Differentiable curve

Differential geometry of curves is the branch of geometry that deals with smooth curves in the plane and the Euclidean space by methods of differential and integral calculus.

Many specific curves have been thoroughly investigated using the synthetic approach. Differential geometry takes another path: curves are represented in a parametrized form, and their geometric properties and various quantities associated with them, such as the curvature and the arc length, are expressed via derivatives and integrals using vector calculus. One of the most important tools used to analyze a curve is the Frenet frame, a moving frame that provides a coordinate system at each point of the curve that is "best adapted" to the curve near that point.

The theory of curves is much simpler and narrower in scope than the theory of surfaces and its higher-dimensional generalizations because a regular curve in a Euclidean space has no intrinsic geometry. Any regular curve may be parametrized by the arc length (the natural parametrization). From the point of view of a theoretical point particle on the curve that does not know anything about the ambient space, all curves would appear the same. Different space curves are only distinguished by how they bend and twist. Quantitatively, this is measured by the differential-geometric invariants called the curvature and the torsion of a curve. The fundamental theorem of curves asserts that the knowledge of these invariants completely determines the curve.

Definitions

A parametric $C r$ -curve or a $C r$ -parametrization is a vector-valued function $\gamma :I\to \mathbb {R} ^{n}$ that is $r$ -times continuously differentiable (that is, the component functions of $γ$ are continuously differentiable), where $n\in \mathbb {N}$ , $r\in \mathbb {N} \cup \{\infty \}$ , and $I$ is a non-empty interval of real numbers. The image of the parametric curve is $\gamma [I]\subseteq \mathbb {R} ^{n}$ . The parametric curve $γ$ and its image $γ [I]$ must be distinguished because a given subset of $\mathbb {R} ^{n}$ can be the image of many distinct parametric curves. The parameter $t$ in $γ (t)$ can be thought of as representing time, and $γ$ the trajectory of a moving point in space. When $I$ is a closed interval $[a, b]$ , $γ (a)$ is called the starting point and $γ (b)$ is the endpoint of $γ$ . If the starting and the end points coincide (that is, $γ (a) = γ (b)$ ), then $γ$ is a closed curve or a loop. To be a $C r$ -loop, the function $γ$ must be $r$ -times continuously differentiable and satisfy $γ (k) (a) = γ (k) (b)$ for $0 \leq k \leq r$ .

The parametric curve is simple if $\gamma |_{(a,b)}:(a,b)\to \mathbb {R} ^{n}$ is injective. It is analytic if each component function of $γ$ is an analytic function, that is, it is of class $C ω$ .

The curve $γ$ is regular of order $m$ (where $m \leq r$ ) if, for every $t \in I$ , $\left\{\gamma '(t),\gamma ''(t),\ldots ,{\gamma ^{(m)}}(t)\right\}$ is a linearly independent subset of $\mathbb {R} ^{n}$ . In particular, a parametric $C 1$ -curve $γ$ is regular if and only if $γ' (t) \neq 0$ for any $t \in I$ .

Re-parametrization and equivalence relation

Given the image of a parametric curve, there are several different parametrizations of the parametric curve. Differential geometry aims to describe the properties of parametric curves that are invariant under certain reparametrizations. A suitable equivalence relation on the set of all parametric curves must be defined. The differential-geometric properties of a parametric curve (such as its length, its Frenet frame, and its generalized curvature) are invariant under reparametrization and therefore properties of the equivalence class itself. The equivalence classes are called $C r$ -curves and are central objects studied in the differential geometry of curves.

Two parametric $C r$ -curves, $\gamma _{1}:I_{1}\to \mathbb {R} ^{n}$ and $\gamma _{2}:I_{2}\to \mathbb {R} ^{n}$ , are said to be equivalent if and only if there exists a bijective $C r$ -map $φ : I 1 \to I 2$ such that $\forall t\in I_{1}:\quad \varphi '(t)\neq 0$ and $\forall t\in I_{1}:\quad \gamma _{2}{\bigl (}\varphi (t){\bigr )}=\gamma _{1}(t).$ $γ 2$ is then said to be a re-parametrization of $γ 1$ .

Re-parametrization defines an equivalence relation on the set of all parametric $C r$ -curves of class $C r$ . The equivalence class of this relation simply a $C r$ -curve.

An even finer equivalence relation of oriented parametric $C r$ -curves can be defined by requiring $φ$ to satisfy $φ' (t) > 0$ .

Equivalent parametric $C r$ -curves have the same image, and equivalent oriented parametric $C r$ -curves even traverse the image in the same direction.

Length and natural parametrization

The length $l$ of a parametric $C 1$ -curve $\gamma :[a,b]\to \mathbb {R} ^{n}$ is defined as $l~{\stackrel {\text{def}}{=}}~\int _{a}^{b}\left\|\gamma '(t)\right\|\,\mathrm {d} {t}.$ The length of a parametric curve is invariant under reparametrization and is therefore a differential-geometric property of the parametric curve.

For each regular parametric $C r$ -curve $\gamma :[a,b]\to \mathbb {R} ^{n}$ , where $r \geq 1$ , the function is defined $\forall t\in [a,b]:\quad s(t)~{\stackrel {\text{def}}{=}}~\int _{a}^{t}\left\|\gamma '(x)\right\|\,\mathrm {d} {x}.$ Writing $γ (s) = γ (t (s))$ , where $t (s)$ is the inverse function of $s (t)$ . This is a re-parametrization $γ$ of $γ$ that is called an arc-length parametrization, natural parametrization, unit-speed parametrization. The parameter $s (t)$ is called the natural parameter of $γ$ .

This parametrization is preferred because the natural parameter $s (t)$ traverses the image of $γ$ at unit speed, so that $\forall t\in I:\quad \left\|{\overline {\gamma }}'{\bigl (}s(t){\bigr )}\right\|=1.$ In practice, it is often very difficult to calculate the natural parametrization of a parametric curve, but it is useful for theoretical arguments.

For a given parametric curve $γ$ , the natural parametrization is unique up to a shift of parameter.

The quantity $E(\gamma )~{\stackrel {\text{def}}{=}}~{\frac {1}{2}}\int _{a}^{b}\left\|\gamma '(t)\right\|^{2}~\mathrm {d} {t}$ is sometimes called the energy or action of the curve; this name is justified because the geodesic equations are the Euler–Lagrange equations of motion for this action.

Frenet frame

An illustration of the Frenet frame for a point on a space curve.

T

is the unit tangent,

P

the unit normal, and

B

the unit binormal.

A Frenet frame is a moving reference frame of $n$ orthonormal vectors $e i (t)$ which are used to describe a curve locally at each point $γ (t)$ . It is the main tool in the differential geometric treatment of curves because it is far easier and more natural to describe local properties (e.g. curvature, torsion) in terms of a local reference system than using a global one such as Euclidean coordinates.

Given a $C n + 1$ -curve $γ$ in $\mathbb {R} ^{n}$ which is regular of order $n$ the Frenet frame for the curve is the set of orthonormal vectors $\mathbf {e} _{1}(t),\ldots ,\mathbf {e} _{n}(t)$ called Frenet vectors. They are constructed from the derivatives of $γ (t)$ using the Gram–Schmidt orthogonalization algorithm with ${\begin{aligned}\mathbf {e} _{1}(t)&={\frac {{\boldsymbol {\gamma }}'(t)}{\left\|{\boldsymbol {\gamma }}'(t)\right\|}}\\[1ex]\mathbf {e} _{j}(t)&={\frac {{\overline {\mathbf {e} _{j}}}(t)}{\left\|{\overline {\mathbf {e} _{j}}}(t)\right\|}},&{\overline {\mathbf {e} _{j}}}(t)&={\boldsymbol {\gamma }}^{(j)}(t)-\sum _{i=1}^{j-1}\left\langle {\boldsymbol {\gamma }}^{(j)}(t),\,\mathbf {e} _{i}(t)\right\rangle \,\mathbf {e} _{i}(t){\vphantom {\Bigg \langle }}\end{aligned}}$

The real-valued functions $χ i (t)$ are called generalized curvatures and are defined as $\chi _{i}(t)={\frac {{\bigl \langle }\mathbf {e} _{i}'(t),\mathbf {e} _{i+1}(t){\bigr \rangle }}{\left\|{\boldsymbol {\gamma }}^{'}(t)\right\|}}$

The Frenet frame and the generalized curvatures are invariant under reparametrization and are therefore differential geometric properties of the curve. For curves in $\mathbb {R} ^{3}$ $\chi _{1}(t)$ is the curvature and $\chi _{2}(t)$ is the torsion.

Bertrand curve

A Bertrand curve is a regular curve in $\mathbb {R} ^{3}$ with the additional property that there is a second curve in $\mathbb {R} ^{3}$ such that the principal normal vectors to these two curves are identical at each corresponding point. In other words, if $γ 1 (t)$ and $γ 2 (t)$ are two curves in $\mathbb {R} ^{3}$ such that for any $t$ , the two principal normals $N 1 (t), N 2 (t)$ are equal, then $γ 1$ and $γ 2$ are Bertrand curves, and $γ 2$ is called the Bertrand mate of $γ 1$ . We can write $γ 2 (t) = γ 1 (t) + r N 1 (t)$ for some constant $r$ .^[1]

According to problem 25 in Kühnel's "Differential Geometry Curves – Surfaces – Manifolds", it is also true that two Bertrand curves that do not lie in the same two-dimensional plane are characterized by the existence of a linear relation $a κ (t) + b τ (t) = 1$ where $κ (t)$ and $τ (t)$ are the curvature and torsion of $γ 1 (t)$ and $a$ and $b$ are real constants with $a \neq 0$ .^[2] Furthermore, the product of torsions of a Bertrand pair of curves is constant.^[3] If $γ 1$ has more than one Bertrand mate then it has infinitely many. This only occurs when $γ 1$ is a circular helix.^[1]

Special Frenet vectors and generalized curvatures

The first three Frenet vectors and generalized curvatures can be visualized in three-dimensional space. They have additional names and more semantic information attached to them.

Tangent vector

If a curve $γ$ represents the path of a particle, then the instantaneous velocity of the particle at a given point $P$ is expressed by a vector, called the tangent vector to the curve at $P$ . Mathematically, given a parametrized $C 1$ curve $γ = γ (t)$ , for every value $t = t 0$ of the parameter, the vector $\gamma '(t_{0})=\left.{\frac {\mathrm {d} }{\mathrm {d} t}}{\boldsymbol {\gamma }}(t)\right|_{t=t_{0}}$ is the tangent vector at the point $P = γ (t 0)$ . Generally speaking, the tangent vector may be zero. The tangent vector's magnitude $\left\|{\boldsymbol {\gamma }}'(t_{0})\right\|$ is the speed at the time $t 0$ .

The first Frenet vector $e 1 (t)$ is the unit tangent vector in the same direction, defined at each regular point of $γ$ : $\mathbf {e} _{1}(t)={\frac {{\boldsymbol {\gamma }}'(t)}{\left\|{\boldsymbol {\gamma }}'(t)\right\|}}.$ If $t = s$ is the natural parameter, then the tangent vector has unit length. The formula simplifies: $\mathbf {e} _{1}(s)={\boldsymbol {\gamma }}'(s).$ The unit tangent vector determines the orientation of the curve, or the forward direction, corresponding to the increasing values of the parameter. The unit tangent vector taken as a curve traces the spherical image of the original curve.

Normal vector or curvature vector

A curve normal vector, sometimes called the curvature vector, indicates the deviance of the curve from being a straight line. It is defined as ${\overline {\mathbf {e} _{2}}}(t)={\boldsymbol {\gamma }}''(t)-{\bigl \langle }{\boldsymbol {\gamma }}''(t),\mathbf {e} _{1}(t){\bigr \rangle }\,\mathbf {e} _{1}(t).$

Its normalized form, the unit normal vector, is the second Frenet vector $e 2 (t)$ and is defined as $\mathbf {e} _{2}(t)={\frac {{\overline {\mathbf {e} _{2}}}(t)}{\left\|{\overline {\mathbf {e} _{2}}}(t)\right\|}}.$

The tangent and the normal vector at point $t$ define the osculating plane at point $t$ .

It can be shown that $ē 2 (t) \propto e' 1 (t)$ . Therefore, $\mathbf {e} _{2}(t)={\frac {\mathbf {e} _{1}'(t)}{\left\|\mathbf {e} _{1}'(t)\right\|}}.$

Curvature

The first generalized curvature $χ 1 (t)$ is called curvature and measures the deviance of $γ$ from being a straight line relative to the osculating plane. It is defined as $\kappa (t)=\chi _{1}(t)={\frac {{\bigl \langle }\mathbf {e} _{1}'(t),\mathbf {e} _{2}(t){\bigr \rangle }}{\left\|{\boldsymbol {\gamma }}'(t)\right\|}}$ and is called the curvature of $γ$ at point $t$ . It can be shown that $\kappa (t)={\frac {\left\|\mathbf {e} _{1}'(t)\right\|}{\left\|{\boldsymbol {\gamma }}'(t)\right\|}}.$

The reciprocal of the curvature ${\frac {1}{\kappa (t)}}$ is called the radius of curvature.

A circle with radius $r$ has a constant curvature of $\kappa (t)={\frac {1}{r}}$ whereas a line has a curvature of 0.

Binormal vector

The unit binormal vector is the third Frenet vector $e 3 (t)$ . It is always orthogonal to the unit tangent and normal vectors at $t$ . It is defined as

$\mathbf {e} _{3}(t)={\frac {{\overline {\mathbf {e} _{3}}}(t)}{\left\|{\overline {\mathbf {e} _{3}}}(t)\right\|}},\quad {\overline {\mathbf {e} _{3}}}(t)={\boldsymbol {\gamma }}'''(t)-{\bigr \langle }{\boldsymbol {\gamma }}'''(t),\mathbf {e} _{1}(t){\bigr \rangle }\,\mathbf {e} _{1}(t)-{\bigl \langle }{\boldsymbol {\gamma }}'''(t),\mathbf {e} _{2}(t){\bigr \rangle }\,\mathbf {e} _{2}(t)$

In 3-dimensional space, the equation simplifies to $\mathbf {e} _{3}(t)=\mathbf {e} _{1}(t)\times \mathbf {e} _{2}(t)$ or to $\mathbf {e} _{3}(t)=-\mathbf {e} _{1}(t)\times \mathbf {e} _{2}(t),$ That either sign may occur is illustrated by the examples of a right-handed helix and a left-handed helix.

Torsion

The second generalized curvature $χ 2 (t)$ is called torsion and measures the deviance of $γ$ from being a plane curve. In other words, if the torsion is zero, the curve lies completely in the same osculating plane (there is only one osculating plane for every point $t$ ). It is defined as $\tau (t)=\chi _{2}(t)={\frac {{\bigl \langle }\mathbf {e} _{2}'(t),\mathbf {e} _{3}(t){\bigr \rangle }}{\left\|{\boldsymbol {\gamma }}'(t)\right\|}}$ and is called the torsion of $γ$ at point $t$ .

Aberrancy

The third derivative may be used to define aberrancy, a metric of non-circularity of a curve.^[4]^[5]^[6]

Main theorem of curve theory

Given $n - 1$ functions: $\chi _{i}\in C^{n-i}([a,b],\mathbb {R} ^{n}),\quad \chi _{i}(t)>0,\quad 1\leq i\leq n-1$ then there exists a unique (up to transformations using the Euclidean group) $C n + 1$ -curve $γ$ which is regular of order $n$ and has the following properties: ${\begin{aligned}\|\gamma '(t)\|&=1&t\in [a,b]\\\chi _{i}(t)&={\frac {\langle \mathbf {e} _{i}'(t),\mathbf {e} _{i+1}(t)\rangle }{\|{\boldsymbol {\gamma }}'(t)\|}}\end{aligned}}$ where the set $\mathbf {e} _{1}(t),\ldots ,\mathbf {e} _{n}(t)$ is the Frenet frame for the curve.

By additionally providing a start $t 0$ in $I$ , a starting point $p 0$ in $\mathbb {R} ^{n}$ and an initial positive orthonormal Frenet frame ${e 1, ..., e n - 1}$ with ${\begin{aligned}{\boldsymbol {\gamma }}(t_{0})&=\mathbf {p} _{0}\\\mathbf {e} _{i}(t_{0})&=\mathbf {e} _{i},\quad 1\leq i\leq n-1\end{aligned}}$ the Euclidean transformations are eliminated to obtain a unique curve $γ$ .

Frenet–Serret formulas

The Frenet–Serret formulas are a set of ordinary differential equations of first order. The solution is the set of Frenet vectors describing the curve specified by the generalized curvature functions $χ i$ .

2 dimensions

${\begin{bmatrix}\mathbf {e} _{1}'(t)\\\mathbf {e} _{2}'(t)\end{bmatrix}}=\left\Vert \gamma '(t)\right\Vert {\begin{bmatrix}0&\kappa (t)\\-\kappa (t)&0\\\end{bmatrix}}{\begin{bmatrix}\mathbf {e} _{1}(t)\\\mathbf {e} _{2}(t)\end{bmatrix}}$

3 dimensions

${\begin{bmatrix}\mathbf {e} _{1}'(t)\\[0.75ex]\mathbf {e} _{2}'(t)\\[0.75ex]\mathbf {e} _{3}'(t)\end{bmatrix}}=\left\Vert \gamma '(t)\right\Vert {\begin{bmatrix}0&\kappa (t)&0\\[1ex]-\kappa (t)&0&\tau (t)\\[1ex]0&-\tau (t)&0\end{bmatrix}}{\begin{bmatrix}\mathbf {e} _{1}(t)\\[1ex]\mathbf {e} _{2}(t)\\[1ex]\mathbf {e} _{3}(t)\end{bmatrix}}$

$n$ dimensions (general formula)

${\begin{bmatrix}\mathbf {e} _{1}'(t)\\[1ex]\mathbf {e} _{2}'(t)\\[1ex]\vdots \\[1ex]\mathbf {e} _{n-1}'(t)\\[1ex]\mathbf {e} _{n}'(t)\\[1ex]\end{bmatrix}}=\left\Vert \gamma '(t)\right\Vert {\begin{bmatrix}0&\chi _{1}(t)&\cdots &0&0\\[1ex]-\chi _{1}(t)&0&\cdots &0&0\\[1ex]\vdots &\vdots &\ddots &\vdots &\vdots \\[1ex]0&0&\cdots &0&\chi _{n-1}(t)\\[1ex]0&0&\cdots &-\chi _{n-1}(t)&0\\[1ex]\end{bmatrix}}{\begin{bmatrix}\mathbf {e} _{1}(t)\\[1ex]\mathbf {e} _{2}(t)\\[1ex]\vdots \\[1ex]\mathbf {e} _{n-1}(t)\\[1ex]\mathbf {e} _{n}(t)\\[1ex]\end{bmatrix}}$

References

^ ^a ^b do Carmo, Manfredo P. (2016). Differential Geometry of Curves and Surfaces (revised & updated 2nd ed.). Mineola, NY: Dover Publications, Inc. pp. 27–28. ISBN 978-0-486-80699-0.
^ Kühnel, Wolfgang (2005). Differential Geometry: Curves, Surfaces, Manifolds. Providence: AMS. p. 53. ISBN 0-8218-3988-8.
^ Weisstein, Eric W. "Bertrand Curves". mathworld.wolfram.com.
^ Schot, Stephen (November 1978). "Aberrancy: Geometry of the Third Derivative". Mathematics Magazine. 5. 51 (5): 259–275. doi:10.2307/2690245. JSTOR 2690245.
^ Cameron Byerley; Russell a. Gordon (2007). "Measures of Aberrancy". Real Analysis Exchange. 32 (1). Michigan State University Press: 233. doi:10.14321/realanalexch.32.1.0233. ISSN 0147-1937.
^ Gordon, Russell A. (2004). "The aberrancy of plane curves". The Mathematical Gazette. 89 (516). Cambridge University Press (CUP): 424–436. doi:10.1017/s0025557200178271. ISSN 0025-5572. S2CID 118533002.

Differentiable curve

Contents

Definitions

Re-parametrization and equivalence relation

Length and natural parametrization