Английская Википедия: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

Шаблон:Main A parametric Шаблон:Math-curve or a Шаблон:Math-parametrization is a vector-valued function

<math>\gamma: I \to \mathbb{R}^{n}</math>

that is Шаблон:Mvar-times continuously differentiable (that is, the component functions of Шаблон:Mvar are continuously differentiable), where <math>n \isin \mathbb{N}</math>, <math>r \isin \mathbb{N} \cup \{\infty\}</math>, and Шаблон:Mvar is a non-empty interval of real numbers. The Шаблон:Em of the parametric curve is <math>\gamma[I] \subseteq \mathbb{R}^n</math>. The parametric curve Шаблон:Mvar and its image Шаблон:Math must be distinguished because a given subset of <math>\mathbb{R}^n</math> can be the image of many distinct parametric curves. The parameter Шаблон:Mvar in Шаблон:Math can be thought of as representing time, and Шаблон:Mvar the trajectory of a moving point in space. When Шаблон:Mvar is a closed interval Шаблон:Math, Шаблон:Math is called the starting point and Шаблон:Math is the endpoint of Шаблон:Mvar. If the starting and the end points coincide (that is, Шаблон:Math), then Шаблон:Mvar is a closed curve or a loop. To be a Шаблон:Math-loop, the function Шаблон:Mvar must be Шаблон:Mvar-times continuously differentiable and satisfy Шаблон:Math for Шаблон:Math.

The parametric curve is Шаблон:Em if

<math> \gamma|_{(a,b)}: (a,b) \to \mathbb{R}^{n} </math>

is injective. It is Шаблон:Em if each component function of Шаблон:Mvar is an analytic function, that is, it is of class Шаблон:Math.

The curve Шаблон:Mvar is regular of order Шаблон:Mvar (where Шаблон:Math) if, for every Шаблон:Math,

<math>\left\{ \gamma'(t),\gamma(t),\ldots,{\gamma^{(m)}}(t) \right\}</math>

is a linearly independent subset of <math>\mathbb{R}^n</math>. In particular, a parametric Шаблон:Math-curve Шаблон:Mvar is Шаблон:Em if and only if Шаблон:Math for any Шаблон:Math.

Re-parametrization and equivalence relation

Шаблон:See also

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 Шаблон:Math-curves and are central objects studied in the differential geometry of curves.

Two parametric Шаблон:Math-curves, <math>\gamma_1 : I_1 \to \mathbb{R}^n</math> and <math>\gamma_2 : I_2 \to \mathbb{R}^n</math>, are said to be Шаблон:Em if and only if there exists a bijective Шаблон:Math-map Шаблон:Math such that

<math>\forall t \in I_1: \quad \varphi'(t) \neq 0</math>

and

<math>\forall t \in I_1: \quad \gamma_2\bigl(\varphi(t)\bigr) = \gamma_1(t).</math>

Шаблон:Math is then said to be a Шаблон:Em of Шаблон:Math.

Re-parametrization defines an equivalence relation on the set of all parametric Шаблон:Math-curves of class Шаблон:Math. The equivalence class of this relation simply a Шаблон:Math-curve.

An even finer equivalence relation of oriented parametric Шаблон:Math-curves can be defined by requiring Шаблон:Mvar to satisfy Шаблон:Math.

Equivalent parametric Шаблон:Math-curves have the same image, and equivalent oriented parametric Шаблон:Math-curves even traverse the image in the same direction.

Length and natural parametrizationШаблон:Anchor

Шаблон:Main Шаблон:See also

The length Шаблон:Mvar of a parametric Шаблон:Math-curve <math>\gamma : [a, b] \to \mathbb{R}^n</math> is defined as

<math>l ~ \stackrel{\text{def}}{=} ~ \int_a^b \left\| \gamma'(t) \right\| \, \mathrm{d}{t}.</math>

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 Шаблон:Math-curve <math>\gamma : [a, b] \to \mathbb{R}^n</math>, where Шаблон:Math, the function is defined

<math>\forall t \in [a,b]: \quad s(t) ~ \stackrel{\text{def}}{=} ~ \int_a^t \left\| \gamma'(x) \right\| \, \mathrm{d}{x}.</math>

Writing Шаблон:Math, where Шаблон:Math is the inverse function of Шаблон:Math. This is a re-parametrization Шаблон:Math of Шаблон:Mvar that is called an Шаблон:Vanchor, natural parametrization, unit-speed parametrization. The parameter Шаблон:Math is called the Шаблон:Em of Шаблон:Mvar.

This parametrization is preferred because the natural parameter Шаблон:Math traverses the image of Шаблон:Mvar at unit speed, so that

<math>\forall t \in I: \quad \left\| \overline{\gamma}'\bigl(s(t)\bigr) \right\| = 1.</math>

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 Шаблон:Mvar, the natural parametrization is unique up to a shift of parameter.

The quantity

<math>E(\gamma) ~ \stackrel{\text{def}}{=} ~ \frac{1}{2} \int_a^b \left\| \gamma'(t) \right\|^2 ~ \mathrm{d}{t}</math>

is sometimes called the Шаблон:Em 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

Шаблон:Main

Файл:Frenet frame.png

An illustration of the Frenet frame for a point on a space curve. Шаблон:Math is the unit tangent, Шаблон:Math the unit normal, and Шаблон:Math the unit binormal.

A Frenet frame is a moving reference frame of Шаблон:Math orthonormal vectors Шаблон:Math which are used to describe a curve locally at each point Шаблон:Math. 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 Шаблон:Math-curve Шаблон:Math in <math>\mathbb{R}^n</math> which is regular of order Шаблон:Math the Frenet frame for the curve is the set of orthonormal vectors

<math>\mathbf{e}_1(t), \ldots, \mathbf{e}_n(t)</math>

called Frenet vectors. They are constructed from the derivatives of Шаблон:Math using the Gram–Schmidt orthogonalization algorithm with

<math>\begin{align}

\mathbf{e}_1(t) &= \frac{\boldsymbol{\gamma}'(t)}{\left\| \boldsymbol{\gamma}'(t) \right\|} \\[8px] \mathbf{e}_{j}(t) &= \frac{\overline{\mathbf{e}_{j}}(t)}{\left\|\overline{\mathbf{e}_{j}}(t) \right\|}, \quad \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) \end{align}</math>

The real-valued functions Шаблон:Math are called generalized curvatures and are defined as

<math>\chi_i(t) = \frac{\bigl\langle \mathbf{e}_i'(t), \mathbf{e}_{i+1}(t) \bigr\rangle}{\left\| \boldsymbol{\gamma}^'(t) \right\|} </math>

The Frenet frame and the generalized curvatures are invariant under reparametrization and are therefore differential geometric properties of the curve. For curves in <math>\mathbb R^3</math> <math>\chi_1(t)</math> is the curvature and <math>\chi_2(t)</math> is the torsion.

Bertrand curve

A Bertrand curve is a regular curve in <math>\mathbb R^3</math> with the additional property that there is a second curve in <math>\mathbb R^3</math> such that the principal normal vectors to these two curves are identical at each corresponding point. In other words, if Шаблон:Math and Шаблон:Math are two curves in <math>\mathbb R^3</math> such that for any Шаблон:Mvar, the two principal normals Шаблон:Math are equal, then Шаблон:Math and Шаблон:Math are Bertrand curves, and Шаблон:Math is called the Bertrand mate of Шаблон:Math. We can write Шаблон:Math for some constant Шаблон:Math.^[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 Шаблон:Math where Шаблон:Math and Шаблон:Math are the curvature and torsion of Шаблон:Math and Шаблон:Mvar and Шаблон:Mvar are real constants with Шаблон:Math.^[2] Furthermore, the product of torsions of a Bertrand pair of curves is constant.^[3] If Шаблон:Math has more than one Bertrand mate then it has infinitely many. This only occurs when Шаблон:Math is a circular helix.^[1]

Special Frenet vectors and generalized curvatures

Шаблон:Main 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 Шаблон:Math represents the path of a particle, then the instantaneous velocity of the particle at a given point Шаблон:Math is expressed by a vector, called the tangent vector to the curve at Шаблон:Math. Mathematically, given a parametrized Шаблон:Math curve Шаблон:Math, for every value Шаблон:Math of the parameter, the vector

<math> \gamma'(t_0) = \left.\frac{\mathrm{d}}{\mathrm{d}t}\boldsymbol{\gamma}(t)\right|_{t=t_0} </math>

is the tangent vector at the point Шаблон:Math. Generally speaking, the tangent vector may be zero. The tangent vector's magnitude

<math>\left\|\boldsymbol{\gamma}'(t_0)\right\|</math>

is the speed at the time Шаблон:Math.

The first Frenet vector Шаблон:Math is the unit tangent vector in the same direction, defined at each regular point of Шаблон:Math:

<math>\mathbf{e}_{1}(t) = \frac{ \boldsymbol{\gamma}'(t) }{ \left\| \boldsymbol{\gamma}'(t) \right\|}.</math>

If Шаблон:Math is the natural parameter, then the tangent vector has unit length. The formula simplifies:

<math>\mathbf{e}_{1}(s) = \boldsymbol{\gamma}'(s)</math>.

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

<math>\overline{\mathbf{e}_2}(t) = \boldsymbol{\gamma}(t) - \bigl\langle \boldsymbol{\gamma}(t), \mathbf{e}_1(t) \bigr\rangle \, \mathbf{e}_1(t).</math>

Its normalized form, the unit normal vector, is the second Frenet vector Шаблон:Math and is defined as

<math>\mathbf{e}_2(t) = \frac{\overline{\mathbf{e}_2}(t)} {\left\| \overline{\mathbf{e}_2}(t) \right\|}.</math>

The tangent and the normal vector at point Шаблон:Math define the osculating plane at point Шаблон:Math.

It can be shown that Шаблон:Math. Therefore,

<math>\mathbf{e}_2(t) = \frac{\mathbf{e}_1'(t)}{\left\| \mathbf{e}_1'(t) \right\|}.</math>

Curvature

Шаблон:Main

The first generalized curvature Шаблон:Math is called curvature and measures the deviance of Шаблон:Math from being a straight line relative to the osculating plane. It is defined as

<math>\kappa(t) = \chi_1(t) = \frac{\bigl\langle \mathbf{e}_1'(t), \mathbf{e}_2(t) \bigr\rangle}{\left\| \boldsymbol{\gamma}'(t) \right\|}</math>

and is called the curvature of Шаблон:Math at point Шаблон:Math. It can be shown that

<math>\kappa(t) = \frac{\left\| \mathbf{e}_1'(t) \right\|}{\left\| \boldsymbol{\gamma}'(t) \right\|}.</math>

The reciprocal of the curvature

<math>\frac{1}{\kappa(t)}</math>

is called the radius of curvature.

A circle with radius Шаблон:Math has a constant curvature of

<math>\kappa(t) = \frac{1}{r}</math>

whereas a line has a curvature of 0.

Binormal vector

The unit binormal vector is the third Frenet vector Шаблон:Math. It is always orthogonal to the unit tangent and normal vectors at Шаблон:Math. It is defined as

<math>\mathbf{e}_3(t) = \frac{\overline{\mathbf{e}_3}(t)} {\| \overline{\mathbf{e}_3}(t) \|}

, \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) </math>

In 3-dimensional space, the equation simplifies to

<math>\mathbf{e}_3(t) = \mathbf{e}_1(t) \times \mathbf{e}_2(t)</math>

or to

<math>\mathbf{e}_3(t) = -\mathbf{e}_1(t) \times \mathbf{e}_2(t),</math>

That either sign may occur is illustrated by the examples of a right-handed helix and a left-handed helix.

Torsion

Шаблон:Main

The second generalized curvature Шаблон:Math is called Шаблон:Em and measures the deviance of Шаблон:Math 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 Шаблон:Math). It is defined as

<math>\tau(t) = \chi_2(t) = \frac{\bigl\langle \mathbf{e}_2'(t), \mathbf{e}_3(t) \bigr\rangle}{\left\| \boldsymbol{\gamma}'(t) \right\|}</math>

and is called the torsion of Шаблон:Math at point Шаблон:Math.

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

Шаблон:Main Given Шаблон:Math functions:

<math>\chi_i \in C^{n-i}([a,b],\mathbb{R}^n) , \quad \chi_i(t) > 0 ,\quad 1 \leq i \leq n-1</math>

then there exists a unique (up to transformations using the Euclidean group) Шаблон:Math-curve Шаблон:Math which is regular of order n and has the following properties:

<math>\begin{align}

\|\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{align}</math>

where the set

<math>\mathbf{e}_1(t), \ldots, \mathbf{e}_n(t)</math>

is the Frenet frame for the curve.

By additionally providing a start Шаблон:Math in Шаблон:Math, a starting point Шаблон:Math in <math>\mathbb{R}^n</math> and an initial positive orthonormal Frenet frame Шаблон:Math with

<math>\begin{align}

\boldsymbol{\gamma}(t_0) &= \mathbf{p}_0 \\ \mathbf{e}_i(t_0) &= \mathbf{e}_i ,\quad 1 \leq i \leq n-1 \end{align}</math> the Euclidean transformations are eliminated to obtain a unique curve Шаблон:Math.

Frenet–Serret formulas

Шаблон:Main

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 Шаблон:Math.

2 dimensions

<math>

\begin{bmatrix}

\mathbf{e}_1'(t) \\
\mathbf{e}_2'(t) \\

\end{bmatrix}

=

\left\Vert \gamma'\left(t\right) \right\Vert

\begin{bmatrix}

        0  & \kappa(t) \\
-\kappa(t) &         0 \\

\end{bmatrix}

\begin{bmatrix} \mathbf{e}_1(t) \\ \mathbf{e}_2(t) \\ \end{bmatrix} </math>

3 dimensions

<math>

\begin{bmatrix}

\mathbf{e}_1'(t) \\
\mathbf{e}_2'(t) \\
\mathbf{e}_3'(t) \\

\end{bmatrix}

=

\left\Vert \gamma'\left(t\right) \right\Vert

\begin{bmatrix}

         0 &  \kappa(t) &        0 \\
-\kappa(t) &          0 & \tau(t)  \\
         0 &   -\tau(t) &        0 \\

\end{bmatrix}

\begin{bmatrix}

\mathbf{e}_1(t) \\
\mathbf{e}_2(t) \\
\mathbf{e}_3(t) \\

\end{bmatrix} </math>

Шаблон:Math dimensions (general formula)

<math>

\begin{bmatrix}

\mathbf{e}_1'(t) \\
\mathbf{e}_2'(t) \\
          \vdots \\
\mathbf{e}_{n-1}'(t) \\
\mathbf{e}_n'(t) \\

\end{bmatrix}

=

\left\Vert \gamma'\left(t\right) \right\Vert

\begin{bmatrix}

         0 &  \chi_1(t) & \cdots &              0 &             0 \\
-\chi_1(t) &          0 & \cdots &              0 &             0 \\
    \vdots &     \vdots & \ddots &         \vdots &        \vdots \\
         0 &          0 & \cdots &              0 & \chi_{n-1}(t) \\
         0 &          0 & \cdots & -\chi_{n-1}(t) &             0 \\

\end{bmatrix}

\begin{bmatrix}

\mathbf{e}_1(t) \\
\mathbf{e}_2(t) \\
         \vdots \\
\mathbf{e}_{n-1}(t) \\
\mathbf{e}_n(t) \\

\end{bmatrix} </math>

References

Шаблон:Reflist

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Английская Википедия:Differentiable curve

Содержание

Definitions

Re-parametrization and equivalence relation

Length and natural parametrizationШаблон:Anchor

Frenet frame

Bertrand curve

Special Frenet vectors and generalized curvatures

Tangent vector

Normal vector or curvature vector

Curvature

Binormal vector

Torsion

Aberrancy

Main theorem of curve theory

Frenet–Serret formulas

2 dimensions

3 dimensions

Шаблон:Math dimensions (general formula)

See also

References

Further reading

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