Differentiable curve

This article is about curves in Euclidean space. For curves in an arbitrary topological space, see 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

Main article: Curve

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がんま 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がんま and its image γがんま[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がんま(t) can be thought of as representing time, and γがんま the trajectory of a moving point in space. When I is a closed interval [a,b], γがんま(a) is called the starting point and γがんま(b) is the endpoint of γがんま. If the starting and the end points coincide (that is, γがんま(a) = γがんま(b)), then γがんま is a closed curve or a loop. To be a C^r-loop, the function γがんま must be r-times continuously differentiable and satisfy γがんま^(k)(a) = γがんま^(k)(b) for 0 ≤ k ≤ 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がんま is an analytic function, that is, it is of class C^ωおめが.

The curve γがんま is regular of order m (where m ≤ r) if, for every t ∈ I,

$$\left\{\gamma '(t),\gamma ''(t),\ldots ,{\gamma ^{(m)}}(t)\right\}$$

is a linearly independent subset of $\mathbb {R} ^{n}$. In particular, a parametric C¹-curve γがんま is regular if and only if γがんま′(t) ≠ 0 for any t ∈ I.

Re-parametrization and equivalence relation

See also: Position vector and Vector-valued function

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ふぁい : I₁ → I₂ 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がんま₂ is then said to be a re-parametrization of γがんま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ふぁい to satisfy φふぁい′(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

Main article: Arc length

See also: Curve § Length of a curve

The length l of a parametric C¹-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 ≥ 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がんま(s) = γがんま(t(s)), where t(s) is the inverse function of s(t). This is a re-parametrization γがんま of γがんま that is called an arc-length parametrization, natural parametrization, unit-speed parametrization. The parameter s(t) is called the natural parameter of γがんま.

This parametrization is preferred because the natural parameter s(t) traverses the image of γがんま 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がんま, 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

Main article: Frenet–Serret formulas

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がんま(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がんま 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がんま(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かい_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がんま₁(t) and γがんま₂(t) are two curves in $\mathbb {R} ^{3}$ such that for any t, the two principal normals N₁(t), N₂(t) are equal, then γがんま₁ and γがんま₂ are Bertrand curves, and γがんま₂ is called the Bertrand mate of γがんま₁. We can write γがんま₂(t) = γがんま₁(t) + r N₁(t) for some constant r.

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かっぱ(t) + b τたう(t) = 1 where κかっぱ(t) and τたう(t) are the curvature and torsion of γがんま₁(t) and a and b are real constants with a ≠ 0. Furthermore, the product of torsions of a Bertrand pair of curves is constant. If γがんま₁ has more than one Bertrand mate then it has infinitely many. This only occurs when γがんま₁ is a circular helix.

Special Frenet vectors and generalized curvatures

Main article: Frenet–Serret formulas

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がんま 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 C¹ curve γがんま = γがんま(t), for every value t = t₀ 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がんま(t₀). 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 t₀.

The first Frenet vector e₁(t) is the unit tangent vector in the same direction, defined at each regular point of γがんま:

$$\mathbf {e} _{1}(t)={\frac {{\boldsymbol {\gamma }}'(t)}{\left\|{\boldsymbol {\gamma }}'(t)\right\|}}.$$

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 e₂(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 ē₂(t) ∝ e′₁(t). Therefore,

$$\mathbf {e} _{2}(t)={\frac {\mathbf {e} _{1}'(t)}{\left\|\mathbf {e} _{1}'(t)\right\|}}.$$

Curvature

Main article: Curvature of space curves

The first generalized curvature χかい₁(t) is called curvature and measures the deviance of γがんま 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がんま 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 e₃(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

Main article: Torsion of a curve

The second generalized curvature χかい₂(t) is called torsion and measures the deviance of γがんま 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がんま at point t.

Aberrancy

The third derivative may be used to define aberrancy, a metric of non-circularity of a curve.

Main theorem of curve theory

Main article: Fundamental theorem of curves

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がんま 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 t₀ in I, a starting point p₀ in $\mathbb {R} ^{n}$ and an initial positive orthonormal Frenet frame {e₁, ..., 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がんま.

Frenet–Serret formulas

Main article: 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かい_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)

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

See also

• List of curves topics

Further reading

• Kreyszig, Erwin (1991). Differential Geometry. New York: Dover Publications. ISBN 0-486-66721-9. Chapter II is a classical treatment of Theory of Curves in 3-dimensions.