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| :''This article is about the curvature of affine plane curves, not to be confused with the [[curvature]] of an [[affine connection]].''
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| '''Special affine curvature''', also known as the '''equi-affine curvature''' or '''affine curvature''', is a particular type of [[curvature]] that is defined on a plane [[curve]] that remains unchanged under a [[special affine group|special affine transformation]] (an [[affine transformation]] that preserves [[area]]). The curves of constant equi-affine curvature ''k'' are precisely all non-singular [[conic section|plane conics]]. Those with ''k'' > 0 are [[ellipse]]s, those with ''k'' = 0 are [[parabola]]s, and those with ''k'' < 0 are [[hyperbola]]s.
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| The usual Euclidean curvature of a curve at a point is the curvature of its [[osculating circle]], the unique circle making second order contact (having three point contact) with the curve at the point. In the same way, | |
| the special affine curvature of a curve at a point ''P'' is the special affine curvature of its '''hyperosculating conic''', which is the unique conic making fourth order [[contact (mathematics)|contact]] (having five point contact) with the curve at ''P''. In other words it is the
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| limiting position of the (unique) conic through ''P'' and four points ''P''<sub>1</sub>, ''P''<sub>2</sub>, ''P''<sub>3</sub>, ''P''<sub>4</sub> on the curve, as each of the points approaches ''P'':
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| :<math>P_1,P_2,P_3,P_4\to P.</math>
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| In some contexts, the '''affine curvature''' refers to a differential invariant κ of the [[affine group|general affine group]], which may readily obtained from the special affine curvature ''k'' by κ = ''k''<sup>−3/2</sup>d''k''/d''s'', where ''s'' is the special affine arc length. Where the general affine group is not used, the special affine curvature ''k'' is sometimes also called the affine curvature {{harv|Shirokov|2001b}}.
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| ==Formal definition==
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| ===Special affine arclength===
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| To define the special affine curvature, it is necessary first to define the '''special affine arclength''' (also called the '''equi-affine arclength'''). Consider an affine plane curve <math>\beta (t)</math>. Choose co-ordinates for the affine plane such that the area of the parallelogram spanned by two vectors <math>a = (a_1, \; a_2)</math> and <math>b = (b_1, \; b_2)</math> is given by the [[determinant]]
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| :<math>\det\left[ a\; b \right] = a_{1} b_{2} - a_{2} b_{1}.</math> | |
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| In particular, the determinant
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| :<math>\det\begin{bmatrix}\frac{d\beta}{dt} & \frac{d^2\beta}{dt^2}\end{bmatrix}</math>
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| is a well-defined invariant of the special affine group, and gives the signed area of the parallelogram spanned by the velocity and acceleration of the curve β. Consider a reparameterization of the curve β, say with a new parameter ''s'' related to ''t'' by means of a regular reparameterization ''s'' = ''s''(''t''). This determinant undergoes then a transformation of the following sort, by the [[chain rule]]:
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| :<math>\begin{align}
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| \det\begin{bmatrix}\frac{d\beta}{dt} & \frac{d^2\beta}{dt^2}\end{bmatrix} &= \det\begin{bmatrix}\frac{d\beta}{ds}\frac{ds}{dt} & \left(\frac{d^2\beta}{ds^2}\left(\frac{ds}{dt}\right)^2+\frac{d\beta}{ds}\frac{d^2s}{dt^2}\right)\end{bmatrix}\\
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| &=\left(\frac{ds}{dt}\right)^3\det\begin{bmatrix}\frac{d\beta}{ds} & \frac{d^2\beta}{ds^2}\end{bmatrix}.
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| \end{align}</math>
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| The reparameterization can be chosen so that
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| :<math>\det\begin{bmatrix}\frac{d\beta}{ds} & \frac{d^2\beta}{ds^2}\end{bmatrix} = 1</math>
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| provided the velocity and acceleration, dβ/d''t'' and d<sup>2</sup>β/d''t''<sup>2</sup> are [[linearly independent]]. Existence and uniqueness of such a parameterization follows by integration:
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| :<math>s(t) = \int_a^t\sqrt[3]{\det\begin{bmatrix}\frac{d\beta}{dt} & \frac{d^2\beta}{dt^2}\end{bmatrix}}\,\,dt. </math>
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| This integral is called the '''special affine arclength''', and a curve carrying this parameterization is said to be parameterized with respect to its special affine arclength.
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| ===Special affine curvature===
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| Suppose that β(''s'') is a curve parameterized with its special affine arclength. Then the '''special affine curvature''' (or '''equi-affine curvature''') is given by
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| :<math>k(s) = \det\begin{bmatrix}\beta''(s) & \beta'''(s) \end{bmatrix}.</math>
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| Here β′ denotes the derivative of β with respect to ''s''.
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| More generally ({{harvnb|Guggenheimer|1977|loc=§7.3}}; {{harvnb|Blaschke|1923|loc=§5}}), for a plane curve with arbitrary parameterization
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| :<math>t \mapsto (x(t), y(t)),</math>
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| the special affine curvature is:
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| :<math>
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| \begin{align}
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| k(t)&=\frac{x''y'''-x'''y''}{(x'y''-x''y')^{5/3}}-\frac{1}{2}\left[\frac{1}{(x'y''-x''y')^{2/3}}\right]''\\
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| &= \frac{4(x''y'''-x'''y'')+(x'y''''-x''''y')}{3(x'y''-x''y')^{5/3}} -\frac{5}{9}\frac{(x'y'''-x'''y')^2}{(x'y''-x''y')^{8/3}}
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| \end{align}</math>
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| provided the first and second derivatives of the curve are linearly independent. In the special case of a graph ''y'' = ''y''(''x''), these formulas reduce to
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| :<math>k=-\frac{1}{2}\left(\frac{1}{(y'')^{2/3}}\right)''=\frac{1}{3}\frac{y''''}{(y'')^{5/3}}-\frac{5}{9}\frac{(y''')^2}{(y'')^{8/3}}</math>
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| where the prime denotes differentiation with respect to ''x'' ({{harvnb|Blaschke|1923|loc=§5}}; {{harvnb|Shirokov|2001a}}).
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| ===Affine curvature===
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| Suppose as above that β(''s'') is a curve parameterized by special affine arclength. There are a pair of invariants of the curve that are invariant under the full general affine group {{harv|Shirokov|2001b}} — the group of all affine motions of the plane, not just those that are area-preserving. The first of these is
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| :<math>\sigma = \int \sqrt{k(s)}\, ds,</math>
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| sometimes called the ''affine arclength'' (although this risks confusion with the special affine arclength described above). The second is referred to as the ''affine curvature'':
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| :<math>\kappa = \frac{1}{k^{3/2}} \frac{dk}{ds}.</math>
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| ==Conics==
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| Suppose that β(''s'') is a curve parameterized by special affine arclength with constant affine curvature ''k''. Let
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| :<math>C_\beta(s) = \begin{bmatrix}\beta'(s) & \beta''(s)\end{bmatrix}.</math>
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| Note that det ''C''<sub>β</sub>, since β is assumed to carry the special affine arclength parameterization, and that
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| :<math>k = \det(C_\beta').\,</math>
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| It follows from the form of ''C''<sub>β</sub> that
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| :<math>C_\beta' = C_\beta\begin{bmatrix}0&-k\\1&0\end{bmatrix}.</math>
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| By applying a suitable special affine transformation, we can arrange that ''C''<sub>β</sub>(0) = ''I'' is the identity matrix. Since ''k'' is constant, it follows that ''C''<sub>β</sub> is given by the [[matrix exponential]]
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| :<math>\begin{align}
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| C_\beta(s) &= \exp\left\{s\cdot\begin{bmatrix}0&-k\\1&0\end{bmatrix}\right\}\\
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| &=\begin{bmatrix}\cos\sqrt{k}s&\sqrt{k}\sin\sqrt{k}s\\ -\frac{1}{\sqrt{k}}\sin\sqrt{k}s&\cos\sqrt{k}s\end{bmatrix}.
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| \end{align}
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| </math>
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| The three cases are now as follows.
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| ;''k'' = 0
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| If the curvature vanishes identically, then upon passing to a limit,
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| :<math>C_\beta(s) = \begin{bmatrix}1&0\\s&1\end{bmatrix}</math>
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| so β'(''s'') = (1,s), and so integration gives
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| :<math>\beta(s)=(s,s^2/2)\,</math>
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| up to an overall constant translation, which is the special affine parameterization of the parabola ''y'' = ''x''<sup>2</sup>/2. | |
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| ;''k'' > 0
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| If the special affine curvature is positive, then it follows that
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| :<math>\beta'(s) = \left(\cos\sqrt{k}s,\frac{1}{\sqrt{k}}\sin\sqrt{k}s\right)</math>
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| so that
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| :<math>\beta(s) = \left(\frac{1}{\sqrt{k}}\sin\sqrt{k}s, -\frac{1}{k}\cos\sqrt{k}s\right)</math>
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| up to a translation, which is the special affine parameterization of the ellipse ''kx''<sup>2</sup> + ''k''<sup>2</sup>''y''<sup>2</sup> = 1.
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| ;''k'' < 0
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| If ''k'' is negative, then the trigonometric functions in ''C''<sub>β</sub> give way to [[hyperbolic function]]s:
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| :<math>C_\beta(s) =\begin{bmatrix}\cosh\sqrt{|k|}s&\sqrt{|k|}\sinh\sqrt{|k|}s\\ \frac{1}{\sqrt{|k|}}\sinh\sqrt{|k|}s&\cosh\sqrt{|k|}s\end{bmatrix}.
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| </math>
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| Thus
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| :<math>\beta(s) = \left(\frac{1}{\sqrt{|k|}}\sinh\sqrt{|k|}s,\frac{1}{|k|}\cosh\sqrt{|k|}s\right)</math>
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| up to a translation, which is the special affine parameterization of the hyperbola
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| :<math>-|k|x^2 + |k|^2y^2 = 1.</math>
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| ==Characterization up to affine congruence==
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| The special affine curvature of an immersed curve is the only (local) invariant of the curve in the following sense:
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| *If two curves have the same special affine curvature at every point, then one curve is obtained from the other by means of a special affine transformation.
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| In fact, a slightly stronger statement holds:
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| *Given any continuous function ''k'' : [''a'',''b''] → '''R''', there exists a curve β whose first and second derivatives are linearly independent, such that the special affine curvature of β relative to the special affine parameterization is equal to the given function ''k''. The curve β is uniquely determined up to a special affine transformation.
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| This is analogous to the fundamental theorem of curves in the classical Euclidean [[differential geometry of curves]], in which the complete classification of plane curves up to Euclidean motion depends on a single function κ, the curvature of the curve. It follows essentially by applying the [[Picard–Lindelöf theorem]] to the system
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| :<math>C_\beta' = C_\beta\begin{bmatrix}0&-k\\1&0\end{bmatrix}</math>
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| where ''C''<sub>β</sub> = [β′ β′′]. An alternative approach, rooted in the theory of [[moving frame]]s, is to apply the existence of a primitive for the [[Darboux derivative]].
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| ==Derivation of the curvature by affine invariance==
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| The special affine curvature can be derived explicitly by techniques of [[invariant theory]]. For simplicity, suppose that an affine plane curve is given in the form of a graph ''y'' = ''y''(''x''). The special affine group acts on the Cartesian plane via transformations of the form
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| :<math>\begin{align}
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| x&\mapsto ax+by + \alpha\\
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| y&\mapsto cx+dy + \beta,
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| \end{align}
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| </math>
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| with ''ad'' − ''bc'' = 1. The following [[vector field]]s span the [[Lie algebra]] of infinitesimal generators of the special affine group:
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| :<math>T_1 = \partial_x, \quad T_2 = \partial y</math>
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| :<math>X_1 = x\partial_y, \quad X_2 = y\partial_x, \quad H=x\partial_x - y\partial_y.</math>
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| An affine transformation not only acts on points, but also on the tangent lines to graphs of the form ''y'' = ''y''(''x''). That is, there is an action of the special affine group on triples of coordinates
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| :<math>(x,y,y').\,</math>
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| The group action is generated by vector fields
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| :<math>T_1^{(1)},T_2^{(1)},X_1^{(1)},X_2^{(1)},H^{(1)}</math>
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| defined on the space of three variables (''x'',''y'',''y''′). These vector fields can be determined by the following two requirements:
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| * Under the projection onto the ''xy''-plane, they must to project to the corresponding original generators of the action <math>T_1,T_2,X_1,X_2,H</math>, respectively.
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| * The vectors must preserve up to scale the [[contact structure]] of the [[jet (mathematics)|jet space]]
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| ::<math>\theta_1 = dy - y'dx.</math>
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| :Concretely, this means that the generators ''X''<sup>(1)</sup> must satisfy
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| ::<math>L_{X^{(1)}}\theta_1 \equiv 0 \pmod{\theta_1}</math> | |
| :where ''L'' is the [[Lie derivative]].
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| Similarly, the action of the group can be extended to the space of any number of derivatives
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| :<math>(x,y,y',y'',\dots,y^{(k)}).</math>
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| The prolonged vector fields generating the action of the special affine group must then inductively satisfy, for each generator ''X'' ∈ {''T''<sub>1</sub>,''T''<sub>2</sub>,''X''<sub>1</sub>,''X''<sub>2</sub>,''H''}:
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| * The projection of ''X''<sup>(k)</sup> onto the space of variables (''x'',''y'',''y''′,…,''y''<sup>(''k''−1)</sup>) is ''X''<sup>(''k''−1)</sup>.
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| * ''X''<sup>(''k'')</sup> preserves the contact ideal:
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| ::<math>L_{X^{(k)}}\theta_k \equiv 0 \pmod{\theta_1,\dots, \theta_k}</math>
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| :where
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| ::<math>\theta_i = dy^{(i-1)} - y^{(i)}dx.</math>
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| Carrying out the inductive construction up to order 4 gives
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| :<math>T_1^{(4)} = \partial_x, \quad T_2^{(4)} = \partial_y</math>
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| :<math>X_1^{(4)} = x\partial_y + \partial_{y'}</math>
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| :<math>\begin{align}X_2^{(4)} = y\partial_x&-y'^2\partial_{y'}-3y'y''\partial_{y''}-(3y''^2+4y'y''')\partial_{y'''}\\
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| &-(10y''y'''+5y'y'''')\partial_{y''''}
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| \end{align}</math>
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| :<math>H^{(4)} = x\partial_x - y\partial_y - 2y'\partial_{y'} - 3y''\partial_{y''}-4y'''\partial_{y'''}-5y''''\partial_{y''''}.</math>
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| The special affine curvature
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| :<math>k=\frac{1}{3}\frac{y''''}{(y'')^{5/3}}-\frac{5}{9}\frac{(y''')^2}{(y'')^{8/3}}</math>
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| does not depend explicitly on ''x'', ''y'', or ''y''′, and so satisfies
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| :<math>T_1^{(4)}k=T_2^{(4)}k=X_1^{(4)}k=0.</math>
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| The vector field ''H'' acts diagonally as a modified [[homogeneity operator]], and it is readily verified that ''H''<sup>(4)</sup>''k'' = 0. Finally,
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| :<math>X_2^{(4)}k = \frac{1}{2}[H,X 1]^{(4)}k = \frac{1}{2}[H^{(4)},X 1^{(4)}]k = 0.</math> | |
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| The five vector fields
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| :<math>T_1^{(4)},T_2^{(4)},X_1^{(4)},X_2^{(4)},H^{(4)}</math>
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| form an involutive distribution on (an open subset of) '''R'''<sup>6</sup> so that, by the [[Frobenius integration theorem]], they integrate locally to give a foliation of '''R'''<sup>6</sup> by five-dimensional leaves. Concretely, each leaf is a local orbit of the special affine group. The function ''k'' parameterizes these leaves.
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| ==Human Motor System==
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| Human curvilinear 2-dimensional drawing movements tend to follow the equi-affine parametrization.<ref name="flashHandzel">{{cite journal|authors=Flash,Tamar; Handzel,Amir A|title = Affine differential geometry analysis of human arm movements|journal = Biological cybernetics|year = 2007|volume = 96|pages=577–601}} </ref> This is more commonly known as the two thirds power law, according to which the hand's speed is proportional to the Euclidean curvature raised to the minus third power.<ref name="lacquaniti">{{cite journal|authors=Lacquaniti, Francesco and Terzuolo, Carlo and Viviani, Paolo|title = The law relating the kinematic and figural aspects of drawing movements|journal = Acta psychologica|year = 1983|volume = 54|pages=115–130}} </ref> Namely,
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| :<math> v = \gamma \kappa^{-\frac{1}{3}}, </math>
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| where <math>v</math> is the speed of the hand, <math>\kappa</math> is the Euclidean curvature and <math>\gamma</math> is a constant termed the velocity gain factor.
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| ==See also==
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| *[[Affine geometry of curves]]
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| *[[Affine sphere]]
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| ==References==
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| <references />
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| *{{Citation | last=Blaschke|first=Wilhelm|authorlink=Wilhelm Blaschke|title=Affine Differentialgeometrie|series=Vorlesungen über Differentialgeometrie und geometrische Grundlagen von Einsteins Relativitätstheorie|volume=II|year=1923|publisher=Springer-Verlag OHG|publication-place=Berlin}}
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| *{{Citation | last1=Guggenheimer | first1=Heinrich | title=Differential Geometry | publisher=[[Dover Publications]] | location=New York | isbn=978-0-486-63433-3 | year=1977}}
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| *{{springer|id=A/a010980|first=A.P.|last= Shirokov|title=Affine curvature|year=2001a}}
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| *{{springer|id=a/a010990|title=Affine differential geometry|first=A.P.|last= Shirokov|year=2001b}}
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| *{{Citation | last1=Spivak | first1=Michael | author1-link=Michael Spivak | title=A Comprehensive introduction to differential geometry (Volume 2) | publisher=Publish or Perish | location=Houston, TX | isbn=978-0-914098-71-3 | year=1999}}
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| {{Reflist}}
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| [[Category:Differential geometry]]
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| [[Category:Curves]]
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| [[Category:Affine geometry]]
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