Area

A numerical characteristic assigned to planar figures of a certain class (for example, polygons) and having the following properties: 1) the area is non-negative; 2) the area is additive (in the case of polygons this means that if a figure $ P \cup Q $ is composed of two figures $ P $ and $ Q $ not having common interior points, then $ \mathop{\rm area} ( P \cup Q ) = \mathop{\rm area} P + \mathop{\rm area} Q $); 3) the area is preserved under displacements; and 4) the area of the unit square is 1. The term "area" is used also in a more general sense as a numerical characteristic of two-dimensional surfaces in three-dimensional space, for $ k $- dimensional surfaces in an $ n $- dimensional Euclidean or Riemannian space $ ( 2 \leq k \leq n ) $, and for the boundaries of sets and other objects, see below.

The area of a planar figure.

Historically, the area was determined initially on the class of polygons (figures having a decomposition into a finite number of triangles without common interior points). It is important that an area with the properties 1)–4) exists and is unique in the class of polygons [1], [2]. One of the consequences of 1)–4) is that the area of an entire figure is not less than the area of its parts.

In Antiquity, the existence and uniqueness of the area with properties 1)–4) was assumed without an explicit description of the class of figures; attention was concentrated on ways of calculating the area. The formula for the area of a rectangle, including one with irrational sides, was based on the method of exhaustion (cf. Exhaustion, method of). The area of a triangle or of any polygon was calculated as the area of the rectangle such that the given triangle or polygon and the rectangle are both built up from the same congruent parts. It has been shown [2] that any polygons of equal area are decomposable into the same congruent parts.

Later, the class of squarable (Jordan-measurable) figures was distinguished. A figure $ M $ in the plane is said to be squarable if for any $ \epsilon > 0 $ there exist polygonal figures $ P $ and $ Q $ such that $ P \subset M \subset Q $ and $ ( \mathop{\rm area} Q - \mathop{\rm area} P ) < \epsilon $. The class of squarable figures is extremely rich. It includes, in particular, all bounded domains in the plane with piecewise-smooth boundaries. However, there are also non-squarable planar figures. In the class of squarable figures, an area with the properties 1)–4) exists and is unique [2].

Historically, before the class of squarable figures was considered, people knew how to calculate the areas of some of them: the disc, a circular sector and segments, various kinds of choral figures, and curvilinear trapezia. These calculations were based on the method of exhaustion by polygons. In some cases, Cavalieri's principle (cf. Cavalieri principle) was used as the basis for such calculations. It states that if two planar figures of this type intersect every line parallel to a given one in segments of identical length, then these figures are equal in area. Techniques in integral calculus (see, for example, [3]) give convenient methods of calculating the area of any planar domain with a piecewise-smooth boundary. The integral calculus justifies Cavalieri's principle.

Attempts to extend the concept of area to more general planar sets while retaining the properties 1)–4) led to measure theory and to the distinction of the class of planar Lebesgue-measurable sets. The transition to even more general classes of sets in the plane leads to non-unique measures with the properties 1)–4).

Oriented area.

If there is a directed closed curve $ l $ in the oriented plane, possibly with self-intersections and overlaps, then for each point in the plane not lying on $ l $ there is an integer function (positive, negative or zero), called the degree of the point with respect on $ l $. It indicates how many times and in what direction the contour $ l $ encircles that point. The integral over the entire plane of this function, if it exists, is called the oriented area enclosed by $ l $. The latter differs from the ordinary area in having a sign. See [4] for simple properties of an oriented area.

The area of surfaces.

The areas of polyhedral surfaces are first and most simply defined as the sums of the areas of the planar faces. There are difficulties in attempts to introduce the concept of area for curved surfaces as the limit of the areas of inscribed polyhedral surfaces (in a similar was as the length of a curve is defined as the limit of inscribed polygonal lines). Even for a very simple curved surface, the areas of inscribed polyhedra with successively smaller faces may have different limits, depending on the choice of the sequence of polyhedra. This is clearly demonstrated by Schwartz' example, in which sequences of inscribed polyhedra with different limits of the area are constructed for the lateral surface of a right circular cylinder [2].

Most frequently, the area of surfaces is defined for the class of piecewise-smooth surfaces with piecewise-smooth boundaries (or without boundary). Usually, this is based on the following construction. The surface is split up into small parts with piecewise-smooth boundaries. In each part one selects a point at which the tangent plane exists and projects the part orthogonally on the tangent plane to the surface at that point; the areas of the resulting planar projections are summed, and finally one passes to the limit for increasingly fine subdivisions (such that the largest of the diameters in the parts of the subdivision tends to zero). This limit always exists in the class of surfaces, and if the surface is given parametrically by a piecewise $ C ^ {1} $- smooth function $ \mathbf r ( u , v ) $, where the parameters $ u $ and $ v $ vary in a domain $ D $ in the $ ( u , v ) $- plane, then the area $ F $ is expressed by the double integral

$$ \tag{1 } F = \int\limits _ { D } \sqrt {g _ {11} g _ {22} - g _ {12} ^ {2} } \ d u d v , $$

where $ g _ {11} = \mathbf r _ {u} ^ {2} , g _ {12} = \mathbf r _ {u} \mathbf r _ {v} , g _ {22} = \mathbf r _ {v} ^ {2} $, while $ \mathbf r _ {u} $ and $ \mathbf r _ {v} $ are the partial derivatives with respect to $ u $ and $ v $. The proof is given, for example, in [3], [5]. In particular, if the surface is the graph of a $ C ^ {1} $- smooth function $ z = f ( x , y ) $ over a domain $ D $ in the $ ( x , y ) $- plane, then

$$ \tag{2 } F = \int\limits _ { D } \sqrt {1 + f _ {x} ^ { 2 } + f _ {y} ^ { 2 } } d x d y . $$

Here $ ( 1 + f _ {x} ^ { 2 } + f _ {y} ^ { 2 } ) ^ {1/2} = \cos \alpha $, where $ \alpha $ is the (acute) angle between the normal to the surface and the $ z $- axis. From (1) and (2) one gets the standard formulas for the area of a sphere or parts of it, as well as methods for calculating the areas of surfaces of rotation, etc.

For two-dimensional piecewise-smooth surfaces in Riemannian manifolds [6], formula (1) serves as a definition of the area, where the roles of $ g _ {11} $, $ g _ {12} $ and $ g _ {22} $ are played by the components of the metric tensor restricted to the surface itself.

It is important that even in the case of a two-dimensional surface, the area is assigned not to the set of points but to the equivalence class of mappings representing the two-dimensional manifold into a space, and thus differs from a measure.

$ k $-dimensional area.

For a piecewise-smooth immersion $ f : M \rightarrow \mathbf R ^ {n} $ of a $ k $- dimensional manifold (with or without boundary) into $ n $- dimensional Euclidean space, $ 2 \leq k < n $, the area is defined by means of a construction that is completely analogous to that described above for piecewise-smooth surfaces. The only difference is that the projection now involves $ k $- dimensional tangent planes and that the summation is over the $ k $- dimensional projection volumes. If one has introduced coordinates $ u ^ {1} \dots u ^ {k} $ in a domain $ D \subset M $, then the area $ F $ of the immersion $ ( D , f \mid _ {D} ) $ is expressed by the integral

$$ \tag{3 } F = \int\limits _ { D } \sqrt { \mathop{\rm det} ( g _ {ij} ) } \ d u ^ {1} \dots d u ^ {k} , $$

where $ g _ {ij} $ are the scalar products $ ( \partial f / \partial u ^ {i} , \partial f / \partial u ^ {j} ) $. If $ k = n - 1 $ and $ D $ is a domain in the coordinate hyperplane $ ( x _ {1} \dots x _ {n-1} ) $, while $ f $ has the explicit form $ x _ {n} = z ( x _ {1} \dots x _ {n-1} ) $, (3) becomes

$$ \tag{4 } F = \int\limits _ { D } \sqrt {1 + \sum _ { i=1 } ^ { n-1 } \left ( \frac{\partial z }{\partial x _ {i} } \right ) ^ {2} } d x _ {1} \dots d x _ {n-1} . $$

If $ f $ is a smooth mapping, then the $ g _ {ij} $ are the coefficients of the metric tensor induced by the immersion, and it follows from (3) that the area defined by the external construction belongs to the intrinsic geometry of the immersed manifold. In general, (3) is taken as the definition of area in the case of an immersion not only into $ \mathbf R ^ {n} $ but also into a Riemannian manifold.

On the class of piecewise-smooth immersed manifolds, the area is: a) non-negative; b) equal to 1 on the $ k $- dimensional unit cube in $ \mathbf R ^ {n} $; c) preserved under orthogonal transformations; d) additive; e) semi-continuous, i.e. $ F (f) \leq \lim\limits \inf _ {i\rightarrow \infty } F ( f _ {i} ) $ if $ f _ {i} \rightarrow f $ uniformly. Further: f) if $ \phi : \mathbf R ^ {n} \rightarrow \mathbf R ^ {n} $ is a non-stretching mapping, then $ F ( \phi \circ f ) \leq F ( f ) $; and g) if $ \{ P _ {i} \} $ is a set of $ ( {} _ {k} ^ {n} ) $ pairwise-orthogonal $ k $- dimensional planes and if $ p _ {i} $ is the projection onto $ P _ {i} $, then

$$ \tag{5 } F ( p _ {i} \circ f ) \leq \ F ( f ) \leq \ \sum _ { i=1 } ^ { {( } {} _ {k} ^ {n} ) } F ( p _ {i} \circ f ) . $$

Further extensions. The theory of area.

The extension of $ F $ to more general objects while retaining some of properties a)–g) is possible in various ways and leads to various results. The theory of area deals with these extensions. See [10] for a review of the relations between various concepts of area.

The boundary between the theory of area and measure theory is not clear-cut. Traditionally, the theory of area relates primarily to the study of the areas of continuous mappings, where multiplicity is taken into account and the preservation of additivity is less important. See, for example, Hausdorff measure; Favard measure or $ k $- dimensional measures in $ n $- dimensional space.

An area can be introduced on the basis of an approximating, integral-geometric, or functional approach or else axiomatically. The most common concepts are given below.

The Lebesgue area.

It (cf. [7], [9]) is defined by

$$ \tag{6 } L ( M , f ) = \ \lim\limits _ {\overline{ {i \rightarrow \infty }}\; } F ( f _ {i} ) , $$

where $ M $ is a finitely-triangulable $ k $- dimensional manifold, $ f _ {i} : M \rightarrow \mathbf R ^ {n} $ are all possible sequences of piecewise-linear mappings with $ f _ {i} \rightarrow f $, and $ F ( f _ {i} ) $ is the $ k $- dimensional area of the corresponding polyhedral surface. The Lebesgue area is the same for Fréchet-equivalent mappings and therefore is a characteristic of a Fréchet surface.

If $ k = 2 $, the condition $ L ( M , f ) < \infty $ implies various useful properties of the surface (for example, the possibility of introducing isothermal parameters); in that case, the Lebesgue area is a convenient tool sufficient to solve the Plateau problem and more general two-dimensional problems in variational calculus. There arise difficulties in the investigation of variational problems for the class of continuous mappings for $ k > 2 $( the compactness problem), which has led to the search for other objects (currents and varifolds) and related characteristics of area type. The use of the Lebesgue area is restricted also by the complexity in establishing its relationships to other concepts of area and certain of its properties (for example, the right-hand inequality in (5)). There are two notable features of $ L ( M , f ) $: for $ L ( M , f ) = 0 $ it may happen that the volume $ V ( f ( M )) > 0 $, and, secondly, even for $ k=2 $, decomposing $ M $ into $ M _ {1} $ and $ M _ {2} $ with a common boundary in the form of a curve, it may happen that

$$ \tag{7 } L ( M , f ) > \ L ( M _ {1} , f \mid _ {M _ {1} } ) + L ( M _ {2} , f \mid _ {M _ {2} } ) . $$

Studies have also been made concerning which of the properties a)–g) can be retained when additivity is replaced by semi-additivity, e.g. assuming an inequality of the type of (7) one still gets the Lebesgue area. Topics of this type have not yet (1983) been examined completely [7].

Integral-geometric areas.

(Cf. [9].) Let $ N ( l , \phi , y ) $ be some multiplicity function for mappings $ \phi : M \rightarrow \mathbf R ^ {n} $ at a point $ y \in \mathbf R ^ {k} $, for example $ N = \mathop{\rm card} \phi ^ {-1} (y) $. Then for $ f : M \rightarrow \mathbf R ^ {n} $, one may define the integral-geometric area

$$ \tag{8 } \int\limits _ {G ( n , k ) } \int\limits _ {\mathbf R ^ {k} } N ( M , p _ \sigma \circ f , y ) dV (y) dv ( \sigma ) , $$

where $ G ( n , k ) $ is the Grassmann manifold of $ k $- dimensional subspaces $ \mathbf R ^ {k} \subset \mathbf R ^ {n} $, $ v $ is the normalized Haar measure on $ G ( n , k ) $ and $ p _ \sigma $ are the orthogonal projections on $ \sigma \in G ( n , k ) $. The areas (8) may differ for various multiplicity functions. A very special choice of $ N ( M , \phi , y ) $ makes (8) coincide with the Lebesgue area for $ k = 2 $ for triangulable $ M $, as well as for $ k > 0 $ if the Hausdorff measure $ H _ {k+1} ( f (M) ) $ vanishes completely [9]. For the case $ k = 2 $, various integral-geometric areas have been introduced also by G. Peano, Hetz and S. Banach [7].

The area of the boundary of a set.

The Minkowski area was introduced in connection with the proof of the Brunn–Minkowski inequality and the classical isoperimetric inequality. It is assigned to a set $ A \subset \mathbf R ^ {n} $, but characterizes the area of the boundary, and is defined by

$$ \tag{9 } F (a) = \lim\limits _ {\overline{ {h \downarrow 0 }}\; } \frac{1}{h} ( V ( A + hB ) - V (A) ) , $$

where $ B $ is the unit sphere in $ \mathbf R ^ {n} $ and $ V $ denotes the volume. For sets $ A $ with a piecewise-smooth boundary and for convex sets $ A $, the ordinary limit in (9) exists; it coincides with the $ ( n-1 ) $- dimensional area of the boundary. Definition (9) still applies to sets in finite-dimensional normed spaces, where the area (9) may differ from the Hausdorff measure $ H _ {n-1} ( \partial A ) $ even for convex sets $ A $.

There is another useful characteristic analogous to area, which is put into correspondence with a set, but which characterizes mainly the boundary, namely the perimeter of a measurable set. This is a particular case of the concept of a current mass.

Intrinsic area.

If $ M $ is metrized and $ f : M \rightarrow \mathbf R ^ {n} $ is a locally isometric mapping, one has to consider the relation between $ L ( M , f ) $ and the Hausdorff measure $ H _ {k} ( M ) $. When $ M $ is a two-dimensional manifold of bounded curvature, one has $ L ( M , f ) = H _ {2} ( M ) $. In general, on the other hand, a continuous $ f : M \rightarrow \mathbf R ^ {n} $ induces a generalized metric $ \rho $ distinguished by the possibility $ \rho ( x , y ) = \infty $ on the connected components $ f ^ {-1} (u) $, $ u \in \mathbf R ^ {n} $. The construction of the $ k $- dimensional Hausdorff measure applied to $ \rho $ gives a characteristic that can be taken as the intrinsic area of an immersion. For $ f $ in a Lipschitz class, this coincides with $ L ( M , f ) $[11].

Current masses and varifolds.

The integration of $ k $- forms with respect to a $ k $- dimensional piecewise-smooth oriented manifold $ M $ imbedded in $ \mathbf R ^ {n} $ leads to the current $ T _ {M} \phi = \int _ {M} ( \phi (x) , \nu (x) ) dF (x) $, being a linear functional of the $ k $- forms $ \phi $ on $ \mathbf R ^ {n} $. Here $ \nu $ is a unit $ k $- vector tangent to $ M $. The linear functional $ T _ {M} $ essentially characterizes $ M $. Also, a non-linear functional can be defined (in the same way and for non-oriented $ M $): the varifold $ V _ {M} \phi = \int _ {M} | ( \phi , \nu ) | dF $. The integral norm (masses) $ \| T _ {M} \| $ and $ \| V _ {M} \| $ coincide with the area, i.e. with $ F (M) $. The inclusion of the class of piecewise-smooth submanifolds of $ \mathbf R ^ {n} $ in more general current and varifold classes plays the same role in variational calculus as do generalized solutions in the theory of partial differential equations.

There are extensive classes of integer currents and varifolds among all of these. These retain many geometric properties of submanifolds. For example, an integer-valued current is a current having a representation $ T = \sum _ {i=1} ^ \infty n _ {i} T _ {M _ {i} } $, where $ n _ {i} $ are integers and $ M _ {i} $ are $ C ^ {1} $- smooth submanifolds, subject to the condition that the mass of the $ ( k-1 ) $- dimensional current $ dT $ defined by $ dT ( \phi ) = T ( d \phi ) $ is finite. The masses of integer-valued currents and varifolds can be considered as an extension of the concept of the area of a surface. Here again, there is a relationship to the Lebesgue area. Let piecewise-smooth mappings $ f _ {i} : M \rightarrow \mathbf R ^ {n} $ converge uniformly, $ f _ {i} \rightarrow f $, and let $ L ( M , f ) = \lim\limits _ {i \rightarrow \infty } L ( M , f _ {i} ) $. Then the corresponding varifolds $ V _ {f _ {i} } $ converge weakly to some integer-valued varifold $ V _ {f} $, where $ \| V _ {f} \| = L ( M , f ) $. Thus, each $ f : M \rightarrow \mathbf R ^ {n} $ with $ L ( M , f ) < \infty $ is naturally put into correspondence with a varifold $ V _ {f} $ with mass $ L ( M , f ) $; see [13] for this in the language of currents.

References

[1]	H. Lebesgue, "Sur la mesure des grandeurs" Monogr. de l'Enseign. Math. , 1 (1935)
[2]	, Encyclopaedia of elementary mathematics , 5 , Moscow (1966) (In Russian)
[3]	V.A. Il'in, E.G. Poznyak, "Fundamentals of mathematical analysis" , 1–2 , MIR (1982) (Translated from Russian)
[4]	A.M. Lopshits, "Calculating the area of oriented figures" , Moscow (1956) (In Russian)
[5]	A.V. Pogorelov, "Differential geometry" , Noordhoff (1959) (Translated from Russian)
[6]	P.K. [P.K. Rashevskii] Rashewski, "Riemannsche Geometrie und Tensoranalyse" , Deutsch. Verlag Wissenschaft. (1959) (Translated from Russian)
[7]	L. Cesari, "Surface area" , Princeton Univ. Press (1956)
[8]	H. Federer, "Geometric measure theory" , Springer (1969)
[9]	H. Federer, "Measure and area" Bull. Amer. Math. Soc. , 58 : 3 (1952) pp. 306–378
[10]	Yu.D. Burago, V.A. Zalgaller, "Geometric inequalities" , Springer (1988) (Translated from Russian)
[11]	H. Busemann, "Intrinsic area" Ann. of Math. Ser. 2 , 48 (1947) pp. 234–267
[12]	F.J. Almgren, "The theory of varifolds" , Princeton Univ. Press (1965)
[13]	H. Federer, "Currents and area" Trans. Amer. Math. Soc. , 98 : 2 (1961) pp. 204–233