Spherical cap

 ( Spherical Geometry ) 

In geometry, a spherical cap or spherical dome is a portion of a sphere or of a ball cut off by a plane. It is also a spherical segment of one base, i.e., bounded by a single plane. If the plane passes through the center of the sphere (forming a great circle), so that the height of the cap is equal to the radius of the sphere, the spherical cap is called a hemisphere.

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Volume and surface area The volume 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 $\backslash theta$ 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 $a\; =\; r\; \backslash sin\; \backslash theta$, $h\; =\; r\; (\; 1\; -\; \backslash cos\; \backslash theta\; )$, $2hr\; =\; a^2\; +\; h^2$, and $2\; h\; a\; =\; (a^2\; +\; h^2)\backslash sin\; \backslash theta$.

If $\backslash phi$ denotes the latitude in geographic coordinates, then $\backslash theta+\backslash phi\; =\; \backslash pi/2\; =\; 90^\backslash circ\backslash ,$, and $\backslash cos\; \backslash theta\; =\; \backslash sin\; \backslash phi$.

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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 $V\_\{sec\}$ of the spherical sector, by an intuitive argument, as

$A\; =\; \backslash frac\{3\}\{r\}V\_\{sec\}\; =\; \backslash frac\{3\}\{r\}\; \backslash frac\{2\backslash pi\; r^2h\}\{3\}\; =\; 2\backslash pi\; rh\backslash ,.$

The intuitive argument is based upon summing the total sector volume from that of infinitesimal Tetrahedron. Utilizing the pyramid (or cone) volume formula of $V\; =\; \backslash frac\{1\}\{3\}\; bh\text{'}$, where $b$ is the infinitesimal area of each pyramidal base (located on the surface of the sphere) and $h\text{'}$ is the height of each pyramid from its base to its apex (at the center of the sphere). Since each $h\text{'}$, 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:

$V\_\{sec\}\; =\; \backslash sum\{V\}\; =\; \backslash sum\backslash frac\{1\}\{3\}\; bh\text{'}\; =\; \backslash sum\backslash frac\{1\}\{3\}\; br\; =\; \backslash frac\{r\}\{3\}\; \backslash sum\; b\; =\; \backslash frac\{r\}\{3\}\; A$

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Deriving the volume and surface area using calculus The volume and area formulas may be derived by examining the rotation of the function

$f(x)=\backslash sqrt\{r^2-(x-r)^2\}=\backslash sqrt\{2rx-x^2\}$

for $x\; \backslash in\; 0,h$, using the formulas the surface of the rotation for the area and the solid of the revolution for the volume. The area is

$A\; =\; 2\backslash pi\backslash int\_0^h\; f(x)\; \backslash sqrt\{1+f\text{'}(x)^2\}\; \backslash ,dx$

The derivative of $f$ is

$f\text{'}(x)\; =\; \backslash frac\{r-x\}\{\backslash sqrt\{2rx-x^2\}\}$

and hence

$1+f\text{'}(x)^2\; =\; \backslash frac\{r^2\}\{2rx-x^2\}$

The formula for the area is therefore

$A\; =\; 2\backslash pi\backslash int\_0^h\; \backslash sqrt\{2rx-x^2\}\; \backslash sqrt\{\backslash frac\{r^2\}\{2rx-x^2\}\}\; \backslash ,dx$

        = 2\pi \int_0^h r\,dx
        = 2\pi r \left[x\right]_0^h
        = 2 \pi r h 
     

The volume is

$V\; =\; \backslash pi\; \backslash int\_0^h\; f(x)^2\; \backslash ,dx$

        = \pi \int_0^h (2rx-x^2) \,dx
        = \pi \left[rx^2-\frac13x^3\right]_0^h
        = \frac{\pi h^2}{3} (3r - h)
     

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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:

$J\_\{zz,\backslash text\{cap\}\}\; =\; \backslash frac\{m\; h\; \backslash left(\; 3\; h^2\; -\; 15\; h\; R\; +\; 20\; R^2\; \backslash right)\}\{10\; \backslash left(\; 3\; R\; -\; h\; \backslash right)\}$

$J\_\{xx,\backslash text\{cap\}\}\; =\; J\_\{yy,\; \backslash text\{cap\}\}\; =\backslash frac\{m\; \backslash left(\; -9\; h^3\; +\; 45\; h^2\; R\; -\; 80\; h\; R^2\; +\; 60\; R^3\; \backslash right)\}\{20\; \backslash left(\; 3\; R\; -\; h\; \backslash right)\}$

where m and h are, respectively, the mass and height of the spherical cap and R is the radius of the entire sphere.

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Applications

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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

$V\; =\; V^\{(1)\}-V^\{(2)\}\backslash ,,$

where

$V^\{(1)\}\; =\; \backslash frac\{4\backslash pi\}\{3\}r\_1^3\; +\backslash frac\{4\backslash pi\}\{3\}r\_2^3$

is the sum of the volumes of the two isolated spheres, and

$V^\{(2)\}\; =\; \backslash frac\{\backslash pi\; h\_1^2\}\{3\}(3r\_1-h\_1)+\backslash frac\{\backslash pi\; h\_2^2\}\{3\}(3r\_2-h\_2)$

the sum of the volumes of the two spherical caps forming their intersection. If $d\; \backslash le\; r\_1+r\_2$ is the distance between the two sphere centers, elimination of the variables $h\_1$ and $h\_2$ leads to

$V^\{(2)\}\; =\; \backslash frac\{\backslash pi\}\{12d\}(r\_1+r\_2-d)^2\; \backslash left(\; d^2+2d(r\_1+r\_2)-3(r\_1-r\_2)^2\; \backslash right)\backslash ,.$

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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 $h$),

This formula is valid only for configurations that satisfy

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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

$|r\_1-r\_2|\backslash leq\; d\; \backslash leq\; r\_1+r\_2$

From the law of cosines, the polar angle of the spherical cap on the sphere of radius $r\_1$ is

$\backslash cos\; \backslash theta\; =\; \backslash frac\{r\_1^2-r\_2^2+d^2\}\{2r\_1d\}$

Using this, the surface area of the spherical cap on the sphere of radius $r\_1$ is

$A\_1\; =\; 2\backslash pi\; r\_1^2\; \backslash left(\; 1+\backslash frac\{r\_2^2-r\_1^2-d^2\}\{2\; r\_1\; d\}\; \backslash right)$

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Surface area bounded by parallel disks The curved surface area of the spherical segment 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

$A=2\; \backslash pi\; r\; |h\_1\; -\; h\_2|\backslash ,,$

or, using geographic coordinates with latitudes $\backslash phi\_1$ and $\backslash phi\_2$,

$A=2\; \backslash pi\; r^2\; |\backslash sin\; \backslash phi\_1\; -\; \backslash sin\; \backslash phi\_2|\backslash ,,$

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.

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Generalizations

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Sections of other solids The spheroidal dome is obtained by sectioning off a portion of a spheroid so that the resulting dome is circularly symmetric (having an axis of rotation), and likewise the ellipsoidal dome is derived from the ellipsoid.

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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: $$V\; =\; \backslash frac\{\backslash pi\; ^\; \{\backslash frac\{n-1\}\{2\}\}\backslash ,\; r^\{n\}\}\{\backslash ,\backslash Gamma\; \backslash left\; (\; \backslash frac\{n+1\}\{2\}\; \backslash right\; )\}\; \backslash int\_\{0\}^\{\backslash arccos\backslash left(\backslash frac\{r-h\}\{r\}\backslash right)\}\backslash sin^n\; (\backslash theta)\; \backslash ,\backslash mathrm\{d\}\backslash theta$$ where $\backslash Gamma$ (the gamma function) is given by $\backslash Gamma(z)\; =\; \backslash int\_0^\backslash infty\; t^\{z-1\}\; \backslash mathrm\{e\}^\{-t\}\backslash ,\backslash mathrm\{d\}t$.

The formula for $V$ can be expressed in terms of the volume of the unit n-ball $C\_n\; =\; \backslash pi^\{n/2\}\; /\; \backslash Gamma1+\backslash frac\{n\}\{2\}$ and the hypergeometric function $\{\}\_\{2\}F\_\{1\}$ or the regularized incomplete beta function $I\_x(a,b)$ as $$V\; =\; C\_\{n\}\; \backslash ,\; r^\{n\}\; \backslash left(\; \backslash frac\{1\}\{2\}\backslash ,\; -\; \backslash ,\backslash frac\{r-h\}\{r\}\; \backslash ,\backslash frac\{\backslash Gamma1+\backslash frac\{n\}\{2\}\}\{\backslash sqrt\{\backslash pi\}\backslash ,\backslash Gamma\backslash frac\{n+1\}\{2\}\}\; \{\backslash ,\backslash ,\}\_\{2\}F\_\{1\}\backslash left(\backslash tfrac\{1\}\{2\},\backslash tfrac\{1-n\}\{2\};\backslash tfrac\{3\}\{2\};\backslash left(\backslash tfrac\{r-h\}\{r\}\backslash right)^\{2\}\backslash right)\backslash right)\; =\; \backslash frac\{1\}\{2\}C\_\{n\}\; \backslash ,\; r^n\; I\_\{(2rh-h^2)/r^2\}\; \backslash left(\backslash frac\{n+1\}\{2\},\; \backslash frac\{1\}\{2\}\; \backslash right),$$

and the area formula $A$ can be expressed in terms of the area of the unit n-ball $A\_\{n\}=\{2\backslash pi^\{n/2\}/\backslash Gamma\backslash frac\{n\}\{2\}\}$ as $$A\; =\backslash frac\{1\}\{2\}A\_\{n\}\; \backslash ,\; r^\{n-1\}\; I\_\{(2rh-h^2)/r^2\}\; \backslash left(\backslash frac\{n-1\}\{2\},\; \backslash frac\{1\}\{2\}\; \backslash right),$$ where $0\backslash le\; h\backslash le\; r$.

A. Chudnov derived the following formulas: $$A\; =\; A\_n\; r^\{n-1\}\; p\_\; \{\; n-2\; \}\; (q),\backslash ,\; V\; =\; C\_n\; r^\{n\}\; p\_n\; (q)\; ,$$ where $$q\; =\; 1-h/r\; (0\; \backslash le\; q\; \backslash le\; 1\; ),\; p\_n\; (q)\; =(1-G\_n(q)/G\_n(1))/2\; ,$$ $$G\; \_n(q)=\; \backslash int\; \_0^q\; (1-t^2)\; ^\; \{\; (n-1)\; /2\; \}\; dt\; .$$

For odd $n=2k+1$: $$G\_n(q)\; =\; \backslash sum\_\{i=0\}^k\; (-1)\; ^i\; \backslash binom\; k\; i\; \backslash frac\; \{q^\{2i+1\}\}\; \{2i+1\}\; .$$

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Asymptotics If $n\; \backslash to\; \backslash infty$ and $q\backslash sqrt\; n\; =\; \backslash text\{const.\}$, then $p\_n\; (q)\; \backslash to\; 1-\; F(\{q\; \backslash sqrt\; n\})$ where $F$ is the integral of the standard normal distribution.

A more quantitative bound is $A/(A\_n\; r^\{n-1\})\; =\; n^\{\backslash Theta(1)\}\; \backslash cdot\; (2-h/r)h/r^\{n/2\}$. For large caps (that is when $(1-h/r)^4\backslash cdot\; n\; =\; O(1)$ as $n\backslash to\; \backslash infty$), the bound simplifies to $n^\{\backslash Theta(1)\}\; \backslash cdot\; e^\{-(1-h/r)^2n/2\}$.

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See also

• Circular segment — the analogous 2D object
• Solid angle — contains formula for n-sphere caps
• Spherical segment
• Spherical sector
• Spherical wedge

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Further reading

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External links

• Derivation and some additional formulas.
• Online calculator for spherical cap volume and area.
• Summary of spherical formulas.

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