Angular distance in the context of "Trigonometry"

Celestial navigation, also known as astronavigation, is the practice of position fixing using stars and other celestial bodies that enables a navigator to accurately determine their actual current physical position in space or on the surface of the Earth without relying solely on estimated positional calculations, commonly known as dead reckoning. Celestial navigation is performed without using satellite navigation or other similar modern electronic or digital positioning means.

Celestial navigation uses "sights," or timed angular measurements, taken typically between a celestial body (e.g., the Sun, the Moon, a planet, or a star) and the visible horizon. Celestial navigation can also take advantage of measurements between celestial bodies without reference to the Earth's horizon, such as when the Moon and other selected bodies are used in the practice called "lunars" or the lunar distance method, used for determining precise time when time is unknown.

Right ascension (abbreviated RA; symbol α) is the angular distance of a particular point measured eastward along the celestial equator from the Sun at the March equinox to the (hour circle of the) point in question above the Earth. When paired with declination, these astronomical coordinates specify the location of a point on the celestial sphere in the equatorial coordinate system.

An old term, right ascension (Latin: ascensio recta) refers to the ascension, or the point on the celestial equator that rises with any celestial object as seen from Earth's equator, where the celestial equator intersects the horizon at a right angle. It contrasts with oblique ascension, the point on the celestial equator that rises with any celestial object as seen from most latitudes on Earth, where the celestial equator intersects the horizon at an oblique angle.

A sextant is a doubly reflecting navigation instrument that measures the angular distance between two visible objects. The primary use of a sextant is to measure the angle between an astronomical object and the horizon for the purposes of celestial navigation.

The estimation of this angle, the altitude, is known as sighting or shooting the object, or taking a sight. The angle, and the time when it was measured, can be used to calculate a position line on a nautical or aeronautical chart—for example, sighting the Sun at noon or Polaris at night (in the Northern Hemisphere) to estimate latitude (with sight reduction). Sighting the height of a landmark can give a measure of distance off and, held horizontally, a sextant can measure angles between objects for a position on a chart. A sextant can also be used to measure the lunar distance between the moon and another celestial object (such as a star or planet) in order to determine Greenwich Mean Time and hence longitude.

In celestial navigation, lunar distance, also called a lunar, is the angular distance between the Moon and another celestial body. The lunar distances method uses this angle and a nautical almanac to calculate Greenwich time if so desired, or by extension any other time. That calculated time can be used in solving a spherical triangle. The theory was first published by Johannes Werner in 1524, before the necessary almanacs had been published. A fuller method was published in 1763 and used until about 1850 when it was superseded by the marine chronometer. A similar method uses the positions of the Galilean moons of Jupiter.

In mathematics, a metric space is a set together with a notion of distance between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are a general setting for studying many of the concepts of mathematical analysis and geometry.

The most familiar example of a metric space is 3-dimensional Euclidean space with its usual notion of distance. Other well-known examples are a sphere equipped with the angular distance and the hyperbolic plane. A metric may correspond to a metaphorical, rather than physical, notion of distance: for example, the set of 100-character Unicode strings can be equipped with the Hamming distance, which measures the number of characters that need to be changed to get from one string to another.

Polaris is a star in the northern circumpolar constellation of Ursa Minor. It is designated α Ursae Minoris (Latinized to Alpha Ursae Minoris) and is commonly called the North Star. With an apparent magnitude that fluctuates around 1.98, it is the brightest star in the constellation and is readily visible to the naked eye at night. The position of the star lies less than 1° away from the north celestial pole, making it the current northern pole star. The stable position of the star in the Northern Sky makes it useful for navigation.

Although appearing to the naked eye as a single point of light, Polaris is a triple star system, composed of the primary, a yellow supergiant designated Polaris Aa, in orbit with a smaller companion, Polaris Ab; the pair is almost certainly in a wider orbit with Polaris B. The outer companion B was discovered in August 1779 by William Herschel, with the inner Aa/Ab pair only confirmed in the early 20th century.

In mathematics, a metric space is a set together with a notion of distance between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are a general setting for studying many of the concepts of mathematical analysis and geometry.

The most familiar example of a metric space is 3-dimensional Euclidean space with its usual notion of distance. Other well-known examples are a sphere equipped with the angular distance and the hyperbolic plane. A metric may correspond to a metaphorical, rather than physical, notion of distance. For example, the set of 100-character Unicode strings can be equipped with the Hamming distance, which measures the number of characters that need to be changed to get from one string to another.

A central angle is an angle whose apex (vertex) is the center O of a circle and whose legs (sides) are radii intersecting the circle in two distinct points A and B. Central angles are subtended by an arc between those two points, and the arc length is the central angle of a circle of radius one (measured in radians). The central angle is also known as the arc's angular distance. The arc length spanned by a central angle on a sphere is called spherical distance.

The size of a central angle Θ is 0° < Θ < 360° or 0 < Θ < 2π (radians). When defining or drawing a central angle, in addition to specifying the points A and B, one must specify whether the angle being defined is the convex angle (<180°) or the reflex angle (>180°). Equivalently, one must specify whether the movement from point A to point B is clockwise or counterclockwise.