallel{du}[/latex]. 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 [latex]t[/latex] is defined to be [latex]{\bf{N}}(t)=\dfrac{{\bf{T}}^{\prime}(t)}{\parallel{\bf{T}}^{\prime}(t)\parallel}[/latex].
- The binormal vector at [latex]t[/latex] is defined as [latex]{\bf{B}}(t)={\bf{T}}(t)\times{\bf{N}}(t)[/latex], where [latex]{\bf{T}}(t)[/latex] 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
[latex]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}[/latex] - Arc-length function
[latex]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[/latex] or [latex]s(t)=\displaystyle\int_{a}^{t} \parallel{\bf{r}}^{\prime}(u)\parallel{du}[/latex] - Curvature
[latex]\kappa=\dfrac{\parallel{\bf{T}}^{\prime}(t)\parallel}{\parallel{\bf{r}}^{\prime}(t)\parallel}[/latex] or [latex]\kappa=\dfrac{\parallel{\bf{r}}^{\prime}(t)\times{\bf{r}}^{\prime\prime}(t)\parallel}{\parallel{\bf{r}}^{\prime}(t)\parallel^{3}}[/latex] or [latex]\kappa=\dfrac{|y^{\sigma}|}{[1+(y^{\prime})^{2}]^{3{/}2}}[/latex] - Principal unit normal vector
[latex]{\bf{N}}(t)=\dfrac{{\bf{T}}^{\prime}(t)}{\parallel{\bf{T}}^{\prime}(t)\parallel}[/latex]
- Binormal vector
[latex]{\bf{B}}(t)={\bf{T}}(t)\times{\bf{N}}(t)[/latex]
Glossary
- arc-length function
- a function [latex]s(t)[/latex] that describes the arc length of curve [latex]C[/latex] as a function of [latex]t[/latex]
- 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 [latex]C[/latex] at a point [latex]P[/latex] 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 [latex]\frac{{\bf{T}}^{\prime}(t)}{\parallel{\bf{T}}^{\prime}(t)\parallel}[/latex]
- radius of curvature
- the reciprocal of the curvature
- smooth
- curves where the vector-valued function [latex]{\bf{r}}(t)[/latex] is differentiable with a non-zero derivative