In
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 ...
, the Frenet–Serret formulas describe the
kinematic
Kinematics is a subfield of physics, developed in classical mechanics, that describes the motion of points, bodies (objects), and systems of bodies (groups of objects) without considering the forces that cause them to move. Kinematics, as a fie ...
properties of a particle moving along a differentiable
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 ...
in three-dimensional
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 ...
, or the geometric properties of the curve itself irrespective of any motion. More specifically, the formulas describe the
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 of the so-called tangent, normal, and binormal
unit vector
In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat").
The term ''direction vecto ...
s in terms of each other. The formulas are named after the two French mathematicians who independently discovered them:
Jean Frédéric Frenet
Jean Frédéric Frenet (; 7 February 1816 – 12 June 1900) was a French mathematician, astronomer, and meteorologist. He was born and died in Périgueux, France.
He is best known for being an independent co-discoverer of the Frenet–Serre ...
, in his thesis of 1847, and
Joseph Alfred Serret
Joseph Alfred Serret (; August 30, 1819 – March 2, 1885) was a French mathematician who was born in Paris, France, and died in Versailles, France.
See also
*Frenet–Serret formulas
In differential geometry, the Frenet–Serret formulas des ...
, in 1851. Vector notation and linear algebra currently used to write these formulas were not yet available at the time of their discovery.
The tangent, normal, and binormal unit vectors, often called T, N, and B, or collectively the Frenet–Serret frame or TNB frame, together form an
orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, ...
spanning and are defined as follows:
* T is the unit vector
tangent
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
to the curve, pointing in the direction of motion.
* N is the
normal Normal(s) or The Normal(s) may refer to:
Film and television
* ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson
* ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie
* ''Norma ...
unit vector, the derivative of T with respect to the
arclength parameter of the curve, divided by its length.
* B is the binormal unit vector, the
cross product
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and is ...
of T and N.
The Frenet–Serret formulas are:
:
where ''d''/''ds'' is the derivative with respect to arclength, ''κ'' is 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 ''τ'' is 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 the curve. The two
scalars ''κ'' and ''τ'' effectively define the curvature and torsion of a space curve. The associated collection, T, N, B, ''κ'', and ''τ'', is called the Frenet–Serret apparatus. Intuitively, curvature measures the failure of a curve to be a straight line, while torsion measures the failure of a curve to be planar.
Definitions
Let r(''t'') be 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.
Intuitively, a curve may be thought of as the trace left by a moving point (ge ...
in
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 ...
, representing the
position vector of the particle as a function of time. The Frenet–Serret formulas apply to curves which are ''non-degenerate'', which roughly means that they have nonzero
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 ...
. More formally, in this situation the
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 ...
vector r′(''t'') and the
acceleration
In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by the ...
vector r′′(''t'') are required not to be proportional.
Let ''s''(''t'') represent 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
* ...
which the particle has moved along the
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 ...
in time ''t''. The quantity ''s'' is used to give the curve traced out by the trajectory of the particle a
natural parametrization by arc length (i.e.
arc-length parametrization
Differential geometry of curves is the branch of geometry that deals with smoothness (mathematics), smooth curves in the Euclidean plane, plane and the Euclidean space by methods of differential calculus, differential and integral calculus.
Many ...
), since many different particle paths may trace out the same geometrical curve by traversing it at different rates. In detail, ''s'' is given by
:
Moreover, since we have assumed that r′ ≠ 0, it follows that ''s''(''t'') is a strictly monotonically increasing function. Therefore, it is possible to solve for ''t'' as a function of ''s'', and thus to write r(''s'') = r(''t''(''s'')). The curve is thus parametrized in a preferred manner by its arc length.
With a non-degenerate curve r(''s''), parameterized by its arc length, it is now possible to define the Frenet–Serret frame (or TNB frame):
-
The tangent unit vector T is defined as
.
-
The normal unit vector N is defined as
from which it follows, since T always has unit
magnitude
Magnitude may refer to:
Mathematics
*Euclidean vector, a quantity defined by both its magnitude and its direction
*Magnitude (mathematics), the relative size of an object
*Norm (mathematics), a term for the size or length of a vector
*Order of ...
, that N (the change of T) is always perpendicular to T, since there is no change in length of T. Note that by calling curvature we automatically obtain the first relation.
-
The binormal unit vector B is defined as the
cross product
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and is ...
of T and N:
,
from which it follows that B is always perpendicular to both T and N. Thus, the three unit vectors T, N, and B are all perpendicular to each other.
The Frenet–Serret formulas are:
:
where
is 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
is 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 ...
.
The Frenet–Serret formulas are also known as ''Frenet–Serret theorem'', and can be stated more concisely using matrix notation:
:
This matrix is
skew-symmetric.
Formulas in ''n'' dimensions
The Frenet–Serret formulas were generalized to higher-dimensional Euclidean spaces by
Camille Jordan
Marie Ennemond Camille Jordan (; 5 January 1838 – 22 January 1922) was a French mathematician, known both for his foundational work in group theory and for his influential ''Cours d'analyse''.
Biography
Jordan was born in Lyon and educated at ...
in 1874.
Suppose that r(''s'') is a smooth curve in
, and that the first ''n'' derivatives of r are linearly independent. The vectors in the Frenet–Serret frame are an
orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, ...
constructed by applying the
Gram-Schmidt process to the vectors (r′(''s''), r′′(''s''), ..., r
(''n'')(''s'')).
In detail, the unit tangent vector is the first Frenet vector ''e''
1(''s'') and is defined as
:
where
:
The normal vector, 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 e
2(''s'') and defined as
:
The tangent and the normal vector at point ''s'' 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 r(''s'').
The remaining vectors in the frame (the binormal, trinormal, etc.) are defined similarly by
:
:
The last vector in the frame is defined by the cross-product of the first
vectors:
:
The real valued functions used below χ
''i''(''s'') are called generalized curvature and are defined as
:
The Frenet–Serret formulas, stated in matrix language, are
:
Notice that as defined here, the generalized curvatures and the frame may differ slightly from the convention found in other sources.
The top curvature
(also called the torsion, in this context) and the last vector in the frame
,
differ by a sign
:
(the orientation of the basis) from the usual torsion.
The Frenet–Serret formulas are invariant under flipping the sign of both
and
, and this change of sign makes the frame positively oriented. As defined above, the frame inherits its orientation from the jet of
.
Proof
Consider the 3 by 3 matrix
:
The rows of this matrix are mutually perpendicular unit vectors: an
orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, ...
of . As a result, the
transpose
In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal;
that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations).
The tr ...
of ''Q'' is equal to the
inverse of ''Q'': ''Q'' is an
orthogonal matrix
In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors.
One way to express this is
Q^\mathrm Q = Q Q^\mathrm = I,
where is the transpose of and is the identity ma ...
. It suffices to show that
:
Note the first row of this equation already holds, by definition of the normal N and curvature ''κ'', as well as the last row by the definition
of torsion. So it suffices to show that ''Q''
T is a
skew-symmetric matrix
In mathematics, particularly in linear algebra, a skew-symmetric (or antisymmetric or antimetric) matrix is a square matrix whose transpose equals its negative. That is, it satisfies the condition
In terms of the entries of the matrix, if a_ ...
. Since ''I'' = ''QQ''
T, taking a derivative and applying the product rule yields
:
which establishes the required skew-symmetry.
Applications and interpretation
Kinematics of the frame
The Frenet–Serret frame consisting of the tangent T, normal N, and binormal B collectively forms an
orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, ...
of 3-space. At each point of the curve, this ''attaches'' a
frame of reference
In physics and astronomy, a frame of reference (or reference frame) is an abstract coordinate system whose origin, orientation, and scale are specified by a set of reference points― geometric points whose position is identified both mathema ...
or
rectilinear coordinate system
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
(see image).
The Frenet–Serret formulas admit a
kinematic
Kinematics is a subfield of physics, developed in classical mechanics, that describes the motion of points, bodies (objects), and systems of bodies (groups of objects) without considering the forces that cause them to move. Kinematics, as a fie ...
interpretation. Imagine that an observer moves along the curve in time, using the attached frame at each point as their coordinate system. The Frenet–Serret formulas mean that this coordinate system is constantly rotating as an observer moves along the curve. Hence, this coordinate system is always
non-inertial. The
angular momentum
In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
of the observer's coordinate system is proportional to the
Darboux vector In differential geometry, especially the theory of space curves, the Darboux vector is the angular velocity vector (geometric), vector of the Frenet-Serret formulas, Frenet frame of a space curve. It is named after Gaston Darboux who discovered it. ...
of the frame.
Concretely, suppose that the observer carries an (inertial)
top
A spinning top, or simply a top, is a toy with a squat body and a sharp point at the bottom, designed to be spun on its vertical axis, balancing on the tip due to the gyroscopic effect.
Once set in motion, a top will usually wobble for a few ...
(or
gyroscope
A gyroscope (from Ancient Greek γῦρος ''gŷros'', "round" and σκοπέω ''skopéō'', "to look") is a device used for measuring or maintaining orientation and angular velocity. It is a spinning wheel or disc in which the axis of rota ...
) with them along the curve. If the axis of the top points along the tangent to the curve, then it will be observed to rotate about its axis with angular velocity -τ relative to the observer's non-inertial coordinate system. If, on the other hand, the axis of the top points in the binormal direction, then it is observed to rotate with angular velocity -κ. This is easily visualized in the case when the curvature is a positive constant and the torsion vanishes. The observer is then in
uniform circular motion
In physics, circular motion is a movement of an object along the circumference of a circle or rotation along a circular path. It can be uniform, with constant angular rate of rotation and constant speed, or non-uniform with a changing rate of rot ...
. If the top points in the direction of the binormal, then by
conservation of angular momentum
In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed system ...
it must rotate in the ''opposite'' direction of the circular motion. In the limiting case when the curvature vanishes, the observer's normal
precess
Precession is a change in the orientation of the rotational axis of a rotating body. In an appropriate reference frame it can be defined as a change in the first Euler angle, whereas the third Euler angle defines the rotation itself. In othe ...
es about the tangent vector, and similarly the top will rotate in the opposite direction of this precession.
The general case is illustrated
below. There are further
illustrations
An illustration is a decoration, interpretation or visual explanation of a text, concept or process, designed for integration in print and digital published media, such as posters, flyers, magazines, books, teaching materials, animations, video ...
on Wikimedia.
Applications
The kinematics of the frame have many applications in the sciences.
* In the
life sciences
This list of life sciences comprises the branches of science that involve the scientific study of life – such as microorganisms, plants, and animals including human beings. This science is one of the two major branches of natural science, the ...
, particularly in models of microbial motion, considerations of the Frenet–Serret frame have been used to explain the mechanism by which a moving organism in a viscous medium changes its direction.
* In physics, the Frenet–Serret frame is useful when it is impossible or inconvenient to assign a natural coordinate system for a trajectory. Such is often the case, for instance, in
relativity theory
The theory of relativity usually encompasses two interrelated theories by Albert Einstein: special relativity and general relativity, proposed and published in 1905 and 1915, respectively. Special relativity applies to all physical phenomena in ...
. Within this setting, Frenet–Serret frames have been used to model the precession of a gyroscope in a gravitational well.
Graphical Illustrations
# Example of a moving Frenet basis (T in blue, N in green, B in purple) along
Viviani's curve
In mathematics, Viviani's curve, also known as Viviani's window, is a figure eight shaped space curve named after the Italian mathematician Vincenzo Viviani. It is the intersection of a sphere with a cylinder that is tangent to the sphere and p ...
.
#
On the example of a torus knot
In knot theory, a torus knot is a special kind of knot that lies on the surface of an unknotted torus in R3. Similarly, a torus link is a link which lies on the surface of a torus in the same way. Each torus knot is specified by a pair of cop ...
, the tangent vector T, the normal vector N, and the binormal vector B, along with the curvature κ(s), and the torsion τ(s) are displayed.
At the peaks of the torsion function the rotation of the Frenet–Serret frame (T,N,B) around the tangent vector is clearly visible.
#
The kinematic significance of the curvature is best illustrated with plane curves (having constant torsion equal to zero). See the page on curvature of plane curves.
Frenet–Serret formulas in calculus
The Frenet–Serret formulas are frequently introduced in courses on
multivariable calculus
Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one variable to calculus with functions of several variables: the differentiation and integration of functions involving several variables, rather th ...
as a companion to the study of space curves such as the
helix
A helix () is a shape like a corkscrew or spiral staircase. It is a type of smooth space curve with tangent lines at a constant angle to a fixed axis. Helices are important in biology, as the DNA molecule is formed as two intertwined helices, ...
. A helix can be characterized by the height 2π''h'' and radius ''r'' of a single turn. The curvature and torsion of a helix (with constant radius) are given by the formulas
:
:
The sign of the torsion is determined by the right-handed or left-handed
sense
A sense is a biological system used by an organism for sensation, the process of gathering information about the world through the detection of Stimulus (physiology), stimuli. (For example, in the human body, the brain which is part of the cen ...
in which the helix twists around its central axis. Explicitly, the parametrization of a single turn of a right-handed helix with height 2π''h'' and radius ''r'' is
: ''x'' = ''r'' cos ''t''
: ''y'' = ''r'' sin ''t''
: ''z'' = ''h'' ''t''
: (0 ≤ t ≤ 2 π)
and, for a left-handed helix,
: ''x'' = ''r'' cos ''t''
: ''y'' = −''r'' sin ''t''
: ''z'' = ''h'' ''t''
: (0 ≤ t ≤ 2 π).
Note that these are not the arc length parametrizations (in which case, each of ''x'', ''y'', and ''z'' would need to be divided by
.)
In his expository writings on the geometry of curves,
Rudy Rucker
Rudolf von Bitter Rucker (; born March 22, 1946) is an American mathematician, computer scientist, science fiction author, and one of the founders of the cyberpunk literary movement. The author of both fiction and non-fiction, he is best known f ...
employs the model of a
slinky
The Slinky is a helical spring toy invented by Richard James in the early 1940s. It can perform a number of tricks, including travelling down a flight of steps end-over-end as it stretches and re-forms itself with the aid of gravity and its ow ...
to explain the meaning of the torsion and curvature. The slinky, he says, is characterized by the property that the quantity
:
remains constant if the slinky is vertically stretched out along its central axis. (Here 2π''h'' is the height of a single twist of the slinky, and ''r'' the radius.) In particular, curvature and torsion are complementary in the sense that the torsion can be increased at the expense of curvature by stretching out the slinky.
Taylor expansion
Repeatedly differentiating the curve and applying the Frenet–Serret formulas gives the following
Taylor approximation
In calculus, Taylor's theorem gives an approximation of a ''k''-times differentiable function around a given point by a polynomial of degree ''k'', called the ''k''th-order Taylor polynomial. For a smooth function, the Taylor polynomial is the t ...
to the curve near ''s'' = 0:
:
For a generic curve with nonvanishing torsion, the projection of the curve onto various coordinate planes in the T, N, B coordinate system at have the following interpretations:
*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 ...
'' is the plane
containing T and N. The projection of the curve onto this plane has the form:
This is a
parabola
In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves.
One descript ...
up to terms of order ''o''(''s''
2), whose curvature at 0 is equal to ''κ''(0).
*The ''
normal plane'' is the plane containing N and B. The projection of the curve onto this plane has the form:
which is a
cuspidal cubic
In mathematics, a cuspidal cubic or semicubical parabola is an algebraic plane curve that has an implicit equation of the form
: y^2 - a^2 x^3 = 0
(with ) in some Cartesian coordinate system.
Solving for leads to the ''explicit form''
: y = \ ...
to order ''o''(''s''
3).
*The rectifying plane is the plane containing T and B. The projection of the curve onto this plane is:
which traces out the graph of a
cubic polynomial
In mathematics, a cubic function is a function of the form f(x)=ax^3+bx^2+cx+d
where the coefficients , , , and are complex numbers, and the variable takes real values, and a\neq 0. In other words, it is both a polynomial function of degree ...
to order ''o''(''s''
3).
Ribbons and tubes
The Frenet–Serret apparatus allows one to define certain optimal ''ribbons'' and ''tubes'' centered around a curve. These have diverse applications in
materials science and
elasticity theory
In physics and materials science, elasticity is the ability of a body to resist a distorting influence and to return to its original size and shape when that influence or force is removed. Solid objects will deform when adequate loads are ...
, as well as to
computer graphics
Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, cell phone and computer displays, and many specialized applications. A great de ...
.
The Frenet ribbon along a curve ''C'' is the surface traced out by sweeping the line segment
minus;N,Ngenerated by the unit normal along the curve. This surface is sometimes confused with the
tangent developable
In the mathematical study of the differential geometry of surfaces, a tangent developable is a particular kind of developable surface obtained from a curve in Euclidean space as the surface swept out by the tangent lines to the curve. Such a surf ...
, which is the
envelope
An envelope is a common packaging item, usually made of thin, flat material. It is designed to contain a flat object, such as a letter or card.
Traditional envelopes are made from sheets of paper cut to one of three shapes: a rhombus, a sh ...
''E'' of the osculating planes of ''C''. This is perhaps because both the Frenet ribbon and ''E'' exhibit similar properties along ''C''. Namely, the tangent planes of both sheets of ''E'', near the singular locus ''C'' where these sheets intersect, approach the osculating planes of ''C''; the tangent planes of the Frenet ribbon along ''C'' are equal to these osculating planes. The Frenet ribbon is in general not developable.
Congruence of curves
In classical
Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small ...
, one is interested in studying the properties of figures in the plane which are ''invariant'' under congruence, so that if two figures are congruent then they must have the same properties. The Frenet–Serret apparatus presents the curvature and torsion as numerical invariants of a space curve.
Roughly speaking, two curves ''C'' and ''C''′ in space are ''congruent'' if one can be rigidly moved to the other. A rigid motion consists of a combination of a translation and a rotation. A translation moves one point of ''C'' to a point of ''C''′. The rotation then adjusts the orientation of the curve ''C'' to line up with that of ''C''′. Such a combination of translation and rotation is called a
Euclidean motion
In mathematics, a rigid transformation (also called Euclidean transformation or Euclidean isometry) is a geometric transformation of a Euclidean space that preserves the Euclidean distance between every pair of points.
The rigid transformation ...
. In terms of the parametrization r(''t'') defining the first curve ''C'', a general Euclidean motion of ''C'' is a composite of the following operations:
* (''Translation'') r(''t'') → r(''t'') + v, where v is a constant vector.
* (''Rotation'') r(''t'') + v → ''M''(r(''t'') + v), where ''M'' is the matrix of a rotation.
The Frenet–Serret frame is particularly well-behaved with regard to Euclidean motions. First, since T, N, and B can all be given as successive derivatives of the parametrization of the curve, each of them is insensitive to the addition of a constant vector to r(''t''). Intuitively, the TNB frame attached to r(''t'') is the same as the TNB frame attached to the new curve .
This leaves only the rotations to consider. Intuitively, if we apply a rotation ''M'' to the curve, then the TNB frame also rotates. More precisely, the matrix ''Q'' whose rows are the TNB vectors of the Frenet–Serret frame changes by the matrix of a rotation
:
''A fortiori'', the matrix ''Q''
T is unaffected by a rotation:
:
since for the matrix of a rotation.
Hence the entries ''κ'' and τ of ''Q''
T are ''invariants'' of the curve under Euclidean motions: if a Euclidean motion is applied to a curve, then the resulting curve has ''the same'' curvature and torsion.
Moreover, using the Frenet–Serret frame, one can also prove the converse: any two curves having the same curvature and torsion functions must be congruent by a Euclidean motion. Roughly speaking, the Frenet–Serret formulas express the
Darboux derivative The Darboux derivative of a map between a manifold and a Lie group is a variant of the standard derivative. It is arguably a more natural generalization of the single-variable derivative. It allows a generalization of the single-variable fundament ...
of the TNB frame. If the Darboux derivatives of two frames are equal, then a version of the
fundamental theorem of calculus
The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each time) with the concept of integrating a function (calculating the area under its graph, or ...
asserts that the curves are congruent. In particular, the curvature and torsion are a ''complete'' set of invariants for a curve in three-dimensions.
Other expressions of the frame
The formulas given above for T, N, and B depend on the curve being given in terms of the arclength parameter. This is a natural assumption in Euclidean geometry, because the arclength is a Euclidean invariant of the curve. In the terminology of physics, the arclength parametrization is a natural choice of
gauge
Gauge ( or ) may refer to:
Measurement
* Gauge (instrument), any of a variety of measuring instruments
* Gauge (firearms)
* Wire gauge, a measure of the size of a wire
** American wire gauge, a common measure of nonferrous wire diameter, ...
. However, it may be awkward to work with in practice. A number of other equivalent expressions are available.
Suppose that the curve is given by r(''t''), where the parameter ''t'' need no longer be arclength. Then the unit tangent vector T may be written as
:
The normal vector N takes the form
:
The binormal B is then
:
An alternative way to arrive at the same expressions is to take the first three derivatives of the curve r′(''t''), r′′(''t''), r′′′(''t''), and to apply the
Gram-Schmidt process. The resulting ordered
orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, ...
is precisely the TNB frame. This procedure also generalizes to produce Frenet frames in higher dimensions.
In terms of the parameter ''t'', the Frenet–Serret formulas pick up an additional factor of , , r′(''t''), , because of the
chain rule
In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ...
:
:
Explicit expressions for the curvature and torsion may be computed. For example,
:
The torsion may be expressed using a
scalar triple product
In geometry and algebra, the triple product is a product of three 3-dimensional vectors, usually Euclidean vectors. The name "triple product" is used for two different products, the scalar-valued scalar triple product and, less often, the vector- ...
as follows,
:
Special cases
If the curvature is always zero then the curve will be a straight line. Here the vectors N, B and the torsion are not well defined.
If the torsion is always zero then the curve will lie in a plane.
A curve may have nonzero curvature and zero torsion. For example, the
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...
of radius ''R'' given by r(''t'')=(''R'' cos ''t'', ''R'' sin ''t'', 0) in the ''z''=0 plane has zero torsion and curvature equal to 1/''R''. The converse, however, is false. That is, a regular curve with nonzero torsion must have nonzero curvature. (This is just the contrapositive of the fact that zero curvature implies zero torsion.)
A
helix
A helix () is a shape like a corkscrew or spiral staircase. It is a type of smooth space curve with tangent lines at a constant angle to a fixed axis. Helices are important in biology, as the DNA molecule is formed as two intertwined helices, ...
has constant curvature and constant torsion.
Plane curves
Given a curve contained on the ''x''-''y'' plane, its tangent vector T is also contained on that plane. Its binormal vector B can be naturally postulated to coincide with the normal ''to the plane'' (along the ''z'' axis). Finally, the curve normal can be found completing the right-handed system, N = B × T.
This form is well-defined even when the curvature is zero; for example, the normal to a straight line in a plane will be perpendicular to the tangent, all co-planar.
See also
*
Affine geometry of curves In the mathematical field of differential geometry, the affine geometry of curves is the study of curves in an affine space, and specifically the properties of such curves which are invariant under the special affine group \mbox(n,\mathbb) \ltimes ...
*
Differential geometry of curves
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 ...
*
Darboux frame In the differential geometry of surfaces, a Darboux frame is a natural moving frame constructed on a surface. It is the analog of the Frenet–Serret frame as applied to surface geometry. A Darboux frame exists at any non-umbilic point of a s ...
*
Kinematics
Kinematics is a subfield of physics, developed in classical mechanics, that describes the Motion (physics), motion of points, Physical object, bodies (objects), and systems of bodies (groups of objects) without considering the forces that cause ...
*
Moving frame
In mathematics, a moving frame is a flexible generalization of the notion of an ordered basis of a vector space often used to study the extrinsic differential geometry of smooth manifolds embedded in a homogeneous space.
Introduction
In lay te ...
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Tangential and normal components
Notes
References
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* . Abstract in ''Journal de Mathématiques Pures et Appliquées'' 17, 1852.
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External links
Create your own animated illustrations of moving Frenet-Serret frames, curvature and torsion functions(
Maple
''Acer'' () is a genus of trees and shrubs commonly known as maples. The genus is placed in the family Sapindaceae.Stevens, P. F. (2001 onwards). Angiosperm Phylogeny Website. Version 9, June 2008 nd more or less continuously updated since http ...
Worksheet)
Rudy Rucker's KappaTau Paper
{{DEFAULTSORT:Frenet-Serret formulas
Differential geometry
Multivariable calculus
Curves
Curvature (mathematics)