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In the [[mathematics|mathematical]] field of [[differential geometry]], the '''affine geometry of curves''' is the study of [[curve]]s in an [[affine space]], and specifically the properties of such curves which are [[invariant (mathematics)|invariant]] under the [[special affine group]] <math> \mbox{SL}(n,\mathbb{R}) \ltimes \mathbb{R}^n.</math>
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In the classical [[Euclidean geometry of curves]], the fundamental tool is the [[Frenet&ndash;Serret frame]]. In affine geometry, the Frenet&ndash;Serret frame is no longer well-defined, but it is possible to define another canonical [[moving frame]] along a curve which plays a similar decisive role. The theory was developed in the early 20th century, largely from the efforts of [[Wilhelm Blaschke]] and [[Jean Favard]].
 
== The affine frame ==
 
Let '''x'''(''t'') be a curve in '''R'''<sup>''n''</sup>.  Assume, as one does in the Euclidean case, that the first ''n'' derivatives of '''x'''(''t'') are [[linearly independent]] so that, in particular, '''x'''(''t'') does not lie in any lower-dimensional affine subspace of '''R'''<sup>n</sup>. Then the curve parameter ''t'' can be normalized by setting [[determinant]]
 
:<math>\det \begin{bmatrix}\dot{\mathbf{x}}, &\ddot{\mathbf{x}}, &\dots, &{\mathbf{x}}^{(n)} \end{bmatrix} = \pm 1.</math>
 
Such a curve is said to be parametrized by its ''[[affine arclength]]''.  For such a parameterization,
 
:<math>t\mapsto [\mathbf{x}(t),\dot{\mathbf{x}}(t),\dots,\mathbf{x}^{(n)}(t)]</math>
 
determines a mapping into the special affine group, known as a special affine frame for the curve.  That is, at each point of the, the quantities <math>\mathbf{x},\dot{\mathbf{x}},\dots,\mathbf{x}^{(n)}</math> define a special [[affine frame]] for the affine space '''R'''<sup>''n''</sup>, consisting of a point '''x''' of the space and a special linear basis <math>\dot{\mathbf{x}},\dots,\mathbf{x}^{(n)}</math> attached to the point at '''x'''. The [[pullback (differential geometry)|pullback]] of the [[Maurer&ndash;Cartan form]] along this map gives a complete set of affine structural invariants of the curve. In the plane, this gives a single scalar invariant, the [[affine curvature]] of the curve.
 
==Discrete invariant==
The normalization of the curve parameter ''s'' was selected above so that
:<math>\det \begin{bmatrix}\dot{\mathbf{x}}, &\ddot{\mathbf{x}}, &\dots, &{\mathbf{x}}^{(n)} \end{bmatrix} = \pm 1.</math>
If ''n''≡0 (mod 4) or ''n''≡3 (mod 4) then the sign of this determinant is a discrete invariant of the curve. A curve is called '''dextrorse''' (right winding, frequently ''weinwendig'' in German) if it is +1, and '''sinistrorse''' (left winding, frequently ''hopfenwendig'' in German) if it is&nbsp;&minus;1.
 
In three-dimensions, a right-handed [[helix]] is dextrorse, and a left-handed helix is sinistrorse.
 
==Curvature==
Suppose that the curve '''x''' in '''R'''<sup>''n''</sup> is parameterized by affine arclength.  Then the '''[[affine curvature]]s''', ''k''<sub>1</sub>, …, ''k''<sub>''n''−1</sub> of '''x''' are defined by
 
:<math>\mathbf{x}^{(n+1)} = k_1\dot{\mathbf{x}} +\cdots + k_{n-1}\mathbf{x}^{(n-1)}.</math>
 
That such an expression is possible follows by computing the derivative of the determinant
 
:<math>0=\det \begin{bmatrix}\dot{\mathbf{x}}, &\ddot{\mathbf{x}}, &\dots, &{\mathbf{x}}^{(n)} \end{bmatrix}\dot{}\, = \det \begin{bmatrix}\dot{\mathbf{x}}, &\ddot{\mathbf{x}}, &\dots, &{\mathbf{x}}^{(n+1)} \end{bmatrix}</math>
 
so that '''x'''<sup>(''n''+1)</sup> is a linear combination of '''x'''′, …, '''x'''<sup>(''n''&minus;1)</sup>.
 
Consider the [[matrix (mathematics)|matrix]]
 
:<math>A = \begin{bmatrix}\dot{\mathbf{x}}, &\ddot{\mathbf{x}}, &\dots, &{\mathbf{x}}^{(n)} \end{bmatrix}</math>
 
whose columns are the first ''n'' derivatives of '''x''' (still parameterized by special affine arclength). Then,
 
:<math>\dot{A} =
\begin{bmatrix}0&1&0&0&\cdots&0&0\\
0&0&1&0&\cdots&0&0\\
\vdots&\vdots&\vdots&\cdots&\cdots&\vdots&\vdots\\
0&0&0&0&\cdots&1&0\\
0&0&0&0&\cdots&0&1\\
k_1&k_2&k_3&k_4&\cdots&k_{n-1}&0
\end{bmatrix}A = CA.</math>
In concrete terms, the matrix ''C'' is the [[pullback (differential geometry)|pullback]] of the Maurer&ndash;Cartan form of the special linear group along the frame given by the first ''n'' derivatives of '''x'''.
 
==See also==
*[[Moving frame]]
*[[Affine sphere]]
 
== References ==
{{reflist}}
* {{cite book|first=Heinrich|last=Guggenheimer|title=Differential Geometry|year=1977|publisher=Dover|isbn=0-486-63433-7}}
* {{cite book|authorlink=Michael Spivak|last=Spivak|first=Michael|title=A Comprehensive introduction to differential geometry (Volume 2)|year=1999|publisher=Publish or Perish|isbn=0-914098-71-3}}
 
[[Category:Curves]]
[[Category:Differential geometry]]
[[Category:Affine geometry]]

Latest revision as of 11:34, 7 January 2015

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