Oriented area

Oriented area is a generalization of the concept of area enclosed within a closed curve in a plane. Unlike the usual square, has a sign.

Definition

If a directed closed curve is located on an oriented plane. $\ell$ ${\ displaystyle \ ell}$ perhaps with self-intersections and inclinations, then for each non-lying $\ell$ ${\ displaystyle \ ell}$ points of a plane, an integral function (positive, negative or zero) is defined, called the index of the point relative to $\ell$ ${\ displaystyle \ ell}$ . It shows how many times and in which direction the contour $\ell$ ${\ displaystyle \ ell}$ bypasses this point. The integral over the entire plane of this function, if it exists, is called the covered $\ell$ ${\ displaystyle \ ell}$ oriented area.

Properties

For oriented square $S$ ${\ displaystyle S}$ enclosed inside a closed polyline $A_{1}\dots A_{n}$ ${\ displaystyle A_ {1} \ dots A_ {n}}$ on the plane equality

S\cdot {\vec {N}}={\vec {OA_{1}}}\times {\vec {A_{1}A_{2}}}+\dots +{\vec {OA_{n}}}\times {\vec {A_{n}A_{1}}},

{\ displaystyle S \ cdot {\ vec {N}} = {\ vec {OA_ {1}}} \ times {\ vec {A_ {1} A_ {2}}} + \ dots + {\ vec {OA_ { n}}} \ times {\ vec {A_ {n} A_ {1}}},}

Where ${\vec {N}}$ ${\ displaystyle {\ vec {N}}}$ denotes the unit vector normal to the plane and $\times$ ${\ displaystyle \ times}$ - vector product .

Literature

Lopshits AM, Calculation of areas of oriented figures, M., 1956;

Oriented area

Definition

Properties

Literature

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