differential geometry of curves

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Differential geometry of curves is the branch of
geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

that deals with smooth
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (geo ...

s in the
plane Plane or planes may refer to: * Airplane or aeroplane or informally plane, a powered, fixed-wing aircraft Arts, entertainment and media *Plane (Dungeons & Dragons), Plane (''Dungeons & Dragons''), a location in the multiverse *Plane (Magic: Th ...
and the
Euclidean space Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimensi ...
by methods of differential and
integral calculus In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with Deriv ...
. Many specific curves have been thoroughly investigated using the synthetic approach.
Differential geometry Differential geometry is a mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast E ...
takes another path: curves are represented in a parametrized form, and their geometric properties and various quantities associated with them, such as the
curvature In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

and the
arc length Arc length is the distance between two points along a section of a curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. In ...

, are expressed via
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its Argument of a function, argument (input value). Derivatives are a fundament ...

s and
integral In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

s using
vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Product differentiation, in ...
. One of the most important tools used to analyze a curve is the , 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

'' and the '' torsion'' of a curve. The
fundamental theorem of curves In differential geometry Differential geometry is a Mathematics, mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The Differential ...
asserts that the knowledge of these invariants completely determines the curve.

# Definitions

A ''parametric'' -''curve'' or a -''parametrization'' is a
vector-valued function A vector-valued function, also referred to as a vector function, is a function (mathematics), mathematical function of one or more variables whose range of a function, range is a set of multidimensional Euclidean vector, vectors or infinite-dimensi ...

:$\gamma: I \to \mathbb^$ that is -times
continuously differentiable In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
(that is, the component functions of are continuously differentiable), where , , and be a non-empty interval of real numbers. The of the parametric curve is . The parametric curve and its image must be distinguished because a given subset of can be the image of several distinct parametric curves. The parameter in can be thought of as representing time, and the
trajectory A trajectory or flight path is the path that an with in follows through as a function of time. In , a trajectory is defined by via ; hence, a complete trajectory is defined by position and momentum, simultaneously. The mass might be a or ...

of a moving point in space. When is a closed interval , is called the starting point and is the endpoint of . If the starting and the end points coincide (that is, ), then is a ''closed curve'' or a ''loop''. For being a -loop, the function must be -times continuously differentiable and satisfy for . The parametric curve is if :$\gamma, _: \left(a,b\right) \to \mathbb^$ is
injective In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
. It is if each component function of is an
analytic function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
, that is, it is of class . The curve is ''regular of order'' (where ) if, for every , :$\left\$ is a
linearly independent In the theory of vector spaces, a set of vectors is said to be if at least one of the vectors in the set can be defined as a linear combinationIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics ...
subset of . In particular, a parametric -curve is if and only if for any .

# 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
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 -curves and are central objects studied in the differential geometry of curves. Two parametric -curves, and , are said to be if and only if there exists a bijective -map such that :$\forall t \in I_1: \quad \varphi\text{'}\left(t\right) \neq 0$ and :$\forall t \in I_1: \quad \gamma_2\bigl\left(\varphi\left(t\right)\bigr\right) = \gamma_1\left(t\right).$ is then said to be a of . Re-parametrization defines an equivalence relation on the set of all parametric -curves of class . The equivalence class of this relation simply a -curve. An even ''finer'' equivalence relation of oriented parametric -curves can be defined by requiring to satisfy . Equivalent parametric -curves have the same image, and equivalent oriented parametric -curves even traverse the image in the same direction.

# Length and natural parametrization

The length of a parametric -curve is defined as :$l ~ \stackrel ~ \int_a^b \left\, \gamma\text{'}\left(t\right) \right\, \, \mathrm.$ 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 -curve , where , the function is defined :$\forall t \in \left[a,b\right]: \quad s\left(t\right) ~ \stackrel ~ \int_a^t \left\, \gamma\text{'}\left(x\right) \right\, \, \mathrm.$ Writing , where is the inverse function of . This is a re-parametrization of that is called an ', ''natural parametrization'', ''unit-speed parametrization''. The parameter is called the of . This parametrization is preferred because the natural parameter traverses the image of at unit speed, so that :$\forall t \in I: \quad \left\, \overline\text{'}\bigl\left(s\left(t\right)\bigr\right) \right\, = 1.$ 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 , the natural parametrization is unique up to a shift of parameter. The quantity :$E\left(\gamma\right) ~ \stackrel ~ \frac \int_a^b \left\, \gamma\text{'}\left(t\right) \right\, ^2 ~ \mathrm$ is sometimes called the or action (physics), action of the curve; this name is justified because the geodesic equations are the Euler–Lagrange equations of motion for this action.

# Frenet frame

A Frenet frame is a Moving frame, moving reference frame of orthonormal vectors which are used to describe a curve locally at each point . 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 -curve in which is regular of order the Frenet frame for the curve is the set of orthonormal vectors :$\mathbf_1\left(t\right), \ldots, \mathbf_n\left(t\right)$ called Frenet–Serret formulas, Frenet vectors. They are constructed from the derivatives of using the Gram–Schmidt process, Gram–Schmidt orthogonalization algorithm with :$\begin \mathbf_1\left(t\right) &= \frac \\\left[8px\right] \mathbf_\left(t\right) &= \frac, \quad \overline\left(t\right) = \boldsymbol^\left(t\right) - \sum _^ \left\langle \boldsymbol^\left(t\right), \mathbf_i\left(t\right) \right\rangle \, \mathbf_i\left(t\right) \end$ The real-valued functions are called generalized curvatures and are defined as :$\chi_i\left(t\right) = \frac$ The Frenet frame and the generalized curvatures are invariant under reparametrization and are therefore differential geometric properties of the curve.

## Bertrand curve

A Bertrand curve is a Frenet curve in $\mathbb R^3$ with the additional property that there is a second curve in $\mathbb R^3$ such that the #Normal or curvature vector, principal normal vectors to these two curves are identical at each corresponding point. In other words, if and are two curves in $\mathbb R^3$ such that for any , , then and are Bertrand curves. For this reason it is common to speak of a Bertrand pair of curves (like and in the previous example). 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 where and are real constants and . Furthermore, the product of #Torsion, torsions of a Bertrand pair of curves is constant.

# 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 represents the path of a particle, then the instantaneous velocity of the particle at a given point is expressed by a Vector (geometric), vector, called the tangent vector to the curve at . Mathematically, given a parametrized curve , for every value of the parameter, the vector : $\gamma\text{'}\left(t_0\right) = \frac\boldsymbol\left(t\right) \ \text \ t=t_0$ is the tangent vector at the point . Generally speaking, the tangent vector may be zero vector, zero. The tangent vector's magnitude :$\left\, \boldsymbol\text{'}\left(t_0\right)\right\,$ is the speed at the time . The first Frenet vector is the unit tangent vector in the same direction, defined at each regular point of : :$\mathbf_\left(t\right) = \frac.$ If is the natural parameter, then the tangent vector has unit length. The formula simplifies: :$\mathbf_\left(s\right) = \boldsymbol\text{'}\left(s\right)$. 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

The ''normal vector'', sometimes called the ''curvature vector'', indicates the deviance of the curve from being a straight line. It is defined as :$\overline\left(t\right) = \boldsymbol\text{'}\text{'}\left(t\right) - \bigl\langle \boldsymbol\text{'}\text{'}\left(t\right), \mathbf_1\left(t\right) \bigr\rangle \, \mathbf_1\left(t\right).$ Its normalized form, the unit normal vector, is the second Frenet vector and is defined as :$\mathbf_2\left(t\right) = \frac .$ The tangent and the normal vector at point define the osculating plane at point . It can be shown that . Therefore, :$\mathbf_2\left(t\right) = \frac.$

## Curvature

The first generalized curvature is called curvature and measures the deviance of from being a straight line relative to the osculating plane. It is defined as :$\kappa\left(t\right) = \chi_1\left(t\right) = \frac$ and is called the
curvature In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

of at point . It can be shown that :$\kappa\left(t\right) = \frac.$ The Multiplicative inverse, reciprocal of the curvature :$\frac$ is called the radius of curvature (mathematics), radius of curvature. A circle with radius has a constant curvature of :$\kappa\left(t\right) = \frac$ whereas a line has a curvature of 0.

## Binormal vector

The unit binormal vector is the third Frenet vector . It is always orthogonal to the unit tangent and normal vectors at . It is defined as :$\mathbf_3\left(t\right) = \frac , \quad \overline\left(t\right) = \boldsymbol\left(t\right) - \bigr\langle \boldsymbol\left(t\right), \mathbf_1\left(t\right) \bigr\rangle \, \mathbf_1\left(t\right) - \bigl\langle \boldsymbol\left(t\right), \mathbf_2\left(t\right) \bigr\rangle \,\mathbf_2\left(t\right)$ In 3-dimensional space, the equation simplifies to :$\mathbf_3\left(t\right) = \mathbf_1\left(t\right) \times \mathbf_2\left(t\right)$ or to :$\mathbf_3\left(t\right) = -\mathbf_1\left(t\right) \times \mathbf_2\left(t\right),$ 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 is called and measures the deviance of 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 ). It is defined as :$\tau\left(t\right) = \chi_2\left(t\right) = \frac$ and is called the torsion (differential geometry), torsion of at point .

## Aberrancy

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

# Main theorem of curve theory

Given functions: :$\chi_i \in C^\left(\left[a,b\right],\mathbb^n\right) , \quad \chi_i\left(t\right) > 0 ,\quad 1 \leq i \leq n-1$ then there exists a unique (up to transformations using the Euclidean group) -curve which is regular of order ''n'' and has the following properties: :$\begin \, \gamma\text{'}\left(t\right)\, &= 1 & t \in \left[a,b\right] \\ \chi_i\left(t\right) &= \frac \end$ where the set :$\mathbf_1\left(t\right), \ldots, \mathbf_n\left(t\right)$ is the Frenet frame for the curve. By additionally providing a start in , a starting point in and an initial positive orthonormal Frenet frame with :$\begin \boldsymbol\left(t_0\right) &= \mathbf_0 \\ \mathbf_i\left(t_0\right) &= \mathbf_i ,\quad 1 \leq i \leq n-1 \end$ the Euclidean transformations are eliminated to obtain a unique curve .

# 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 .

## 2 dimensions

:$\begin \mathbf_1\text{'}\left(t\right) \\ \mathbf_2\text{'}\left(t\right) \\ \end = \left\Vert \gamma\text{'}\left\left(t\right\right) \right\Vert \begin 0 & \kappa\left(t\right) \\ -\kappa\left(t\right) & 0 \\ \end \begin \mathbf_1\left(t\right) \\ \mathbf_2\left(t\right) \\ \end$

## 3 dimensions

:$\begin \mathbf_1\text{'}\left(t\right) \\ \mathbf_2\text{'}\left(t\right) \\ \mathbf_3\text{'}\left(t\right) \\ \end = \left\Vert \gamma\text{'}\left\left(t\right\right) \right\Vert \begin 0 & \kappa\left(t\right) & 0 \\ -\kappa\left(t\right) & 0 & \tau\left(t\right) \\ 0 & -\tau\left(t\right) & 0 \\ \end \begin \mathbf_1\left(t\right) \\ \mathbf_2\left(t\right) \\ \mathbf_3\left(t\right) \\ \end$

## dimensions (general formula)

:$\begin \mathbf_1\text{'}\left(t\right) \\ \mathbf_2\text{'}\left(t\right) \\ \vdots \\ \mathbf_\text{'}\left(t\right) \\ \mathbf_n\text{'}\left(t\right) \\ \end = \left\Vert \gamma\text{'}\left\left(t\right\right) \right\Vert \begin 0 & \chi_1\left(t\right) & \cdots & 0 & 0 \\ -\chi_1\left(t\right) & 0 & \cdots & 0 & 0 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \cdots & 0 & \chi_\left(t\right) \\ 0 & 0 & \cdots & -\chi_\left(t\right) & 0 \\ \end \begin \mathbf_1\left(t\right) \\ \mathbf_2\left(t\right) \\ \vdots \\ \mathbf_\left(t\right) \\ \mathbf_n\left(t\right) \\ \end$