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Wikipedia / Spherical cap

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.

Volume and surface area

The volume of the spherical cap and the area of the curved surface may be calculated using combinations of

  • of the sphere

  • of the base of the cap

  • of the cap

  • The polar angle 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.

. denotes the latitude in geographic coordinates.

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 V_{sec} of the spherical sector, by an intuitive argument, as

The intuitive argument is based upon summing the total sector volume from that of infinitesimal triangular pyramids. Utilizing the pyramid (or cone) volume is the infinitesimal area of the sphere, the sum of the infinitesimal pyramidal bases would equal the area of the spherical sector, and:

Deriving the volume and surface area using calculus

The volume and area formulas may be derived by examining the rotation of the function

, using the formulas the surface of the rotation for the area and the solid of the revolution for the volume. The area is

is

and hence

The formula for the area is therefore

The volume is

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:

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 is

where

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

leads to

Volume of a spherical cap with a curved base

),

, 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

. They intersect if

From the law of cosines is

is

Surface area bounded by parallel disks

The curved surface area of the spherical segment, the area is

,

For example, assuming the Earth is a sphere of radius 6371km, 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.

Generalizations

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.

Hyperspherical cap

-dimensional Euclidean space is given by: (the gamma function.

can be expressed in terms of the volume of the unit n-ball and the hypergeometric function or the regularized incomplete beta function

.

A. Chudnov

Asymptotics

is the integral of the standard normal distribution.

.

See also

References

Further reading

  • Provides formulas for intersection sphere caps.