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In
geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, a spherical cap or spherical dome is a portion of a
sphere A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
or of a
ball A ball is a round object (usually spherical, but sometimes ovoid) with several uses. It is used in ball games, where the play of the game follows the state of the ball as it is hit, kicked or thrown by players. Balls can also be used for s ...
cut off by a plane. It is also a
spherical segment In geometry, a spherical segment is the solid defined by cutting a sphere or a ball with a pair of parallel planes. It can be thought of as a spherical cap with the top truncated, and so it corresponds to a spherical frustum. The surface o ...
of one base, i.e., bounded by a single plane. If the plane passes through the center of the sphere (forming a
great circle In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point. Discussion Any arc of a great circle is a geodesic of the sphere, so that great circles in spher ...
), so that the height of the cap is equal to the
radius In classical geometry, a radius (: radii or radiuses) of a circle or sphere is any of the line segments from its Centre (geometry), center to its perimeter, and in more modern usage, it is also their length. The radius of a regular polygon is th ...
of the sphere, the spherical cap is called a ''
hemisphere Hemisphere may refer to: In geometry * Hemisphere (geometry), a half of a sphere As half of Earth or any spherical astronomical object * A hemisphere of Earth ** Northern Hemisphere ** Southern Hemisphere ** Eastern Hemisphere ** Western Hemi ...
''.


Volume and surface area

The
volume Volume is a measure of regions in three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch) ...
of the spherical cap and the area of the curved surface may be calculated using combinations of * The radius r of the sphere * The radius a of the base of the cap * The height h of the cap * The polar angle \theta between the rays from the center of the sphere to the apex of the cap (the pole) and the edge of the disk forming the base of the cap. These variables are inter-related through the formulas a = r \sin \theta, h = r ( 1 - \cos \theta ), 2hr = a^2 + h^2, and 2 h a = (a^2 + h^2)\sin \theta. If \phi denotes the
latitude In geography, latitude is a geographic coordinate system, geographic coordinate that specifies the north-south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from −90° at t ...
in
geographic coordinates A geographic coordinate system (GCS) is a spherical or geodetic coordinate system for measuring and communicating positions directly on Earth as latitude and longitude. It is the simplest, oldest, and most widely used type of the various ...
, then \theta+\phi = \pi/2 = 90^\circ\,, and \cos \theta = \sin \phi.


Deriving the surface area intuitively from the spherical sector volume

Note that aside from the calculus based argument below, the area of the spherical cap may be derived from the volume V_ of the spherical sector, by an intuitive argument, as :A = \fracV_ = \frac \frac = 2\pi rh\,. The intuitive argument is based upon summing the total sector volume from that of infinitesimal triangular pyramids. Utilizing the pyramid (or cone) volume formula of V = \frac bh', where b is the infinitesimal
area Area is the measure of a region's size on a surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an open surface or the boundary of a three-di ...
of each pyramidal base (located on the surface of the sphere) and h' is the height of each pyramid from its base to its apex (at the center of the sphere). Since each h', in the limit, is constant and equivalent to the radius r of the sphere, the sum of the infinitesimal pyramidal bases would equal the area of the spherical sector, and: :V_ = \sum = \sum\frac bh' = \sum\frac br = \frac \sum b = \frac A


Deriving the volume and surface area using calculus

The volume and area formulas may be derived by examining the rotation of the function :f(x)=\sqrt=\sqrt for x \in ,h/math>, using the formulas the surface of the rotation for the area and the solid of the revolution for the volume. The area is :A = 2\pi\int_0^h f(x) \sqrt \,dx The derivative of f is :f'(x) = \frac and hence :1+f'(x)^2 = \frac The formula for the area is therefore :A = 2\pi\int_0^h \sqrt \sqrt \,dx = 2\pi \int_0^h r\,dx = 2\pi r \left \right0^h = 2 \pi r h The volume is :V = \pi \int_0^h f(x)^2 \,dx = \pi \int_0^h (2rx-x^2) \,dx = \pi \left x^2-\frac13x^3\right0^h = \frac (3r - h)


Moment of inertia

The moments of inertia of a spherical cap (where the z-axis is the symmetrical axis) about the principal axes (center) of the sphere are: :J_ = \frac :J_ = J_ =\frac where ''m'' and ''h'' are, respectively, the mass and height of the spherical cap and ''R'' is the radius of the entire sphere.


Applications


Volumes of union and intersection of two intersecting spheres

The volume of the union of two intersecting spheres of radii r_1 and r_2 is : V = V^-V^\,, where :V^ = \fracr_1^3 +\fracr_2^3 is the sum of the volumes of the two isolated spheres, and :V^ = \frac(3r_1-h_1)+\frac(3r_2-h_2) the sum of the volumes of the two spherical caps forming their intersection. If d \le r_1+r_2 is the distance between the two sphere centers, elimination of the variables h_1 and h_2 leads to :V^ = \frac(r_1+r_2-d)^2 \left( d^2+2d(r_1+r_2)-3(r_1-r_2)^2 \right)\,.


Volume of a spherical cap with a curved base

The volume of a spherical cap with a curved base can be calculated by considering two spheres with radii r_1 and r_2, separated by some distance d, and for which their surfaces intersect at x=h. That is, the curvature of the base comes from sphere 2. The volume is thus the difference between sphere 2's cap (with height (r_2-r_1)-(d-h)) and sphere 1's cap (with height h), \begin V & = \frac(3r_1-h) - \frac
r_2-((r_2-r_1)-(d-h)) R, or r, is the eighteenth letter of the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''ar'' (pronounced ), plural ''ars''. The lette ...
,, \\ V & = \frac(3r_1-h) - \frac(d-h)^3\left(\frac-1\right)^2\left frac+1\right,. \end This formula is valid only for configurations that satisfy 0 and d-(r_2-r_1). If sphere 2 is very large such that r_2\gg r_1, hence d \gg h and r_2\approx d, which is the case for a spherical cap with a base that has a negligible curvature, the above equation is equal to the volume of a spherical cap with a flat base, as expected.


Areas of intersecting spheres

Consider two intersecting spheres of radii r_1 and r_2, with their centers separated by distance d. They intersect if :, r_1-r_2, \leq d \leq r_1+r_2 From the law of cosines, the polar angle of the spherical cap on the sphere of radius r_1 is :\cos \theta = \frac Using this, the surface area of the spherical cap on the sphere of radius r_1 is :A_1 = 2\pi r_1^2 \left( 1+\frac \right)


Surface area bounded by parallel disks

The curved surface area of the
spherical segment In geometry, a spherical segment is the solid defined by cutting a sphere or a ball with a pair of parallel planes. It can be thought of as a spherical cap with the top truncated, and so it corresponds to a spherical frustum. The surface o ...
bounded by two parallel disks is the difference of surface areas of their respective spherical caps. For a sphere of radius r, and caps with heights h_1 and h_2, the area is :A=2 \pi r , h_1 - h_2, \,, or, using geographic coordinates with latitudes \phi_1 and \phi_2, :A=2 \pi r^2 , \sin \phi_1 - \sin \phi_2, \,, For example, assuming the Earth is a sphere of radius 6371 km, the surface area of the arctic (north of the Arctic Circle, at latitude 66.56° as of August 2016) is = , or = 4.125% of the total surface area of the Earth. This formula can also be used to demonstrate that half the surface area of the Earth lies between latitudes 30° South and 30° North in a spherical zone which encompasses all of the
tropics The tropics are the regions of Earth surrounding the equator, where the sun may shine directly overhead. This contrasts with the temperate or polar regions of Earth, where the Sun can never be directly overhead. This is because of Earth's ax ...
.


Generalizations


Sections of other solids

The spheroidal dome is obtained by sectioning off a portion of a
spheroid A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface (mathematics), surface obtained by Surface of revolution, rotating an ellipse about one of its principal axes; in other words, an ellipsoid with t ...
so that the resulting dome is circularly symmetric (having an axis of rotation), and likewise the ellipsoidal dome is derived from the
ellipsoid An ellipsoid is a surface that can be obtained from a sphere by deforming it by means of directional Scaling (geometry), scalings, or more generally, of an affine transformation. An ellipsoid is a quadric surface;  that is, a Surface (mathemat ...
.


Hyperspherical cap

Generally, the n-dimensional volume of a hyperspherical cap of height h and radius r in n-dimensional Euclidean space is given by: V = \frac \int_^\sin^n (\theta) \,\mathrm\theta where \Gamma (the
gamma function In mathematics, the gamma function (represented by Γ, capital Greek alphabet, Greek letter gamma) is the most common extension of the factorial function to complex numbers. Derived by Daniel Bernoulli, the gamma function \Gamma(z) is defined ...
) is given by \Gamma(z) = \int_0^\infty t^ \mathrm^\,\mathrmt . The formula for V can be expressed in terms of the volume of the unit n-ball C_n = \pi^ / \Gamma +\frac/math> and the
hypergeometric function In mathematics, the Gaussian or ordinary hypergeometric function 2''F''1(''a'',''b'';''c'';''z'') is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases. It is ...
_F_ or the regularized incomplete beta function I_x(a,b) as V = C_ \, r^ \left( \frac\, - \,\frac \,\frac _F_\left(\tfrac,\tfrac;\tfrac;\left(\tfrac\right)^\right)\right) = \fracC_ \, r^n I_ \left(\frac, \frac \right), and the area formula A can be expressed in terms of the area of the unit n-ball A_= as A =\fracA_ \, r^ I_ \left(\frac, \frac \right), where 0\le h\le r . A. Chudnov derived the following formulas: A = A_n r^ p_ (q),\, V = C_n r^ p_n (q) , where q = 1-h/r (0 \le q \le 1 ), p_n (q) =(1-G_n(q)/G_n(1))/2 , G _n(q)= \int _0^q (1-t^2) ^ dt . For odd n=2k+1 : G_n(q) = \sum_^k (-1) ^i \binom k i \frac .


Asymptotics

If n \to \infty and q\sqrt n = \text, then p_n (q) \to 1- F() where F() is the integral of the
standard normal distribution In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is f(x) = \frac e^ ...
. A more quantitative bound is A/(A_n r^) = n^ \cdot 2-h/r)h/r . For large caps (that is when (1-h/r)^4\cdot n = O(1) as n\to \infty), the bound simplifies to n^ \cdot e^ .


See also

*
Circular segment In geometry, a circular segment or disk segment (symbol: ) is a region of a disk which is "cut off" from the rest of the disk by a straight line. The complete line is known as a '' secant'', and the section inside the disk as a '' chord''. More ...
— the analogous 2D object *
Solid angle In geometry, a solid angle (symbol: ) is a measure of the amount of the field of view from some particular point that a given object covers. That is, it is a measure of how large the object appears to an observer looking from that point. The poin ...
— contains formula for n-sphere caps *
Spherical segment In geometry, a spherical segment is the solid defined by cutting a sphere or a ball with a pair of parallel planes. It can be thought of as a spherical cap with the top truncated, and so it corresponds to a spherical frustum. The surface o ...
*
Spherical sector In geometry, a spherical sector, also known as a spherical cone, is a portion of a sphere or of a ball defined by a conical boundary with apex at the center of the sphere. It can be described as the union of a spherical cap and the cone formed ...
*
Spherical wedge A sphere (from Greek , ) is a surface analogous to the circle, a curve. In solid geometry, a sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ''center' ...


References


Further reading

* * * * * * *


External links

* {{MathWorld , id=SphericalCap , title=Spherical cap Derivation and some additional formulas.
Online calculator for spherical cap volume and area


Spherical geometry