Affinity length

Affine length is a parameter of a flat curve that is retained during equi-affine transformations (i.e. affine transformations that preserve area ).

Content

Definition

For a flat curve $\gamma \colon [a,b]\to \mathbb {R} ^{2}$ ${\ displaystyle \ gamma \ colon [a, b] \ to \ mathbb {R} ^ {2}}$ affine length is calculated by the formula

l=\int \limits _{a}^{b}|{\dot {\gamma }}(t)\times {\ddot {\gamma }}(t)|^{1/3}dt,

{\ displaystyle l = \ int \ limits _ {a} ^ {b} | {\ dot {\ gamma}} (t) \ times {\ ddot {\ gamma}} (t) | ^ {1/3} dt ,}

Where $\times$ ${\ displaystyle \ times}$ denotes a vector product , and ${\dot {\gamma }}(t)$ ${\ displaystyle {\ dot {\ gamma}} (t)}$ and ${\ddot {\gamma }}(t)$ ${\ displaystyle {\ ddot {\ gamma}} (t)}$ - the first and second derivative.

Special Cases

Affinity Graph Length $y=f(x)$ ${\ displaystyle y = f (x)}$ functions $f$ ${\ displaystyle f}$ is set as
$l=\int \limits _{a}^{b}{\sqrt[{3}]{|f''(x)|}}dx,$ ${\ displaystyle l = \ int \ limits _ {a} ^ {b} {\ sqrt [{3}] {| f '' (x) |}} dx,}$
For curve $\gamma (s)$ ${\ displaystyle \ gamma (s)}$ with natural parameter and curvature $\varkappa (s)$ ${\ displaystyle \ varkappa (s)}$
$l=\int \limits _{a}^{b}{\sqrt[{3}]{|\varkappa (s)|}}ds.$ ${\ displaystyle l = \ int \ limits _ {a} ^ {b} {\ sqrt [{3}] {| \ varkappa (s) |}} ds.}$

Properties

The affine length of the arc of a parabola is equal to $2{\sqrt[{3}]{S}},$ ${\ displaystyle 2 {\ sqrt [{3}] {S}},}$ where S is the area of the triangle formed by the arc chord and the tangents to the parabola at the ends of the arc.
Among convex closed curves with a fixed affine length, ellipses (and only they) limit the smallest area.

Variations and generalizations

There are also generalizations of the affine length to the case of spatial curves and for a common affine group, as well as its other subgroups.

Literature

L. Feyesh Toth, Positioning on a plane, on a sphere and in space , M., Fizmatlit, 1958. - 364 p.