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Central angle
 
 
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A central angle is an angle
Angle

In geometry and trigonometry, an angle is the figure formed by two Ray sharing a common endpoint, called the vertex of the angle . The magnitude of the angle is the "amount of rotation" that separates the two rays, and can be measured by considering the length of circular arc swept out when one ray is rotated about the vertex to coincide...
 whose vertex is the center of a circle
Circle

A circle is a simple shape of Euclidean geometry consisting of those point in a plane which are the same distance from a given point called the center....
, and whose sides pass through a pair of points on the circle, thereby subtending an arc
Arc (geometry)

In geometry, an arc is a closed set segment of a differentiable curve in the two-dimensional manifold; for example, a circular arc is a segment of the circumference of a circle....
 between those two points whose angle is (by definition) equal to the central angle itself.






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Sector Central Angle Arc
A central angle is an angle
Angle

In geometry and trigonometry, an angle is the figure formed by two Ray sharing a common endpoint, called the vertex of the angle . The magnitude of the angle is the "amount of rotation" that separates the two rays, and can be measured by considering the length of circular arc swept out when one ray is rotated about the vertex to coincide...
 whose vertex is the center of a circle
Circle

A circle is a simple shape of Euclidean geometry consisting of those point in a plane which are the same distance from a given point called the center....
, and whose sides pass through a pair of points on the circle, thereby subtending an arc
Arc (geometry)

In geometry, an arc is a closed set segment of a differentiable curve in the two-dimensional manifold; for example, a circular arc is a segment of the circumference of a circle....
 between those two points whose angle is (by definition) equal to the central angle itself. It is also known as the arc segment
Line segment

In geometry, a line segment is a part of a line that is bounded by two end Point , and contains every point on the line between its end points....
's angular distance
Angular distance

In mathematics and all natural sciences , the angular distance between two point objects, as observed from a location different from either of these objects, is the size of the angle between the two directions originating from the observer and pointing towards these two objects....
.

Coordinates

On a sphere
Sphere

A sphere is a symmetrical geometrical object. In non-mathematical usage, the term is used to refer either to a round ball or to its two-dimensional surface....
 or ellipsoid
Ellipsoid

An ellipsoid is a type of Quadric that is a higher dimensional analogue of an ellipse. The equation of a standard axis-aligned ellipsoid body in an xyz-Cartesian coordinate system is...
, the central angle is delineated along a great circle
Great circle

A great circle of a sphere is a circle that runs along the surface of that sphere so as to cut it into two equal halves. The great circle therefore has both the same circumference and the same center as the sphere....
. The usually provided coordinates of a point on a sphere/ellipsoid is its common latitude
Latitude

Latitude, usually denoted symbolically by the Greek letter phi gives the location of a place on Earth north or south of the equator. Lines of Latitude are the horizontal lines shown running east-to-west on maps ....
 ("Lat"), , and longitude
Longitude

Longitude , symbolized by the Greek character lambda , is the geographic coordinate most commonly used in cartography and global navigation for east-west measurement....
 ("Long"), . The "point", , is actually——relative to the great circle it is being measured on——the transverse colatitude ("TvL"), and the central angle/angular distance is the difference between two TvLs, .

Calculation of TvL

The calculation of and can be found using a common subroutine:

 



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Each point has at least two values, both a forward and reverse value.

Occupying great circle

The arc path, , tracing the great circle that a central angle occupies, is measured as that great circle's azimuth at the equator, introducing an important property of spherical geometry, Clairaut's constant
Clairaut's theorem

Clairaut's theorem, published in 1743 by Alexis Clairaut in his Th?orie de la figure de la terre, tir?e des principes de l'hydrostatique, synthesized physical and geodetic evidence that the earth is an oblate rotational ellipsoid....
:

From this and relationships to ,

Angular distance formulary

The angular distance can be calculated either directly as the TvL difference, or via the common coordinates (here, either SAw, SBw value set can be used):

and, using half-angles,

   


There is also a logarithmical form:

See also

  • Inscribed angle
    Inscribed angle

    In geometry, an inscribed angle is formed when two secant lines of a circle intersect on the circle.Typically, it is easiest to think of an inscribed angle as being defined by two Chord of the circle sharing an endpoint....


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