Differential geometry of curves
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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 γ where the vector γ(t) is always perpendicular to the tangent vector γ‘(t). Or written as an inner product
- <math>\langle \mathbf{\gamma}(t), \mathbf{\gamma}'(t) \rangle = 0<math>
The differential properties of many classical curves have been studied thoroughly: see the list of curves for details. The main contemporary application is in physics as part of vector calculus. In general relativity for example a world line is a curve in spacetime.
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.
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Definitions
Let n be a natural number, r an natural number or ∞, I be a non-empty interval of real numbers and t in I. A vector valued function
- <math>\mathbf{\gamma}:I \to R^n<math>
of class Cr (i.e. γ is r times continuously differentiable) is called a parametric curve of class Cr or a Cr parametrization of the curve γ. t is called the parameter of the curve γ. γ(I) is called the image of the curve.
It is important to distinguish between a curve γ and the image of a curve γ(I) because a given image can be described by several different Cr curves.
One may think of the parameter t as representing time and the curve γ(t) as the trajectory of a moving particle in space.
If I is a closed interval [a, b] we call γ(a) the starting point and γ(b) the endpoint of the curve γ.
If γ(a) = γ(b) we say γ is closed or a loop. Furthermore we call γ a closed Cr-curve if γ(k)(a) = γ(k)(b) for all k ≤ r.
If γ:(a,b) → Rn is injective, we call the curve simple.
If γ is a parametric curve which can be locally described as a power series, we call the curve analytic or of class <math>C^\omega<math>.
We write -γ to say the curve is traversed in opposite direction.
A Ck-curve
- <math>\gamma:[a,b] \rightarrow \mathbb{R}^n<math>
is called regular of order m if
- <math>\lbrace \gamma'(t), \gamma''(t), ...,\gamma^{(m)}(t) \rbrace \mbox {, } m \leq k<math>
are linearly independent in Rn.
Examples
Reparametrization and equivalence relation
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 of 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
- <math> \mathbf{\gamma_1}:I_1 \to R^n<math>
and
- <math> \mathbf{\gamma_2}:I_2 \to R^n<math>
are said to be equivalent if there exists a bijective Cr map
- <math> \phi :I_1 \to I_2<math>
such that
- <math> \phi'(t) \neq 0 \qquad (t \in I_1)<math>
and
- <math> \mathbf{\gamma_2}(\phi(t)) = \mathbf{\gamma_1}(t) \qquad (t \in I_1)<math>
γ2 is said to be a reparametrisation of γ1. This reparametrisation of γ1 defines the equivalence relation on the set of all parametric Cr curves. The equivalence class is called a Cr curve.
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.
Length and natural parametrization
The length l of a smooth curve γ : [a, b] → Rn can be defined as
- <math>l = \int_a^b \vert \mathbf{\gamma}'(t) \vert dt<math>
The length of a curve is invariant under reparametrization and therefore a differential geometric property of the curve.
For each regular Cr-curve γ: [a, b] → Rn we can define a function
- <math>s(t) = \int_{t_0}^t \vert \mathbf{\gamma}'(t) \vert dt<math>
Writing
- <math>\mathbf{\gamma}(t) = \bar\mathbf{\gamma}(s(t))<math>
we get a reparametrization <math> \bar \gamma<math>of γ which is called natural, arc-length or unit speed parametrization.
s(t) is called the natural parameter of γ.
We prefer this parametrization because the natural parameter s(t) traverses the image of γ at unit speed so that
- <math>\vert \bar\mathbf{\gamma}'(s(t)) \vert = 1 \qquad (t \in I)<math>
In practice it is often very difficult to calculate the natural parametrization of a curve, but it is useful for theoretical arguments.
For a given parametrized curve γ(t) the natural parametrization is unique up to shift of parameter.
The quantity
- <math>E(\gamma) = \frac{1}{2}\int_a^b \vert \mathbf{\gamma}'(t) \vert^2 dt<math>
is sometimes called the energy 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
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 γ(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.
Given a Cn+1-curve γ in Rn which is regular of order n 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 γ(t) using the Gram-Schmidt orthogonalization algorithm with
- <math>\mathbf{e}_1(t) = \frac{\mathbf{\gamma}'(t)}{\| \mathbf{\gamma}'(t) \|}<math>
- <math>
\mathbf{e}_{n}(t) = \frac{\overline{\mathbf{e}_{n}}(t)}{\|\overline{\mathbf{e}_{n}}(t) \|} \mbox{, } \overline{\mathbf{e}_{n}}(t) = \mathbf{\gamma}^{(n)}(t) - \sum _{i=1}^{n-1} \langle \mathbf{\gamma}^{(n)}(t), \mathbf{e}_i(t) \rangle \, \mathbf{e}_i(t) <math>
The real valued functions χi(t) are called generalized curvature and are defined as
- <math>\chi_i(t) = \frac{\langle \mathbf{e}_i'(t), \mathbf{e}_{i+1}(t) \rangle}{\| \mathbf{\gamma}^'(t) \|} <math>
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
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
At every point of a C1 curve we can define a tangent vector. If γ is interpreted as the path of a particle then the tangent vector can be visualized as the path that the particle takes when free from outer force.
The tangent vector is the first Frenet vector e1(t) and is defined as
- <math>\mathbf{e}_{1}(t) = \frac{ \mathbf{\gamma}'(t) }{ \| \mathbf{\gamma}'(t) \|}<math>
If γ has a natural parameter then the equation simplifies to
- <math>\mathbf{e}_{1}(t) = \mathbf{\gamma}'(t)<math>
The scalar magnitude of the tangent vector
- <math>v = \|\mathbf{\gamma}'(t)\|<math>
is called the speed v of γ at point t. If γ has a natural parameter the speed is 1.
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.
Normal or curvature vector
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
- <math>\mathbf{e}_2(t) = \frac{\overline{\mathbf{e}_2}(t)} {\| \overline{\mathbf{e}_2}(t) \|}
\mbox{, } \overline{\mathbf{e}_2}(t) = \mathbf{\gamma}''(t) - \langle \mathbf{\gamma}''(t), \mathbf{e}_1(t) \rangle \, \mathbf{e}_1(t)<math>
The tangent and the normal vector at point t define the osculating plane at point t.
Curvature
The first generalized curvature χ1(t) is called curvature and measures the deviance of γ from being a straight line relative to the osculating plane. It is defined as
- <math>\kappa(t) = \chi_1(t) = \frac{\langle \mathbf{e}_1'(t), \mathbf{e}_2(t) \rangle}{\| \mathbf{\gamma}^'(t) \|}<math>
and is called the curvature of γ at point t.
The reciprocal of the curvature
- <math>\frac{1}{\kappa(t)}<math>
is called the curvature radius
A circle with radius r has a constant curvature of
- <math>\kappa(t) = \frac{1}{r}<math>
whereas a line has a curvature of 0.
Binormal vector
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
- <math>\mathbf{e}_3(t) = \frac{\overline{\mathbf{e}_3}(t)} {\| \overline{\mathbf{e}_3}(t) \|}
\mbox{, } \overline{\mathbf{e}_3}(t) = \mathbf{\gamma}'''(t) - \langle \mathbf{\gamma}'''(t), \mathbf{e}_1(t) \rangle \, \mathbf{e}_1(t) - \langle \mathbf{\gamma}'''(t), \mathbf{e}_2(t) \rangle \,\mathbf{e}_2(t) <math>
In 3 dimensional space the equation simplifies to
- <math>\mathbf{e}_3(t) = \mathbf{e}_2(t) \times \mathbf{e}_1(t)<math>
Torsion
The second generalized curvature χ2(t) is called torsion and measures the deviance of γ from being a plane curve. Or, in other words, if the torsion is zero the curve lies completely in the osculating plane.
- <math>\tau(t) = \chi_2(t) = \frac{\langle \mathbf{e}_2'(t), \mathbf{e}_3(t) \rangle}{\| \mathbf{\gamma}'(t) \|}<math>
and is called the torsion of γ at point t..
Main theorem of curve theory
Given n functions
- <math>\chi_i \in C^{n-i}([a,b]) \mbox{, } 1 \leq i \leq n<math>
with
- <math>\chi_i(t) > 0 \mbox{, } 1 \leq i \leq n-1<math>
then there exists a unique (up to transformations using the Euclidean group) Cn+1-curve γ which is regular of order n and has the following properties
- <math>\|\gamma'(t)\| = 1 \mbox{ } (t \in [a,b])<math>
- <math>\chi_i(t) = \frac{ \langle \mathbf{e}_i'(t), \mathbf{e}_{i+1}(t) \rangle}{\| \mathbf{\gamma}'(t) \|}<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 t0 in I, a starting point p0 in Rn and an initial positive orthonormal Frenet frame {e1, ..., en-1} with
- <math>\mathbf{\gamma}(t_0) = \mathbf{p}_0<math>
- <math>\mathbf{e}_i(t_0) = \mathbf{e}_i \mbox{, } 1 \leq i \leq n-1<math>
we can eliminate the Euclidean transformations and get unique curve γ.
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 χi
For a proof of the 3-dimensional case see Frenet-Serret formulas.
2-dimensions
- <math>
\begin{bmatrix}
\mathbf{e}_1'(t)\\
\mathbf{e}_2'(t) \\
\end{bmatrix}
=
\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}
=
\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>
n-dimensions (general formula)
- <math>
\begin{bmatrix}
\mathbf{e}_1'(t)\\
\vdots \\
\mathbf{e}_n'(t) \\
\end{bmatrix}
=
\begin{bmatrix}
0 & \chi_1(t) & & 0 \\
-\chi_1(t) & \ddots & \ddots & \\
& \ddots & 0 & \chi_{n-1}(t) \\
0 & & -\chi_{n-1}(t) & 0 \\
\end{bmatrix}
\begin{bmatrix}
\mathbf{e}_1(t) \\
\vdots \\
\mathbf{e}_n(t) \\
\end{bmatrix} <math>
See also
- Osculating circle
- Curve
- Curvature
- Torsion
- Arc
- Parameter, parametrization
- Implicit function
- Tangent, contact, subtangent
- Frenet-Serret formulas
- Envelope, evolute, involute, pedal curve, roulette
- Four-vertex theorem
- Geodesic
- geodesic curvature
- Isoperimetry
- Moving frame
- List of curve topics
- List of curves