Differential geometry of curves is the branch of
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
that deals with
smooth
Smooth may refer to:
Mathematics
* Smooth function, a function that is infinitely differentiable; used in calculus and topology
* Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions
* Smooth algebrai ...
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 (ge ...
s in the
plane
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* ''Planes' ...
and the
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
by methods of
differential and
integral calculus
In mathematics, an integral assigns numbers to Function (mathematics), functions in a way that describes Displacement (geometry), displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding ...
.
Many
specific curves have been thoroughly investigated using the
synthetic approach.
Differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
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, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.
For curves, the canonic ...
and the
arc length
ARC may refer to:
Business
* Aircraft Radio Corporation, a major avionics manufacturer from the 1920s to the '50s
* Airlines Reporting Corporation, an airline-owned company that provides ticket distribution, reporting, and settlement services
* ...
, 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 (input value). Derivatives are a fundamental tool of calculus. F ...
s and
integral
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
s using
vector calculus
Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for the broader subject ...
. 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
In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.
For curves, the canonic ...
'' and the ''
torsion
Torsion may refer to:
Science
* Torsion (mechanics), the twisting of an object due to an applied torque
* Torsion of spacetime, the field used in Einstein–Cartan theory and
** Alternatives to general relativity
* Torsion angle, in chemistry
Bi ...
'' of a curve. The
fundamental theorem of curves
In differential geometry, the fundamental theorem of space curves states that every regular curve in three-dimensional space, with non-zero curvature, has its shape (and size or scale) completely determined by its curvature and torsion..
Use
A ...
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 mathematical function of one or more variables whose range is a set of multidimensional vectors or infinite-dimensional vectors. The input of a vector-valued function could ...
:
that is -times
continuously differentiable (that is, the component functions of are continuously differentiable), where
,
, and is 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 many 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 object with mass in motion follows through space as a function of time. In classical mechanics, a trajectory is defined by Hamiltonian mechanics via canonical coordinates; hence, a complete traj ...
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''. To be a -loop, the function must be -times continuously differentiable and satisfy for .
The parametric curve is if
:
is
injective
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
. It is if each component function of is an
analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex an ...
, that is, it is of class .
The curve is ''regular of order'' (where ) if, for every ,
:
is a
linearly independent
In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts are ...
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, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation.
Each 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
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
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
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
-map such that
:
and
:
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
:
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
:
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
:
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
:
is sometimes called the or
action
Action may refer to:
* Action (narrative), a literary mode
* Action fiction, a type of genre fiction
* Action game, a genre of video game
Film
* Action film, a genre of film
* ''Action'' (1921 film), a film by John Ford
* ''Action'' (1980 fil ...
of the curve; this name is justified because the
geodesic
In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
equations are the
Euler–Lagrange equation
In the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional. The equations were discovered ...
s of motion for this action.
Frenet frame
![Frenet frame](https://upload.wikimedia.org/wikipedia/commons/3/32/Frenet_frame.png)
A Frenet frame is a
moving reference frame of
orthonormal
In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal (or perpendicular along a line) unit vectors. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of un ...
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
:
called
Frenet vectors. They are constructed from the derivatives of using the
Gram–Schmidt orthogonalization algorithm with
:
The real-valued functions are called generalized curvatures and are defined as
:
The Frenet frame and the generalized curvatures are invariant under reparametrization and are therefore differential geometric properties of the curve. For curves in
is the curvature and
is the torsion.
Bertrand curve
A Bertrand curve is a regular curve in
with the additional property that there is a second curve in
such that the
principal normal vectors to these two curves are identical at each corresponding point. In other words, if and are two curves in
such that for any , the two principal normals are equal, then and are Bertrand curves, and is called the Bertrand mate of . We can write for some constant .
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 the curvature and torsion of and and are real constants with . Furthermore, the product of
torsion
Torsion may refer to:
Science
* Torsion (mechanics), the twisting of an object due to an applied torque
* Torsion of spacetime, the field used in Einstein–Cartan theory and
** Alternatives to general relativity
* Torsion angle, in chemistry
Bi ...
s of a Bertrand pair of curves is constant.
If has more than one Bertrand mate then it has infinitely many. This only occurs when is a circular helix.
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
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity is a ...
of the particle at a given point is expressed by a
vector
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
, called the ''
tangent vector
In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point. Tangent vectors are described in the differential geometry of curves in the context of curves in R''n''. More generally, tangent vectors are eleme ...
'' to the curve at . Mathematically, given a parametrized curve , for every value of the parameter, the vector
:
is the tangent vector at the point . Generally speaking, the tangent vector may be
zero
0 (zero) is a number representing an empty quantity. In place-value notation
Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the Hindu–Arabic numeral system (or ...
. The tangent vector's magnitude
:
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 :
:
If is the natural parameter, then the tangent vector has unit length. The formula simplifies:
:
.
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
In differential geometry, the spherical image of a unit-speed curve is given by taking the curve's tangent vectors as points, all of which must lie on the unit sphere
In mathematics, a unit sphere is simply a sphere of radius one around a give ...
of the original curve.
Normal vector or curvature vector
A curve ''
normal vector
In geometry, a normal is an object such as a line, ray, or vector that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the (infinite) line perpendicular to the tangent line to the curve at ...
'', sometimes called the curvature vector, indicates the deviance of the curve from being a straight line.
It is defined as
:
Its normalized form, the unit normal vector, is the second Frenet vector and is defined as
:
The tangent and the normal vector at point define the
osculating plane {{Unreferenced, date=May 2019, bot=noref (GreenC bot)
In mathematics, particularly in differential geometry, an osculating plane is a plane in a Euclidean space or affine space which meets a submanifold at a point in such a way as to have a second ...
at point .
It can be shown that . Therefore,
:
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
:
and is called the
curvature
In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.
For curves, the canonic ...
of at point . It can be shown that
:
The
reciprocal
Reciprocal may refer to:
In mathematics
* Multiplicative inverse, in mathematics, the number 1/''x'', which multiplied by ''x'' gives the product 1, also known as a ''reciprocal''
* Reciprocal polynomial, a polynomial obtained from another pol ...
of the curvature
:
is called the
radius of curvature
In differential geometry, the radius of curvature, , is the reciprocal of the curvature. For a curve, it equals the radius of the circular arc which best approximates the curve at that point. For surfaces, the radius of curvature is the radius o ...
.
A circle with radius has a constant curvature of
:
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
:
In 3-dimensional space, the equation simplifies to
:
or to
:
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 mathematics, a plane curve is a curve in a plane that may be either a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic pla ...
. 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
:
and is called the
torsion
Torsion may refer to:
Science
* Torsion (mechanics), the twisting of an object due to an applied torque
* Torsion of spacetime, the field used in Einstein–Cartan theory and
** Alternatives to general relativity
* Torsion angle, in chemistry
Bi ...
of at point .
Aberrancy
The
third derivative
In calculus, a branch of mathematics, the third derivative is the rate at which the second derivative, or the rate of change of the rate of change, is changing. The third derivative of a function y = f(x) can be denoted by
:\frac,\quad f(x),\qu ...
may be used to define aberrancy, a metric of
non-circularity of a curve.
Main theorem of curve theory
Given functions:
:
then there exists a unique (up to transformations using the
Euclidean group) -curve which is regular of order ''n'' and has the following properties:
:
where the set
:
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
:
the Euclidean transformations are eliminated to obtain a unique curve .
Frenet–Serret formulas
The Frenet–Serret formulas are a set of
ordinary differential equations
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast w ...
of first order. The solution is the set of Frenet vectors describing the curve specified by the generalized curvature functions .
2 dimensions
:
3 dimensions
:
dimensions (general formula)
:
See also
*
List of curves topics
This is an alphabetical index of articles related to curves used in mathematics.
* Acnode
* Algebraic curve
* Arc
* Asymptote
* Asymptotic curve
* Barbier's theorem
* Bézier curve
* Bézout's theorem
* Birch and Swinnerton-Dyer conjecture
* Bi ...
References
Further reading
* Chapter II is a classical treatment of ''Theory of Curves'' in 3-dimensions.
{{tensors
Differential geometry
Curves