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| :''This article considers only curves in Euclidean space. Most of the notions presented here have analogues for curves in [[Riemannian manifold|Riemannian]] and [[pseudo-Riemannian manifold|pseudo-Riemannian]] [[differentiable manifold|manifolds]]. For a discussion of curves in an arbitrary [[topological space]], see the main article on [[curve]]s.''
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| '''Differential geometry of curves''' is the branch of [[geometry]] that deals
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| with smooth [[curve]]s in the [[Euclidean plane|plane]] and in the [[Euclidean space]] by methods of [[differential calculus|differential]] and [[integral calculus]].
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| Starting in antiquity, many [[list of curves|concrete curves]] have been thoroughly investigated using the synthetic approach. {{Clarify|reason=what is the synthetic approach?|date=May 2011}} [[Differential geometry]] takes another path: curves are represented in a [[parametric equation|parametrized form]], and their geometric properties and various quantities associated with them, such as the [[curvature]] and the [[arc length]], are expressed via [[derivative]]s and [[integral]]s using [[vector calculus]]. 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.
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| The theory of curves is much simpler and narrower in scope than the [[differential geometry of surfaces|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'') and from the point of view of a bug 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 the way in which they bend and twist. Quantitatively, this is measured by the differential-geometric invariants called the ''[[curvature]]'' and the ''[[torsion of curves|torsion]]'' of a curve. The [[fundamental theorem of curves]] asserts that the knowledge of these invariants completely determines the curve.
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| == Definitions ==
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| {{main|Curve}}
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| Let ''n'' be a natural number, ''r'' a natural number or ∞, ''I'' be a [[Set (mathematics)|non-empty]] [[Interval (mathematics)|interval]] of real numbers and ''t'' in ''I''. A [[vector-valued function]]
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| :<math>\mathbf{\gamma}:I \to {\mathbb R}^n</math>
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| of class ''C''<sup>''r''</sup> (i.e. γ is ''r'' times [[Smooth function|continuously differentiable]]) is called a '''parametric curve of class C<sup>r</sup>''' or a ''C''<sup>''r''</sup> 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 ''C''<sup>''r''</sup> curves.
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| One may think of the parameter ''t'' as representing time and the curve γ(''t'') as the [[trajectory]] of a moving particle in space.
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| If ''I'' is a closed interval [''a'', ''b''], we call γ(''a'') the '''starting point''' and γ(''b'') the '''endpoint''' of the curve γ.
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| If γ(''a'') = γ(''b''), we say γ is '''closed''' or a '''loop'''. Furthermore, we call γ a '''closed C<sup>r</sup>-curve''' if γ<sup>(''k'')</sup>(a) = γ<sup>(''k'')</sup>(''b'') for all ''k'' ≤ ''r''.
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| If γ:(''a'',''b'') → '''R'''<sup>''n''</sup> is [[injective]], we call the curve '''simple'''.
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| 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>.
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| We write -γ to say the curve is traversed in opposite direction.
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| A ''C''<sup>''k''</sup>-curve
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| :<math>\gamma:[a,b] \rightarrow \mathbb{R}^n</math>
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| is called '''regular of order m''' if for any ''t'' in interval ''I''
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| :<math>\lbrace \gamma'(t), \gamma''(t), ...,\gamma^{(m)}(t) \rbrace \mbox {, } m \leq k</math>
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| are [[linear independence|linearly independent]] in '''R'''<sup>''n''</sup>.
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| In particular, a ''C''<sup>1</sup>-curve ''γ'' is '''regular''' if
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| :<math>\gamma'(t) \neq 0</math> for any <math>t \in I</math>.
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| == Reparametrization and equivalence relation ==
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| {{See also|Position vector|Vector-valued function}}
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| Given the image of a curve one can define several different parameterizations 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|Frenet frame]] and generalized curvature) are invariant under reparametrization and therefore properties of the [[equivalence class]].The equivalence classes are called '''C<sup>r</sup> curves''' and are central objects studied in the differential geometry of curves.
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| Two parametric curves of class ''C''<sup>''r''</sup>
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| :<math> \mathbf{\gamma_1}:I_1 \to R^n</math>
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| and
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| :<math> \mathbf{\gamma_2}:I_2 \to R^n</math>
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| are said to be '''equivalent''' if there exists a bijective ''C''<sup>''r''</sup> map
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| :<math> \phi :I_1 \to I_2</math>
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| such that
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| :<math> \phi'(t) \neq 0 \qquad (t \in I_1)</math>
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| and
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| :<math> \mathbf{\gamma_2}(\phi(t)) = \mathbf{\gamma_1}(t) \qquad (t \in I_1)</math>
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| γ<sub>2</sub> is said to be a '''reparametrisation''' of γ<sub>1</sub>. This reparametrisation of γ<sub>1</sub> defines the equivalence relation on the set of all parametric ''C''<sup>''r''</sup> curves. The equivalence class is called a '''C<sup>r</sup> curve'''.
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| We can define an even ''finer'' equivalence relation of '''oriented C<sup>r</sup> curves''' by requiring φ to be φ‘(''t'') > 0.
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| Equivalent ''C''<sup>''r''</sup> curves have the same image. And equivalent oriented ''C''<sup>''r''</sup> curves even traverse the image in the same direction.
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| ==Length and natural parametrization==
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| {{main|Arc length}}
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| :''See also: [[Curve#Lengths of curves|Lengths of Curves]]''
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| The length ''l'' of a curve ''γ'' : [''a'', ''b''] → '''R'''<sup>''n''</sup> of class ''C''<sup>1</sup> can be defined as
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| :<math>l = \int_a^b \vert \mathbf{\gamma}'(t) \vert dt.</math>
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| The length of a curve is invariant under reparametrization and therefore a differential geometric property of the curve.
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| For each regular ''C''<sup>''r''</sup>-curve (''r'' at least 1) ''γ'': [''a'', ''b''] → '''R'''<sup>''n''</sup> we can define a function
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| :<math>s(t) = \int_{t_0}^t \vert \mathbf{\gamma}'(x) \vert dx.</math>
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| Writing
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| :<math>\bar{\mathbf{\gamma}}(s) = \gamma(t(s))</math>
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| where ''t''(''s'') is the inverse of ''s''(''t''), we get a reparametrization <math> \bar{\gamma}</math> of γ which is called '''natural''', '''arc-length''' or '''unit speed''' parametrization. The parameter ''s''(''t'') is called the '''natural parameter''' of γ.
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| This parametrization is preferred because the natural parameter ''s''(''t'') traverses the image of γ at unit speed so that
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| :<math>\vert \bar{\mathbf{\gamma}}'(s(t)) \vert = 1 \qquad (t \in I).</math>
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| In practice it is often very difficult to calculate the natural parametrization of a curve, but it is useful for theoretical arguments.
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| For a given parametrized curve γ(''t'') the natural parametrization is unique up to shift of parameter.
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| The quantity
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| :<math>E(\gamma) = \frac{1}{2}\int_a^b \vert \mathbf{\gamma}'(t) \vert^2 dt</math>
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| is sometimes called the '''energy''' or [[action (physics)|action]] of the curve; this name is justified because the [[geodesic]] equations are the [[Euler–Lagrange equation]]s of motion for this action.
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| == Frenet frame ==
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| {{see also|Frenet–Serret formulas}}
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| [[Image:Frenet frame.png|thumb|right|An illustration of the Frenet frame for a point on a space curve. T is the unit tangent, P the unit normal, and B the unit binormal.]]
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| A '''Frenet frame''' is a [[Moving frame|moving reference frame]] of ''n'' [[orthonormal]] vectors ''e''<sub>''i''</sub>(''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.
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| Given a ''C''<sup>''n''+1</sup>-curve γ in '''R'''<sup>''n''</sup> which is regular of order ''n'' the '''Frenet frame''' for the curve is the set of orthonormal vectors
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| :<math>\mathbf{e}_1(t), \ldots, \mathbf{e}_n(t)</math>
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| called '''[[Frenet–Serret formulas|Frenet vectors]]'''. They are constructed from the derivatives of γ(''t'') using the [[Gram–Schmidt process|Gram–Schmidt orthogonalization algorithm]] with
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| :<math>\mathbf{e}_1(t) = \frac{\mathbf{\gamma}'(t)}{\| \mathbf{\gamma}'(t) \|}</math> | |
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| :<math>
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| \mathbf{e}_{j}(t) = \frac{\overline{\mathbf{e}_{j}}(t)}{\|\overline{\mathbf{e}_{j}}(t) \|}
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| \mbox{, }
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| \overline{\mathbf{e}_{j}}(t) = \mathbf{\gamma}^{(j)}(t) - \sum _{i=1}^{j-1} \langle \mathbf{\gamma}^{(j)}(t), \mathbf{e}_i(t) \rangle \, \mathbf{e}_i(t)
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| </math>
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| The real-valued functions χ<sub>''i''</sub>(''t'') are called '''generalized curvatures''' and are defined as
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| :<math>\chi_i(t) = \frac{\langle \mathbf{e}_i'(t), \mathbf{e}_{i+1}(t) \rangle}{\| \mathbf{\gamma}^'(t) \|} </math>
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| The Frenet frame and the generalized curvatures are invariant under reparametrization and are therefore differential geometric properties of the curve.
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| == Special Frenet vectors and generalized curvatures ==
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| {{move section portions|Frenet–Serret formulas|date=July 2013}}
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| 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.
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| === Tangent vector ===
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| If a curve γ represents the path of a particle then the instantaneous [[velocity]] of the particle at a given point ''P'' is expressed by a [[Vector (geometric)|vector]], called the '''tangent vector''' to the curve at ''P''. Mathematically, given a parametrized ''C''<sup>1</sup> curve γ = γ(''t''), for every value ''t'' = ''t''<sub>0</sub> of the parameter, the vector
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| : <math> \gamma'(t_0) = \frac{d}{d\,t}\mathbf{\gamma}(t)</math> at <math> {t=t_0} </math>
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| is the tangent vector at the point ''P'' = γ(''t''<sub>0</sub>). Generally speaking, the tangent vector may be [[zero vector|zero]]. The magnitude of the tangent vector,
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| :<math>\|\mathbf{\gamma}'(t_0)\|,</math>
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| is the speed at the time ''t''<sub>0</sub>.
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| The first Frenet vector ''e''<sub>1</sub>(''t'') is the '''unit tangent vector''' in the same direction, defined at each regular point of γ:
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| :<math>\mathbf{e}_{1}(t) = \frac{ \mathbf{\gamma}'(t) }{ \| \mathbf{\gamma}'(t) \|}.</math>
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| If ''t'' = ''s'' is the natural parameter then the tangent vector has unit length, so that the formula simplifies:
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| :<math>\mathbf{e}_{1}(s) = \mathbf{\gamma}'(s).</math>
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| 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.
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| === Normal or curvature vector ===
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| The '''normal vector''', sometimes called the '''curvature vector''', indicates the deviance of the curve from being a straight line.
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| It is defined as
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| :<math>\overline{\mathbf{e}_2}(t) = \mathbf{\gamma}''(t) - \langle \mathbf{\gamma}''(t), \mathbf{e}_1(t) \rangle \, \mathbf{e}_1(t).</math>
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| Its normalized form, the '''unit normal vector''', is the second Frenet vector ''e''<sub>2</sub>(''t'') and defined as
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| :<math>\mathbf{e}_2(t) = \frac{\overline{\mathbf{e}_2}(t)} {\| \overline{\mathbf{e}_2}(t) \|}.
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| </math>
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| The tangent and the normal vector at point ''t'' define the [[osculating plane]] at point ''t''.
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| ===Curvature===
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| {{main|Curvature}}
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| The first generalized curvature χ<sub>1</sub>(''t'') is called '''curvature''' and measures the deviance of γ from being a straight line relative to the osculating plane. It is defined as
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| :<math>\kappa(t) = \chi_1(t) = \frac{\langle \mathbf{e}_1'(t), \mathbf{e}_2(t) \rangle}{\| \mathbf{\gamma}^'(t) \|}</math>
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| and is called the [[curvature]] of γ at point ''t''.
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| The [[Multiplicative inverse|reciprocal]] of the curvature
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| :<math>\frac{1}{\kappa(t)}</math>
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| is called the '''radius of curvature'''.
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| A circle with radius ''r'' has a constant curvature of
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| :<math>\kappa(t) = \frac{1}{r}</math>
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| whereas a line has a curvature of 0.
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| === Binormal vector ===
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| The '''binormal vector''' is the third Frenet vector ''e''<sub>3</sub>(''t'').
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| It is always orthogonal to the '''unit''' tangent and normal vectors at ''t'', and is defined as
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| :<math>\mathbf{e}_3(t) = \frac{\overline{\mathbf{e}_3}(t)} {\| \overline{\mathbf{e}_3}(t) \|} | |
| \mbox{, }
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| \overline{\mathbf{e}_3}(t) = \mathbf{\gamma}'''(t) - \langle \mathbf{\gamma}'''(t), \mathbf{e}_1(t) \rangle \, \mathbf{e}_1(t)
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| - \langle \mathbf{\gamma}'''(t), \mathbf{e}_2(t) \rangle \,\mathbf{e}_2(t)
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| </math>
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| In 3-dimensional space the equation simplifies to
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| :<math>\mathbf{e}_3(t) = \mathbf{e}_1(t) \times \mathbf{e}_2(t)</math> | |
| or to
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| :<math>\mathbf{e}_3(t) = -\mathbf{e}_1(t) \times \mathbf{e}_2(t)</math>
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| That either sign may occur is illustrated by the examples of a right handed helix and a left handed helix.
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| === Torsion ===
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| {{main|Torsion of a curve}}
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| The second generalized curvature χ<sub>2</sub>(''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 same osculating plane (there is only one osculating plane for every point ''t''). It is defined as
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| :<math>\tau(t) = \chi_2(t) = \frac{\langle \mathbf{e}_2'(t), \mathbf{e}_3(t) \rangle}{\| \mathbf{\gamma}'(t) \|}</math>
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| and is called the [[torsion (differential geometry)|torsion]] of γ at point ''t''.
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| == Main theorem of curve theory ==
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| {{main|Fundamental theorem of curves}}
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| Given ''(n-1)'' functions:
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| <math>\chi_i \in C^{n-i}([a,b],\mathbb{R}^n) \mbox{, } 1 \leq i \leq n-1</math>
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| 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]]) ''C''<sup>''n''+1</sup>-curve γ which is regular of order ''n'' and has the following properties
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| :<math>\|\gamma'(t)\| = 1 \mbox{ } (t \in [a,b])</math>
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| :<math>\chi_i(t) = \frac{ \langle \mathbf{e}_i'(t), \mathbf{e}_{i+1}(t) \rangle}{\| \mathbf{\gamma}'(t) \|}</math>
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| where the set
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| :<math>\mathbf{e}_1(t), \ldots, \mathbf{e}_n(t)</math>
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| is the Frenet frame for the curve. | |
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| By additionally providing a start ''t''<sub>0</sub> in ''I'', a starting point ''p''<sub>0</sub> in '''R'''<sup>''n''</sup> and an initial positive orthonormal Frenet frame {''e''<sub>1</sub>, ..., ''e''<sub>''n''-1</sub>} with
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| :<math>\mathbf{\gamma}(t_0) = \mathbf{p}_0</math>
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| :<math>\mathbf{e}_i(t_0) = \mathbf{e}_i \mbox{, } 1 \leq i \leq n-1</math>
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| we can eliminate the Euclidean transformations and get unique curve γ.
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| == Frenet–Serret formulas ==
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| {{main|Frenet–Serret formulas}}
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| 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 χ<sub>''i''</sub>
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| === 2 dimensions ===
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| :<math>
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| \begin{bmatrix}
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| \mathbf{e}_1'(t) \\
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| \mathbf{e}_2'(t) \\
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| \end{bmatrix}
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| =
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| \left\Vert \gamma'\left(t\right) \right\Vert
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| \begin{bmatrix}
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| 0 & \kappa(t) \\
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| -\kappa(t) & 0 \\
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| \end{bmatrix}
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| \begin{bmatrix}
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| \mathbf{e}_1(t) \\
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| \mathbf{e}_2(t) \\
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| \end{bmatrix}
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| </math>
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| === 3 dimensions ===
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| :<math> | |
| \begin{bmatrix}
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| \mathbf{e}_1'(t) \\
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| \mathbf{e}_2'(t) \\
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| \mathbf{e}_3'(t) \\
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| \end{bmatrix}
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| =
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| \left\Vert \gamma'\left(t\right) \right\Vert
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| \begin{bmatrix}
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| 0 & \kappa(t) & 0 \\
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| -\kappa(t) & 0 & \tau(t) \\
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| 0 & -\tau(t) & 0 \\
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| \end{bmatrix}
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| \begin{bmatrix}
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| \mathbf{e}_1(t) \\
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| \mathbf{e}_2(t) \\
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| \mathbf{e}_3(t) \\
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| \end{bmatrix}
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| </math>
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| === ''n'' dimensions (general formula) ===
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| :<math>
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| \begin{bmatrix}
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| \mathbf{e}_1'(t) \\
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| \mathbf{e}_2'(t) \\
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| \vdots \\
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| \mathbf{e}_{n-1}'(t) \\
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| \mathbf{e}_n'(t) \\
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| \end{bmatrix}
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| =
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| \left\Vert \gamma'\left(t\right) \right\Vert
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| \begin{bmatrix}
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| 0 & \chi_1(t) & \cdots & 0 & 0 \\
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| -\chi_1(t) & 0 & \cdots & 0 & 0 \\
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| \vdots & \vdots & \ddots & \vdots & \vdots \\
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| 0 & 0 & \cdots & 0 & \chi_{n-1}(t) \\
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| 0 & 0 & \cdots & -\chi_{n-1}(t) & 0 \\
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| \end{bmatrix}
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| \begin{bmatrix}
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| \mathbf{e}_1(t) \\
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| \mathbf{e}_2(t) \\
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| \vdots \\
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| \mathbf{e}_{n-1}(t) \\
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| \mathbf{e}_n(t) \\
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| \end{bmatrix}
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| </math>
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| == See also ==
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| <!--- Many wrong or ambiguous links, links duplication with the main text, self-links--->
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| <div style="-moz-column-count:3; column-count:3;">
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| *'''[[List of curve topics]]'''
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| </div>
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| ==Additional reading==
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| *Erwin Kreyszig, ''Differential Geometry'', Dover Publications, New York, 1991, ISBN 0-486-66721-9. Chapter II is a classical treatment of ''Theory of Curves'' in 3-dimensions.
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| {{Differential transforms of plane curves}}
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| {{Curvature}}
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| [[Category:Differential geometry]]
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| [[Category:Curves]]
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