Differential geometry of curves
In mathematics, the differential geometry of curves provides definitions and methods to analyze smooth curves in Riemannian manifolds and Pseudo-Riemannian manifolds (and in particular in Euclidean space) using differential and integral calculus.
For example, circle in the plane can be defined as the curve c where the vector c(t)-v is always perpendicular to the tangent vector c‘(t). Or written as an inner product
To simplify the presentation we only consider curves in Euclidean space, it is straightforward to generalize these notions for Riemannian and Pseudo-Riemannian manifolds. For a more abstract curve definition in an arbitrary topological space see the main article on curves.
Let n be a natural number, r an natural number or ∞ and I be a non-empty interval of real numbers. A vector valued function
Definitions
of class Cr (i.e. c is r times continuously differentiable) is called a parametric curve of class Cr or a Cr parametrization of the curve c. t is called the parameter of the curve c.
One may think of the parameter t as representing time and the curve c(t) as the trajectory of a moving particle in space.
If I is a closed interval [a, b] we call c(a) the starting point and c(b) the endpoint of the curve c.
If c(a) = c(b) we say c is closed or a loop. Furthermore we call c a closed Cr-curve if c(k)(a) = c(k)(b) for all k ≤ r.
If c:[a,b) → Rn is injective, we call the curve simple.
If c is a parametric curve which can be locally described as a power series we call the curve analytic or of class .
We write -c to say the curve is traversed in opposite direction.
A Ck-curve
Look at curves in differential geometry to see the definitions in action.
Given the image of a curve one can define several different parametrizations of the curve. Differential geometry aims to describe properties of curves invariant under certain reparametrizations. So we have to define a suitable equivalence relation on the set all parametric curves. The differential geometric properties of a curve (length, frenet frame and generalized curvature) are invariant under reparametrization and therefore properties of the equivalence class.The equivalence classes are called Cr curves and are central objects studied in the differential geometry of curves.
Two parametric curves of class Cr
We can define an even finer equivalence relation of oriented Cr curves by requiring φ to be φ'(t) > 0.
Equivalent Cr curves have the same image. And equivalent oriented Cr curves even traverse the image in the same direction.
The length l of a smooth curve c : [a, b] → Rn can be defined as
For each regular Cr-curve c: [a, b] → Rn we can define a function
s(t) is called the natural parameter of c.
We prefer this paramtrization because the natural parameter s(t) traverses the image of c at unit speed so that
For a given parametrized curve c(t) the natural parametrization is unique up to shift of parameter.
A Frenet frame is a moving reference frame of n orthonormal vectors ei(t) which are used to describe a curve locally at each point c(t). It is the main tool in the differential geometric treatment of curves as 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 like the euclidean coordinates.
Give a Cn+1-curve c in Rn which is regular of order n the Frenet Frame for the curve is the set of orthonormal vectors
The first three Frenet vectors and generalized curvatures can be visualized in three dimensional space. The have additional names and more semantic information attached to them.
At every point of a C1 curve we can define a tangent vector. If c is interpreted as the path of a particle then the tangent vector can be visualized as path the particle takes when free from outer force.
The tangent vector is the first Frenet vector e1(t) and defined as
Since it points along the forward direction of the curve (the direction of increasing parameter), the unit tangent vector introduces an orientation of the curve.
The normal vector, sometimes called the curvature vector, indicates the deviance of the curve from being a straight line.
It is the second Frenet vector e2(t) and defined as
The first generalized curvature χ1(t) is called curvature and measures the deviance of c from being a straight line relative to the osculating plane. It is defined as
The reciprocal of the curvature
A circle with radius r has a constant curvature of
The binormal vector is the third Frenet vector e3(t)
It is always orthogonal to the unit tangent and normal vectors at t, and is defined as
The second generalized curvature χ2(t) is called torsion and measures the deviance of c from being a plane curve. Or, in other words, if the torsion is zero the curve lies completely in the osculating plane.
Reparametrization and equivalence relation
and
are said to be equivalent if there exists a bijective Cr map
such that
and
c2 is said to be a reparametrisation of c1. This reparametrisation of c1 defines the equivalence relation on the set of all parametric Cr curves. The equivalence class is called a Cr curve.The image of a curve can have two different Cr curves.
Length and natural parametrization
The length of a curve is invariant under reparametrization and therefore a differential geometric property of the curve.
Writing
we get a reparametrization of c which is called natural, arc-length or unit speed parametrization.
In practise it is often very difficult to calculate the natural paramtrization of a curve, but it is useful for theoretical arguments.Frenet frame
called Frenet vectors. They are constructed from the derivates of c(t) using the Gram-Schmidt orthogonalization algorithm with
The real valued functions χi(t) are called generalized curvature and are defined as
The frenet frame and the generalized curvatures are invariant under reparametrization and therefore differential geometric properties of the curve.Special Frenet vectors and generalized curvatures
Tangent vector
If c has a natural parameter then the equation simplifies to
The scalar magnitude of the tangent vector
is called the speed v of c at point t. If c has a natural parameter the speed is 1.Normal or curvature vector
The tangent and the normal vector at point t define the osculating plane at point t.Curvature
and is called the curvature of c at point t.
is called the curvature radius
whereas a line has a curvature of 0.Binormal vector
In 3 dimensional space the equation simplifies to
Torsion
and is called the torsion of c at point t..
Main theorem of curve theory
Given n functions
By additionally providing a start t0 in I, a starting point p0 in Rn and an inital positive orthonormal frenet frame {e1, ... , en-1} with
 = \\mathbf{p}_0.png)
The Frenet-Serrat formulas are a set of ordinary differential equations of first order. The solution is the set of frenet vectors describing the curve specificied by the generalized curvature functions χi
For a proof of the 3-dimensional case see Frenet-Serret formulas.
Frenet-Serret Formulas
2-dimensions
3-dimensions
n-dimensions (general formula)
See also