Spherical trigonometry

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The spherical trigonometry is a branch of the spherical geometry (spherical geometry). It mainly deals with the calculation of side lengths and angles in spherical triangles .

Important areas of application are:

Historical background

There are indications that the Babylonians and Egyptians already dealt with problems of spherical trigonometry 4000 years ago in order to calculate the course of the stars . However, they couldn't solve it. The history of spherical trigonometry is therefore closely linked to astronomy . Approx. In 350 BC the Greeks thought about spherical geometry, which became an auxiliary science of astronomers .

The oldest writing on spheres comes from this time: it contains sentences about spherical circles; its author is the Greek Autolykos von Pitane . Hipparchus of Nicaea found around 140 BC Both computational and graphical methods to create star maps and perform new calculations. Menelaus of Alexandria then found 98 BC. The theorem about the sum of the angles in the spherical triangle and for the first time transferred formulas of the plane triangle to spherical triangles.

Ptolemy of Alexandria discovered the methods of calculating right-angled and oblique triangles between AD 125 and 150. The first approaches to the cosine law come from India . Based on the Indian and Greek research, Arab mathematicians continued to develop spherical trigonometry; Al-Battani (around 900 AD) and Nasir Eddin Tusi (around 1250 AD), who first used the sine law and the polar triangle , are worth mentioning included in mathematical considerations. At the time of the great voyages of discovery in the 15th century, research in spherical trigonometry was pushed again, as the location at sea should be improved, including the creation of new sea routes to India. Johannes Müller expanded his knowledge from the Greek, Indian and Arab times with the tangent function and the side cosine law.

In the 16th century, Vieta found the angle cosine theorem using the polar triangle . John Napier (Neper, 1550–1617) brought the trigonometric theorems into more easily applicable forms (e.g. the Neper rule). Leonhard Euler (1707–1783) finally summarized the sentences of spherical trigonometry in today's clear form.

In addition to Euler, numerous other mathematicians have further expanded the sphere and established many new relationships between the sides and angles of a spherical triangle, including Simon L'Huilier (1750-1840), Jean-Baptiste Joseph Delambre (1749-1822), Carl Friedrich Gauß (1777 –1855), Adrien-Marie Legendre (1752–1833) and David Hilbert (1862–1943).

Through further mathematical developments such as the logarithm , many new methods and applications of spherical geometry were discovered, for example in land surveying and cartography . In the 19th and 20th centuries, other non-Euclidean geometries were developed, and spherical trigonometry was also used in the theory of relativity .

Spherical triangle

If you connect three points on the spherical surface that are not all on a great circle with three great circle arcs, you get eight spherical triangles and six points of intersection, namely the end points of three spherical diameters. So these points are the corners and the arcs are the sides of the triangles.

Such a triangle is dealt with in the following, the angles α, β and γ lie in the respective corner points of the triangle ABC, all angles are explained in radians. For the definition of the sides and surfaces see spherical geometry .

Right-angled spherical triangle

In the right-angled spherical triangle (an angle is 90 °), the formulas for Euclidean triangles can usually be used in a slightly modified form.

Formulas for the right-angled spherical triangle

A spherical triangle with is required .

The following formulas apply to this:

Right-angled spherical triangle

The first of these laws replaces the Pythagorean theorem of plane geometry.

All these formulas for right-angled spherical triangles are summarized in the Neper rule ( Neper 1550–1617): If the pieces of the spherical triangle are arranged next to each other on a circle, the right angle is deleted and the complements are written for the legs : The cosine of a piece is equal to the product of the cotangent of the adjacent pieces or the product of the sine of the opposite pieces.

Formulas for the right-hand spherical triangle

In a right-hand spherical triangle, one side is 90 ° long. In the following formulas is assumed.

These formulas also result from the above-mentioned Neper rule : The cosine of a piece is equal to the product of the cotangent of the adjacent pieces or equal to the product of the sine of the opposite pieces. The 90 ° side is removed from the pieces arranged in a circle and the angles adjacent to it are replaced by their complementary angles, the opposite angle by its supplementary angle.

Sentences for the general spherical triangle

The following applies to all formulas:

Sphere radius
Half circumference
Half excess

Angle sum

The following applies to the sum of the angles of a spherical Eulerian triangle


where is the area of ​​the triangle. The sum of the angles of a spherical triangle on the unit sphere fluctuates between and , depending on the size of the triangle , which corresponds to 180 ° to 540 °.

Sine law

“In every triangle the ratio of one side to the opposite angle is constant. This ratio is called the modulus of the triangle. "(Hammer 1916, p. 447)

Side cosine law

Angle cosine law

Sine-cosine theorem

Tangent theorem

Cotangent theorem (cotangent theorem)

Nepersche equations

Delambresche (also Mollweid or Gaussian) equations

Half-angle set

Half-page set

in which

L'Huilier's formula


The solid angle results from the so-called spherical excess:


The absolute area is therefore:


Spherical and plane trigonometry

In the case of "small" spherical triangles, the curvature is largely negligible and the sentences of spherical trigonometry merge into the sentences of plane trigonometry:
The sentence that applies to right-angled spherical triangles corresponds to the Pythagorean theorem (see above). The sine law of spherical trigonometry changes into the
sine law of plane trigonometry. The side cosine law of spherical trigonometry changes into the cosine law of plane trigonometry. The angle cosine law of spherical trigonometry changes into the theorem of the angle sum for plane triangles. For the flattening of spherical triangles, see Legendre's theorem .


earth sciences

See Higher Geodesy , Mathematical Geography, and Map Projection .


See astronomical coordinate systems .


The celestial equator is projected from the earth's equator and the earth's axis is extended to the world axis. In this way one creates a coordinate system for the sky from the earth. The zenith is the point in the sky that is exactly above the observer. Nadir is the name of the counterpoint to the zenith on the celestial sphere. The observer is at a point on the surface of the earth. The earth is assumed to be a sphere which is surrounded by the celestial sphere. The calculations are based on the assumption that you can see half the celestial sphere from the observation point, i.e. up to the true horizon. The true horizon is a plane that bisects both spheres, with their normal vector pointing from the center of the earth to the zenith. The observer is not in the center of the earth, but on the surface, and his apparent horizon is described by a tangential plane to the globe that goes through his position. Due to the fact that the stars are practically infinitely distant in relation to the radius of the earth, the apparent and the true horizon are practically identical. The celestial meridian goes through the zenith and both poles. All stars in the sky describe circular orbits through the rotation of the earth's axis. Each star covers 360 ° measured horizontally per sidereal day . There is the phenomenon of circumpolar stars, which are always visible from an observation point. You are near the celestial pole. The size of the circumpolar area measured away from the pole corresponds to the latitude of the observer. At a pole there are therefore only circumpolar stars, which move on orbits parallel to the equator. There are no circumpolar stars at the equator , and the day arcs of all stars are semicircles there. The arc that a star describes from the point of rise to the point of setting is called the day arc . The intersection of the diary with the meridian is the highest point of the star and is also known as the upper culmination point. Circumpolar stars also have a lowest point on the day arc, which is called the lower culmination point.

Coordinate systems

Horizon system

The base circle lies in the plane of the observer. The altitude on the celestial sphere is measured in degrees. The horizon is at 0 °, the zenith at 90 ° and the nadir at −90 °. The zenith distance , which results from 90 ° minus the height, is often used instead of the height . The south point is selected as the zero point and the second position angle, the azimuth , can be measured from there. The azimuth is the angle between the celestial meridian and the vertical plane of the star. The azimuth is measured clockwise from 0 to 360 °. The pole height at a location is equal to the geographical latitude . In the northern hemisphere, the Pole Star facilitates measurements. The advantage of the horizon system is that you can measure the height of an object even if you cannot precisely determine the horizon. Because the direction to the zenith coincides with the direction of gravity . Two measuring instruments that were very common in the past make use of the properties of the horizon system: the theodolite and the sextant .

Equatorial system

In addition to the horizon system, in which the coordinates of a star constantly change due to the rotation of the earth, there is also the equator system. The celestial equator serves as the base circle for this system. The height above the celestial equator is called the declination . It can assume values ​​between 90 ° ( north celestial pole ) and –90 ° ( south celestial pole ). The other coordinate of the equatorial system is the right ascension , which is measured counterclockwise from the vernal equator along the celestial equator. The right ascension is related to the hour angle . This is counted clockwise from the point of intersection of the celestial equator with the celestial meridian from 0 ° to 360 ° or from 0 h to 24 h.

Nautical triangle

The nautical triangle is used to convert the two systems. It is a triangle on the surface of the celestial sphere with the corners pole, zenith and apparent star location. Using the cosine and sine theorem, formulas for conversion can be derived.


The sun as a timepiece

Due to the rotation of the earth , the sun apparently moves around the earth once within a day. The earth orbits the sun once in the course of a year. When defining a sunny day as the period from one culmination to the next, it is also taken into account that the earth has to make a little more than one full revolution in order to reach the appropriate position. A sidereal day begins with the upper meridian passage of the vernal equinox, which is fixed in the firmament. Since the earth's orbit is not taken into account, a sidereal day only has 23 h 56 min. There is one more sidereal day per year, because every day from Earth you have the same view of the stars as the day before, only 4 minutes earlier. Viewed from the earth, the sun passes through the ecliptic, i.e. the intersection of the celestial sphere and the plane of the earth's orbit, within a year. The seasons arise from the inclination of the earth's axis to the orbit plane by 23 ° 27 ′. The solar ephemeris indicates the slight fluctuations in the coordinates of the sun. The solar declination has the smallest value at the time of the winter solstice and the largest during the summer solstice. At the equinox, the sun rises exactly in the east and sets in the west. The east point-observer- rising point angle is called the morning distance. The angle West Point observer destruction point is in accordance with evening length. The nautical triangle pole-zenith-setting point can be used to calculate the length of a day.

The time of sunset (from the time of culmination) and the location of sunset (from the south point) can be calculated from the height of the pole (or latitude) of the location and the sun declination. When measuring time , a day is assumed to be the time between two culminations of the sun. But since the earth's orbit is not a circle, and due to other factors, there are considerable fluctuations in the “true sun”. Due to the inclination of the earth's axis, a sundial does not work either. In order to compensate for these disadvantages of the true sun, the mean sun is used as a calculation variable. One assumes a fictional sun moving along the equator . The true local time results from the hour angle of the true sun less than twelve hours. The mean local time can be calculated from the hour angle of the mean sun minus twelve hours. The difference between true local time and mean local time is called the equation of time , it has the value 0 four times a year. The values ​​of the equation of time can be found in tables. Since the local times are only the same on the same longitude, the difference is staggered. The international time zones result from this. The local time at the prime meridian is known as Greenwich Mean Time or world time . The longitude on which one is located can be determined by measuring the local time. Then you subtract the local time from the local time in Greenwich and get the longitude.

Stars as a timepiece

The time can be determined from the current position of a star (or vice versa). The sidereal time is defined as the hour angle of the vernal equinox , i.e. as the angle between the local meridian (the great circle on which the zenith, the north point and the south point of the horizon lie) and the declination circle of the vernal equinox (the great circle on which the vernal equinox and the two celestial poles are). This angle is counted on the celestial equator , from the local meridian in the direction of SWNE to the vernal equinox. 0 o'clock sidereal time means that the vernal equinox is just passing through the local meridian, i.e. it is exactly in the south for an observer in the northern hemisphere or exactly in the north for an observer in the southern hemisphere. One hour of sidereal time is obviously equated with 15 ° (angle in degrees), so that 24 sidereal time hours correspond to a 360 ° angle. A sidereal day is the period between two successive meridian passages of the vernal equinox. It is only slightly (0.0084 s) shorter than the Earth's rotation time, which is around 23 h 56 min 4 s. With the help of the last entry, sidereal time and solar time (civil time) can be converted into each other.

A direct determination of the sidereal time from the position of the vernal equinox is not possible, since the vernal equinox is only an imaginary point on the celestial sphere. No star occupies this exact position. Therefore one measures the hour angle for any star of known right ascension and calculates the sidereal time accordingly .

In a sense, stars are also timepieces for very long periods of time. Due to the gyration of the earth's axis, the vernal equinox shifts by approx. 50 ″ per year. Within a platonic year , that is approximately 26,000 years, it passes through the entire ecliptic once . This phenomenon is called precession .

Angle of incidence on solar panels

Calculation of the angle of incidence

If the position of the sun in the sky is known (see above), then the angle of incidence of the sun on plan collectors can be calculated thanks to spherical trigonometry, as follows:


where and are the azimuth angle of the sun and the azimuth angle of the collector, and and are the vertical angle of the sun and the vertical angle of the collector . And is the angle of incidence.


Web links

Commons : Spherical Trigonometry  - collection of images, videos and audio files

Individual evidence

  1. For the historical background, cf. Kern / Rung 1986, pp. 120-125.