Affine geometry of curves explained

SL (n, R ) \ltimes R ^{n.}

In the classical Euclidean geometry of curves, the fundamental tool is the Frenet - Serret frame. In affine geometry, the Frenet - 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 ^{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 R ^{2} . Then the curve parameter t can be normalized by setting determinant\det \begin{bmatrix} • x , &\ddot{ x }, &\dots, &^ \end = \pm 1.

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

t\mapsto [x(t), • x (t), ... , 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

• x , x , ... , x ^{(n)} define a special affine frame for the affine space R ^{n} , consisting of a point x of the space and a special linear basis • x , ... , 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.

Discrete invariant

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

\det \begin{bmatrix} • x , &\ddot{ x }, &\dots, &^ \end = \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.

Curvature

Suppose that the curve x in

R ^{n} is parameterized by affine arclength. Then the affine curvatures, k₁, …, k_n−1 of x are defined byx ^{(n+1)} = k _{• 1 x} + … + k _{n-1} x ^{(n-1)} .

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

0= \det \begin{bmatrix} • x , &\ddot{ x }, &\dots, &^ \end\dot\, = \det \begin\dot, &\ddot, &\dots, &^ \end

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

Consider the matrix

A = \begin{bmatrix} • x , &\ddot{ x }, &\dots, &^ \end

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

• A = \begin{bmatrix} 0 & 1 & 0 & 0 & … & 0 & 0 \\ 0 & 0 & 1 & 0 & … & 0 & 0 \\ \vdots&\vdots&\vdots& … & … &\vdots&\vdots\\ 0 & 0 & 0 & 0 & … & 1 & 0 \\ 0 & 0 & 0 & 0 & … & 0 & 1 \\ k _{1&k} _{2&k} _{3&k} _{4& … &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.

See also

• Moving frame
• Affine sphere

References

• Book: Guggenheimer, Heinrich. Differential Geometry. 1977. Dover. 0-486-63433-7.
• Book: Spivak, Michael. Michael Spivak

. Michael Spivak. A Comprehensive introduction to differential geometry (Volume 2). 1999. Publish or Perish. 0-914098-71-3.

This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Affine geometry of curves".

Except where otherwise indicated, Everything.Explained.Today is © Copyright 2009-2024, A B Cryer, All Rights Reserved. Cookie policy.