Affine geometry of curves

Let x(t) be a curve in ${\displaystyle \mathbb {R} ^{n}}$. 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 ${\displaystyle \mathbb {R} ^{2}}$. Then the curve parameter t can be normalized by setting determinant

${\displaystyle \det {\begin{bmatrix}{\dot {\mathbf {x} }},&{\ddot {\mathbf {x} }},&\dots ,&{\mathbf {x} }^{(n)}\end{bmatrix}}=\pm 1.}$

Such a curve is said to be parametrized by its affine arclength. For such a parameterization,

${\displaystyle t\mapsto [\mathbf {x} (t),{\dot {\mathbf {x} }}(t),\dots ,\mathbf {x} ^{(n)}(t)]}$

determines a mapping into the special affine group, known as a special affine frame for the curve. That is, at each point of the quantities ${\displaystyle \mathbf {x} ,{\dot {\mathbf {x} }},\dots ,\mathbf {x} ^{(n)}}$ define a special affine frame for the affine space ${\displaystyle \mathbb {R} ^{n}}$, consisting of a point x of the space and a special linear basis ${\displaystyle {\dot {\mathbf {x} }},\dots ,\mathbf {x} ^{(n)}}$ attached to the point at x. The pullback of the Maurer–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.

The normalization of the curve parameter s was selected above so that

${\displaystyle \det {\begin{bmatrix}{\dot {\mathbf {x} }},&{\ddot {\mathbf {x} }},&\dots ,&{\mathbf {x} }^{(n)}\end{bmatrix}}=\pm 1.}$

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 −1.

In three-dimensions, a right-handed helix is dextrorse, and a left-handed helix is sinistrorse.

Suppose that the curve x in ${\displaystyle \mathbb {R} ^{n}}$ is parameterized by affine arclength. Then the affine curvatures, k₁, …, k_n−1 of x are defined by

${\displaystyle \mathbf {x} ^{(n+1)}=k_{1}{\dot {\mathbf {x} }}+\cdots +k_{n-1}\mathbf {x} ^{(n-1)}.}$

That such an expression is possible follows by computing the derivative of the determinant

${\displaystyle 0=\det {\begin{bmatrix}{\dot {\mathbf {x} }},&{\ddot {\mathbf {x} }},&\dots ,&{\mathbf {x} }^{(n)}\end{bmatrix}}{\dot {}}\,=\det {\begin{bmatrix}{\dot {\mathbf {x} }},&{\ddot {\mathbf {x} }},&\dots ,&{\mathbf {x} }^{(n+1)}\end{bmatrix}}}$

so that x⁽ⁿ⁺¹⁾ is a linear combination of x′, …, x⁽ⁿ⁻¹⁾.

Consider the matrix

${\displaystyle A={\begin{bmatrix}{\dot {\mathbf {x} }},&{\ddot {\mathbf {x} }},&\dots ,&{\mathbf {x} }^{(n)}\end{bmatrix}}}$

whose columns are the first n derivatives of x (still parameterized by special affine arclength). Then,

${\displaystyle {\dot {A}}={\begin{bmatrix}0&1&0&0&\cdots &0&0\\0&0&1&0&\cdots &0&0\\\vdots &\vdots &\vdots &\cdots &\cdots &\vdots &\vdots \\0&0&0&0&\cdots &1&0\\0&0&0&0&\cdots &0&1\\k_{1}&k_{2}&k_{3}&k_{4}&\cdots &k_{n-1}&0\end{bmatrix}}A=CA.}$

In concrete terms, the matrix C is the pullback of the Maurer–Cartan form of the special linear group along the frame given by the first n derivatives of x.

• Moving frame
• Affine sphere

• Guggenheimer, Heinrich (1977). Differential Geometry. Dover. ISBN 0-486-63433-7.
• Spivak, Michael (1999). A Comprehensive introduction to differential geometry (Volume 2). Publish or Perish. ISBN 0-914098-71-3.