Tangent circulation number

In mathematics , the tangent rotation number or tangent speed (English: turning number ) measures the number of revolutions of a closed curve .

definition

Tangent directions

A closed regular curve in the plane is a map

${\displaystyle c\colon S^{1}\to \mathbb {R} ^{2}}$

with for everyone . ${\displaystyle f^{\prime }(x)\not =(0,0)}$ ${\displaystyle x}$

Your tangent revolution number is the revolution number of the tangent as an image

${\displaystyle {\dot {c}}\colon S^{1}\to \mathbb {R} ^{2}\setminus \left\{(0,0)\right\}}$

in relation to the zero point , i.e. the number of times it has been circled counterclockwise around the zero point. ${\displaystyle (0,0)}$ ${\displaystyle {\dot {c}}}$

Tangent revolution number and total curvature

The tangent revolution number can be expressed as the total curvature of the curve divided by ${\displaystyle 2\pi }$

${\displaystyle {\frac {1}{2\pi }}\int _{S^{1}}k(\theta )d\theta }$,

where denotes the curvature of the curve at the point . ${\displaystyle k(\theta )}$ ${\displaystyle f(\theta )}$

A curve with a number of revolutions> 1 must have a colon.

Classification of regular curves except for regular homotopy

The set of Whitney Gray Stone says that two regular curves if and in the plane homotopic regularly , if they have the same tangent circulation rate.

Sales rate

The circulation rate of Heinz Hopf states that the tangent revolution number to be a curve without self-intersections 1 or -1 needs.

literature

• Christian Bär : Elementary Differential Geometry. De Gruyter Studium, 2002. ISBN 978-3-11-022459-7