The navigational triangle or PZX triangle is a

spherical triangle Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, geodesics are gre ...

used in

astronavigation 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 ...

to determine the observer's position on the

globe A globe is a spherical model of Earth, of some other celestial body, or of the celestial sphere. Globes serve purposes similar to maps, but unlike maps, they do not distort the surface that they portray except to scale it down. A model glo ...

. It is composed of three reference points on the

celestial sphere In astronomy and navigation, the celestial sphere is an abstract sphere that has an arbitrarily large radius and is concentric to Earth. All objects in the sky can be conceived as being projected upon the inner surface of the celestial sphere, ...

: * P is the Celestial Pole (either

North North is one of the four compass points or cardinal directions. It is the opposite of south and is perpendicular to east and west. ''North'' is a noun, adjective, or adverb indicating direction or geography. Etymology The word ''no ...

or South). It is a fixed point. * Z is the observer's

zenith The zenith (, ) is an imaginary point directly "above" a particular location, on the celestial sphere. "Above" means in the vertical direction ( plumb line) opposite to the gravity direction at that location ( nadir). The zenith is the "high ...

, or their position on the celestial sphere. * X is the position of a celestial body, such as the

sun The Sun is the star at the center of the Solar System. It is a nearly perfect ball of hot plasma, heated to incandescence by nuclear fusion reactions in its core. The Sun radiates this energy mainly as light, ultraviolet, and infrared rad ...

moon The Moon is Earth's only natural satellite. It is the fifth largest satellite in the Solar System and the largest and most massive relative to its parent planet, with a diameter about one-quarter that of Earth (comparable to the width ...

, a

planet A planet is a large, rounded astronomical body that is neither a star nor its remnant. The best available theory of planet formation is the nebular hypothesis, which posits that an interstellar cloud collapses out of a nebula to create a ...

, or a

star A star is an astronomical object comprising a luminous spheroid of plasma held together by its gravity. The nearest star to Earth is the Sun. Many other stars are visible to the naked eye at night, but their immense distances from Earth make ...

. The position of Z or X is described via its

declination In astronomy, declination (abbreviated dec; symbol ''δ'') is one of the two angles that locate a point on the celestial sphere in the equatorial coordinate system, the other being hour angle. Declination's angle is measured north or south of t ...

—the angular distance north or south of the equator (corresponding to its

latitude In geography, latitude is a coordinate that specifies the north– south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north po ...

)—and the

hour angle In astronomy and celestial navigation, the hour angle is the angle between two planes: one containing Earth's axis and the zenith (the ''meridian plane''), and the other containing Earth's axis and a given point of interest (the ''hour circle'') ...

—the angle between its

meridian Meridian or a meridian line (from Latin ''meridies'' via Old French ''meridiane'', meaning “midday”) may refer to Science * Meridian (astronomy), imaginary circle in a plane perpendicular to the planes of the celestial equator and horizon * ...

and the

Greenwich meridian The historic prime meridian or Greenwich meridian is a geographical reference line that passes through the Royal Observatory, Greenwich, in London, England. The modern IERS Reference Meridian widely used today is based on the Greenwich mer ...

(corresponding to its

longitude Longitude (, ) is a geographic coordinate that specifies the east– west position of a point on the surface of the Earth, or another celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek let ...

). If the observer knows the angles subtended by P, Z, and X, they can calculate their position on the globe. By measuring the angle of the celestial body in the sky, the observer can get the

local hour angle In astronomy and celestial navigation, the hour angle is the angle between two planes: one containing Earth's axis and the zenith (the '' meridian plane''), and the other containing Earth's axis and a given point of interest (the ''hour circle'' ...

(LHA) of X, which is the angle subtended at P between Z and X (the angle between the Z and X's meridians) and calculate the longitude by subtracting from the Greenwich hour angle of the celestial body. Finding the latitude requires measuring the vertical angle (altitude) of X from the horizon using a sextant, the declination of X from a reference book, and a set of

sight reduction In astronavigation, sight reduction is the process of deriving from a sight, (in celestial navigation usually obtained using a sextant), the information needed for establishing a line of position, generally by intercept method. Sight is defined as ...

Tables. The sun, moon, and planets move relative to the celestial sphere, but the only the stars' hour angles change with the rotation of the earth, completing a full 360 degrees every solar day.

Gallery

Первый астрономический треугольник.svg, PZX triangle. Due to the Russian-language source, the "X" point is marked as "R". Второй астрономический треугольник.svg, RPM triangle. Here "M" is the celestial body ("X") and "R" is the ecliptic north pole; "P" is still the Earth rotational north pole. This is not for navigation, but for showing the difference between

astronomical coordinate systems Astronomical coordinate systems are organized arrangements for specifying positions of satellites, planets, stars, galaxies, and other celestial objects relative to physical reference points available to a situated observer (e.g. the true horizon ...

– ecliptic and equatorial in this case. Третий астрономический треугольник.svg, GPR triangle. "R" is the body in question, "G" is the Earth's north pole, and "P" is the galactic north pole. This is not for navigation, but for showing the difference between astronomical coordinate systems – galactic and equatorial in this case.

References

{{reflist
Altitudes and the PZX
at opensacandaga.com

at icys.co.uk Spherical astronomy Navigation