Solids {sphere} can result when semicircle rotates around its diameter. Equation is x^2 + y^2 + z^2 <= r^2, where r is radius.
area
Area is 4 * pi * r^2, where r is radius.
volume
Volume is 4 * pi * r^3 / 3, where r is radius.
imaginary circle
Two spheres share imaginary circle.
secondaries
Great circles can pass through poles.
spherical distance
Geodesic has length {spherical distance}.
spherical polygon
Great-circle arcs can bound spherical surface region {spherical polygon}.
spherical surface
x^2 + y^2 + z^2 = r^2 defines spherical surface. Area is 4 * pi * r^2, where r is radius.
diameter
Diameter is perpendicular to sphere at both endpoints.
coordinates
Sphere coordinates are longitude (360 degrees) and latitude (180 degrees), because they define unique points. Longitudes are perpendicular to latitudes. For spinning spheres, longitudes are along general direction of spherical axis, and latitudes are perpendicular to spherical axis.
Spherical coordinates can be vertical and horizontal latitude, so axes are perpendicular, but two latitudes define two different points, so points must have one more coordinate, such as north or south. Spherical coordinates can be vertical and horizontal longitudes, with axes not always perpendicular, but two longitudes can define the same great circle, so points must have one more coordinate. Therefore, only longitude and latitude define sphere points using two numbers.
Diameter intersects sphere at two points {antipodes, geometry}|.
Spheres have points {pole, sphere} where meridians meet.
Radii can make an angle {azimuth}| {polar angle} with polar axis.
Dihedral angles {spherical angle} are at diameter where great-circle planes intersect.
1/720 {spherical degree}| of sphere surface is solid-angle unit. One spherical degree is the birectangular spherical triangle whose third angle is one degree.
Difference {spherical excess} between spherical-polygon angle sum and plane angle sum is (n - 2) * (180 degrees), where n is number of angles.
Sphere has solid angle 4 * pi {steradian, solid angle}|.
Spherical areas {cap, sphere} can go from pole down to circle where plane intersects sphere. Cap area is pi * d * h, where d is diameter, and h is cap height from center.
Sphere halves {hemisphere}| are solids.
Two great circles not in perpendicular planes make two major and two minor spherical-surface regions {lune}.
Sectors {segment, sphere} {major sector, sphere} {major segment, sphere} can be greater than hemisphere. Sectors {minor sector, sphere} {minor segment, sphere} can be less than hemisphere.
Planes of two great circles intersect at diameter {spherical wedge} and divide sphere into four parts. Spherical-wedge volume is (A / (3 * pi / 2)) * pi * r^2, where A is angle between planes, and r is radius.
Sphere and two parallel planes intersect to make a solid figure {zone, sphere region}. Sphere and plane intersection has area pi * d * h, where d is diameter, and h is height from center.
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Date Modified: 2022.0225