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EspañolLUNE (MATHEMATICS)
A 'lune' is either of two figures, both shaped roughly like a crescent Moon. The word "lune" derives from ''luna'', the Latin word for Moon.
In plane geometry, a lune is a convex area bounded by two intersecting circles. In the special case when the circles have the same radius, the figure is called a ''lens''. Formally, a lune is the relative complement of one circle in another (where they intersect but neither is a subset of the other).[1]
In spherical geometry, a lune is an area on a sphere bounded by two half great circles.[1] Such circles are the largest possible circles on a sphere; each great circle divides the surface of the sphere into two equal halves. Two great circles always intersect at two polar opposite points. Common examples of great circles are lines of longitude (''meridians''), which meet at the North and South Poles. Thus, the area between two meridians of longitude is a lune. The area of a spherical lune is 2θ ''R''2, where ''R'' is the radius of the sphere and θ is the dihedral angle between the two half great circles. When this angle equals 2π — i.e., when the second half great circle has moved a full circle, and the lune inbetween covers the sphere — the area formula for the spherical lune gives 4π''R''2, the surface area of the sphere.
The lighted portion of the Moon is a spherical lune. The first of the two intersecting great circles is the boundary separating the lighted half of the Moon from the dark half. The second great circle is that which separates the half visible from the Earth from the invisible half. Seen face on, this lighted spherical lune produces the familiar crescent shape of the Moon seen from Earth, as illustrated in the Figure at the left.
★ Arbelos
★ Spherical geometry
★ Gauss-Bonnet theorem
1.
2.
In plane geometry, the crescent shape formed from two intersecting circles is called a ''lune''.
| Contents |
| Plane geometry |
| Spherical geometry |
| See also |
| References |
Plane geometry
In plane geometry, a lune is a convex area bounded by two intersecting circles. In the special case when the circles have the same radius, the figure is called a ''lens''. Formally, a lune is the relative complement of one circle in another (where they intersect but neither is a subset of the other).[1]
Spherical geometry
A spherical lune. The two great circles are shown as thin black lines, whereas the lune itself (shown in green) is outlined in thick black lines, corresponding to its defining half great circles. The great circles intersect at two polar opposite points, such as the North and South poles.
In spherical geometry, a lune is an area on a sphere bounded by two half great circles.[1] Such circles are the largest possible circles on a sphere; each great circle divides the surface of the sphere into two equal halves. Two great circles always intersect at two polar opposite points. Common examples of great circles are lines of longitude (''meridians''), which meet at the North and South Poles. Thus, the area between two meridians of longitude is a lune. The area of a spherical lune is 2θ ''R''2, where ''R'' is the radius of the sphere and θ is the dihedral angle between the two half great circles. When this angle equals 2π — i.e., when the second half great circle has moved a full circle, and the lune inbetween covers the sphere — the area formula for the spherical lune gives 4π''R''2, the surface area of the sphere.
The crescent Moon is a spherical lune, as described in the text. Here, the blue and red portions may be taken equally as the lighted and dark portions of the Moon visible from Earth, or vice versa.
The lighted portion of the Moon is a spherical lune. The first of the two intersecting great circles is the boundary separating the lighted half of the Moon from the dark half. The second great circle is that which separates the half visible from the Earth from the invisible half. Seen face on, this lighted spherical lune produces the familiar crescent shape of the Moon seen from Earth, as illustrated in the Figure at the left.
See also
★ Arbelos
★ Spherical geometry
★ Gauss-Bonnet theorem
References
1.
2.
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