Affine curvature

Template:Distinguish Special affine curvature, also known as the equiaffine curvature or affine curvature, is a particular type of curvature that is defined on a plane curve that remains unchanged under a special affine transformation (an affine transformation that preserves area). The curves of constant equiaffine curvature Template:Mvar are precisely all non-singular plane conics. Those with Template:Math are ellipses, those with Template:Math are parabolae, and those with Template:Math are hyperbolae.

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 Template:Mvar is the special affine curvature of its hyperosculating conic, which is the unique conic making fourth order contact (having five point contact) with the curve at Template:Mvar. In other words, it is the limiting position of the (unique) conic through Template:Mvar and four points Template:Math on the curve, as each of the points approaches Template:Mvar:

$P_1,P_2,P_3,P_4\to P.$

In some contexts, the affine curvature refers to a differential invariant Template:Mvar of the general affine group, which may readily obtained from the special affine curvature Template:Mvar by Template:Math, where Template:Mvar is the special affine arc length. Where the general affine group is not used, the special affine curvature Template:Mvar is sometimes also called the affine curvature.Template:Sfn

Formal definition

Special affine arclength

To define the special affine curvature, it is necessary first to define the special affine arclength (also called the equiaffine arclength). Consider an affine plane curve Template:Math. Choose coordinates for the affine plane such that the area of the parallelogram spanned by two vectors Template:Math and Template:Math is given by the determinant

$\det \left[\begin{array}{cc}a& b\end{array}\right]=a_1{b}_2-a_2{b}_1.$

In particular, the determinant

$\det \left[\begin{array}{cc}\frac{d\beta }{dt}& \frac{d^2\beta }{d{t}^2}\end{array}\right]$

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 Template:Mvar. Consider a reparameterization of the curve Template:Mvar, say with a new parameter Template:Mvar related to Template:Mvar by means of a regular reparameterization Template:Math. This determinant undergoes then a transformation of the following sort, by the chain rule:

$\begin{array}{cc}\det \left[\begin{array}{cc}\frac{d\beta }{dt}& \frac{d^2\beta }{d{t}^2}\end{array}\right]& =\det \left[\begin{array}{cc}\frac{d\beta }{ds}\frac{ds}{dt}& (\frac{d^2\beta }{d{s}^2}{\left(\frac{ds}{dt}\right)}^2+\frac{d\beta }{ds}\frac{d^{2}s}{d{t}^2})\end{array}\right]\\ & ={\left(\frac{ds}{dt}\right)}^3\det \left[\begin{array}{cc}\frac{d\beta }{ds}& \frac{d^2\beta }{d{s}^2}\end{array}\right].\end{array}$

The reparameterization can be chosen so that

$\det \left[\begin{array}{cc}\frac{d\beta }{ds}& \frac{d^2\beta }{d{s}^2}\end{array}\right]=1$

provided the velocity and acceleration, Template:Math and Template:Math are linearly independent. Existence and uniqueness of such a parameterization follows by integration:

$s(t)=\underset{a}{\overset{t}{\int }}\sqrt[3]{\det \left[\begin{array}{cc}\frac{d\beta }{dt}& \frac{d^2\beta }{d{t}^2}\end{array}\right]}dt.$

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.

Special affine curvature

Suppose that Template:Math is a curve parameterized with its special affine arclength. Then the special affine curvature (or equiaffine curvature) is given by

$k(s)=\det \left[\begin{array}{cc}{\beta }^{″}(s)& {\beta }^{‴}(s)\end{array}\right].$

Here Template:Math denotes the derivative of Template:Mvar with respect to Template:Mvar.

More generally,Template:SfnTemplate:Sfn for a plane curve with arbitrary parameterization

$t\mapsto (x(t),y(t)),$

the special affine curvature is:

$\begin{array}{cc}k(t)& =\frac{x^{″}{y}^{‴}-x^{‴}{y}^{″}}{{(x^{\prime }{y}^{″}-x^{″}{y}^{\prime })}^{\frac{5}{3}}}-\frac{1}{2}\left(\frac{1}{{(x^{\prime }{y}^{″}-x^{″}{y}^{\prime })}^{\frac{2}{3}}}\right)\\ [6px]& =\frac{4(x^{″}{y}^{‴}-x^{‴}{y}^{″})+(x^{\prime }{y}^{⁗}-x^{⁗}{y}^{\prime })}{3{(x^{\prime }{y}^{″}-x^{″}{y}^{\prime })}^{\frac{5}{3}}}-\frac{5{(x^{\prime }{y}^{‴}-x^{‴}{y}^{\prime })}^2}{9{(x^{\prime }{y}^{″}-x^{″}{y}^{\prime })}^{\frac{8}{3}}}\end{array}$

provided the first and second derivatives of the curve are linearly independent. In the special case of a graph Template:Math, these formulas reduce to

$k=-\frac{1}{2}\left(\frac{1}{{\left(y^{″}\right)}^{\frac{2}{3}}}\right)=\frac{y^{⁗}}{3{\left(y^{″}\right)}^{\frac{5}{3}}}-\frac{5{\left(y^{‴}\right)}^2}{9{\left(y^{″}\right)}^{\frac{8}{3}}}$

where the prime denotes differentiation with respect to Template:Mvar.Template:SfnTemplate:Sfn

Affine curvature

Suppose as above that Template:Math 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 groupTemplate:Sfn — the group of all affine motions of the plane, not just those that are area-preserving. The first of these is

$\sigma =\int \sqrt{k(s)}ds,$

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:

$\kappa =k^{-\frac{3}{2}}\frac{dk}{ds}.$

Conics

Suppose that Template:Math is a curve parameterized by special affine arclength with constant affine curvature Template:Mvar. Let

$C_{\beta }(s)=\left[\begin{array}{cc}{\beta }^{\prime }(s)& {\beta }^{″}(s)\end{array}\right].$

Note that Template:Math since Template:Mvar is assumed to carry the special affine arclength parameterization, and that

$k=\det \left({C_{\beta }}^{\prime }\right).$

It follows from the form of Template:Math that

${C_{\beta }}^{\prime }=C_{\beta }\left[\begin{array}{cc}0& -k\\ 1& 0\end{array}\right].$

By applying a suitable special affine transformation, we can arrange that Template:Math is the identity matrix. Since Template:Mvar is constant, it follows that Template:Math is given by the matrix exponential

$\begin{array}{cc}{C}_{\beta }(s)& =\exp \{s\cdot \left[\begin{array}{cc}0& -k\\ 1& 0\end{array}\right]\}\\ & =\left[\begin{array}{cc}\cos \sqrt{k}s& \sqrt{k}\sin \sqrt{k}s\\ -\frac{1}{\sqrt{k}}\sin \sqrt{k}s& \cos \sqrt{k}s\end{array}\right].\end{array}$

The three cases are now as follows.

Template:Math

If the curvature vanishes identically, then upon passing to a limit,

$C_{\beta }(s)=\left[\begin{array}{cc}1& 0\\ s& 1\end{array}\right]$

so Template:Math, and so integration gives

$\beta (s)=(s,\frac{s^2}{2})$

up to an overall constant translation, which is the special affine parameterization of the parabola Template:Math.

Template:Math

If the special affine curvature is positive, then it follows that

${\beta }^{\prime }(s)=(\cos \sqrt{k}s,\frac{1}{\sqrt{k}}\sin \sqrt{k}s)$

so that

$\beta (s)=(\frac{1}{\sqrt{k}}\sin \sqrt{k}s,-\frac{1}{k}\cos \sqrt{k}s)$

up to a translation, which is the special affine parameterization of the ellipse Template:Math.

Template:Math

If Template:Mvar is negative, then the trigonometric functions in Template:Math give way to hyperbolic functions:

$C_{\beta }(s)=\left[\begin{array}{cc}\cosh \sqrt{|k|}s& \sqrt{|k|}\sinh \sqrt{|k|}s\\ \frac{1}{\sqrt{|k|}}\sinh \sqrt{|k|}s& \cosh \sqrt{|k|}s\end{array}\right].$

Thus

$\beta (s)=(\frac{1}{\sqrt{|k|}}\sinh \sqrt{|k|}s,\frac{1}{|k|}\cosh \sqrt{|k|}s)$

up to a translation, which is the special affine parameterization of the hyperbola

$-|k|x^2+|k{|}^2{y}^2=1.$

Characterization up to affine congruence

The special affine curvature of an immersed curve is the only (local) invariant of the curve in the following sense:

• 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.

In fact, a slightly stronger statement holds:

• Given any continuous function Template:Math, there exists a curve Template:Mvar whose first and second derivatives are linearly independent, such that the special affine curvature of Template:Mvar relative to the special affine parameterization is equal to the given function Template:Mvar. The curve Template:Mvar is uniquely determined up to a special affine transformation.

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 Template:Mvar, the curvature of the curve. It follows essentially by applying the Picard–Lindelöf theorem to the system

${C_{\beta }}^{\prime }=C_{\beta }\left[\begin{array}{cc}0& -k\\ 1& 0\end{array}\right]$

where Template:Math. An alternative approach, rooted in the theory of moving frames, is to apply the existence of a primitive for the Darboux derivative.

Derivation of the curvature by affine invariance

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 Template:Math. The special affine group acts on the Cartesian plane via transformations of the form

$\begin{array}{cc}x& \mapsto ax+by+\alpha \\ y& \mapsto cx+dy+\beta ,\end{array}$

with Template:Math. The following vector fields span the Lie algebra of infinitesimal generators of the special affine group:

$\begin{array}{cccc}{T}_1& ={\partial }_x,& T_2& ={\partial }_y\\ X_1& =x{\partial }_y,& X_2& =y{\partial }_x,& H& =x{\partial }_x-y{\partial }_y.\end{array}$

An affine transformation not only acts on points, but also on the tangent lines to graphs of the form Template:Math. That is, there is an action of the special affine group on triples of coordinates Template:Math. The group action is generated by vector fields

$T_1^{(1)},T_2^{(1)},X_1^{(1)},X_2^{(1)},H^{(1)}$

defined on the space of three variables Template:Math. These vector fields can be determined by the following two requirements:

• Under the projection onto the Template:Mvar-plane, they must to project to the corresponding original generators of the action Template:Math, respectively.
• The vectors must preserve up to scale the contact structure of the jet space

${\theta }_1=dy-y^{\prime }dx.$

Concretely, this means that the generators Template:Math must satisfy

$L_{X^{(1)}}{\theta }_1\equiv 0(\mathrm{mod}{\theta }_1)$

where Template:Mvar is the Lie derivative.

Similarly, the action of the group can be extended to the space of any number of derivatives Template:Math.

The prolonged vector fields generating the action of the special affine group must then inductively satisfy, for each generator Template:Math:

• The projection of Template:Math onto the space of variables Template:Math is Template:Math.
• Template:Math preserves the contact ideal:

$L_{X^{(k)}}{\theta }_k\equiv 0(\mathrm{mod}{\theta }_1,\dots ,{\theta }_k)$

where

${\theta }_i=d{y}^{(i-1)}-y^{(i)}dx.$

Carrying out the inductive construction up to order 4 gives

$\begin{array}{cc}{T}_1^{(4)}& ={\partial }_x,T_2^{(4)}={\partial }_y\\ X_1^{(4)}& =x{\partial }_y+{\partial }_{y^{\prime }}\\ X_2^{(4)}& =y{\partial }_x-y{\text{'}}^2{\partial }_{y^{\prime }}-3y^{\prime }{y}^{″}{\partial }_{y^{″}}-(3y^{\prime }{\text{'}}^2+4y^{\prime }{y}^{‴}){\partial }_{y^{‴}}-(10y^{″}{y}^{‴}+5y^{\prime }{y}^{⁗}){\partial }_{y^{⁗}}\\ H^{(4)}& =x{\partial }_x-y{\partial }_y-2y^{\prime }{\partial }_{y^{\prime }}-3y^{″}{\partial }_{y^{″}}-4y^{‴}{\partial }_{y^{‴}}-5y^{⁗}{\partial }_{y^{⁗}}.\end{array}$

The special affine curvature

$k=\frac{y^{⁗}}{3{\left(y^{″}\right)}^{\frac{5}{3}}}-\frac{5{\left(y^{‴}\right)}^2}{9{\left(y^{″}\right)}^{\frac{8}{3}}}$

does not depend explicitly on Template:Math, Template:Math, or Template:Math, and so satisfies

$T_1^{(4)}k=T_2^{(4)}k=X_1^{(4)}k=0.$

The vector field Template:Mvar acts diagonally as a modified homogeneity operator, and it is readily verified that Template:Math. Finally,

$X_2^{(4)}k=\frac{1}{2}{[H,X_1]}^{(4)}k=\frac{1}{2}[H^{(4)},X_1^{(4)}]k=0.$

The five vector fields

$T_1^{(4)},T_2^{(4)},X_1^{(4)},X_2^{(4)},H^{(4)}$

form an involutive distribution on (an open subset of) Template:Math so that, by the Frobenius integration theorem, they integrate locally to give a foliation of Template:Math by five-dimensional leaves. Concretely, each leaf is a local orbit of the special affine group. The function Template:Mvar parameterizes these leaves.

Human motor system

Human curvilinear 2-dimensional drawing movements tend to follow the equiaffine parametrization. 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. Namely,

$v=\gamma {\kappa }^{-\frac{1}{3}},$

where Template:Mvar is the speed of the hand, Template:Mvar is the Euclidean curvature and Template:Mvar is a constant termed the velocity gain factor.

See also

• Affine geometry of curves
• Affine sphere

References

Sources

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