Spherical geometry

The spherical geometry , also spherical geometry or geometry on the sphere , deals with points and point sets on the sphere . It is originally motivated by geometrical observations on the globe (see cartography ) and the celestial sphere (see astrometry ). It is of particular interest within geometry , since with a suitable definition of the point on the sphere it represents both a model for elliptical geometry and fulfills the axioms of projective geometry .

The spherical geometry differs greatly from the plane Euclidean geometry in some points . It has no parallels, since two great circles, the analogue of the straight line on the sphere, always intersect. Many theorems known from Euclidean geometry , such as the 180 ° angle sum in a triangle or the Pythagorean theorem , are not valid on the sphere. However, they are available in an adapted form.

Basic concepts

The starting terms for plane geometries are the point and the straight line. These are defined on the sphere as follows:

Straight

The role of the straight line is played by the great circles in spherical geometry . Great circles are circles on the sphere whose (Euclidean) center is the center of the sphere. Examples of great circles on the globe are the equator and the meridians . A great circle is obtained by intersecting the surface of the sphere with a plane containing the center of the sphere.

Point

A circle is obtained by intersecting the sphere with a Euclidean plane. If the distance between the center of the sphere and the intersecting plane is equal to the radius of the sphere, the section describes a circle with radius 0, i.e. a point on the sphere.

Geographic point

In the geographical view of spherical geometry, the definition of the point is taken from Euclidean geometry, i.e. H. the set of spherical points is defined as the set of all points in three-dimensional Euclidean space that are located on the surface of the sphere.

Elliptical point

From a geometrical point of view, the geographical definition of the point has a serious disadvantage. In geometrical axiom systems it is generally required that two points define exactly one straight line. With the above definition, this is not the case when one considers counterpoints on the sphere. Opposite points are points whose Euclidean connecting line runs through the center of the sphere. (They relate to one another like the north and south poles on the globe.) An infinite number of great circles run through counterpoints (corresponding to the longitude circles on the globe). Every great circle through a point also runs through its counterpoint. It therefore makes sense to combine pairs of opposing points to form one point.

Since the elliptical definition of the point identifies every point with its opposite point, every figure (set of points) on the sphere is also identified with its opposite figure. (In particular, a triangle, for example, consists of two counter-triangles.)

route

Lines are great arcs on the sphere. The distance between two points A and B on the sphere is identical to the length of the shortest great circle arc from A to B. On the unit sphere with the center point M, its length is identical to the angle in radians . Lengths can also be specified as angles on a sphere with any radius r. The actual spherical length d is then calculated from the angle in radians as . ${\displaystyle \angle AMB}$ ${\displaystyle d=\angle AMB\cdot r}$

With an elliptical definition of the point, the smaller of the two angles between the Euclidean straight line connecting the counterpoints corresponds to the spherical distance on the unit sphere. The distance is therefore never greater than . ${\displaystyle \pi /2}$

circle

A circle is obtained by intersecting the sphere with a Euclidean plane. In spherical geometry, straight lines (sections of the sphere with Euclidean planes that contain the center of the sphere) are nothing more than special circles ( great circles ). The circle of intersection of the sphere with a plane that does not contain the center of the sphere is called the small circle. (On the globe, for example, with the exception of the equator, all latitudes are small circles.)

Area calculation

Spherical triangle

→ Main article : Spherical triangle

Two great circles with the points of intersection P and P 'divide the spherical surface into four spherical polygons. A spherical triangle is limited by two arcs of these great circles that connect P and P '. The area of a spherical triangle is related to the total surface of the sphere like its opening angle to the full angle: ${\displaystyle A_{Z}}$ ${\displaystyle A_{K}}$ ${\displaystyle \alpha }$

${\displaystyle A_{Z}={\frac {\alpha }{2\pi }}\cdot A_{K}={\frac {\alpha }{2\pi }}\cdot 4\pi r^{2}=2\alpha r^{2}}$.

In particular, the following applies on the unit sphere

${\displaystyle A_{Z}=2\alpha }$.

Spherical triangle

→ Main article : spherical triangle

The area of a spherical triangle with the angles and is calculated from its angles: ${\displaystyle A_{D}}$ ${\displaystyle \alpha ,\beta }$ ${\displaystyle \gamma }$

${\displaystyle A_{D}=(\alpha +\beta +\gamma -\pi )\cdot r^{2}}$

Since the area is always greater than zero, the sum of the three interior angles of a spherical triangle must be greater than (or 180 °): ${\displaystyle \pi }$

${\displaystyle \alpha +\beta +\gamma >\pi }$

The excess of the sum of the angles over the sum of the angles of a Euclidean triangle is called the spherical excess . The spherical excess of a triangle is proportional to its area (and even equal on the unit sphere with the proportionality factor 1).

The sphere as a projective plane, duality and polarity

Duality of point and line on the sphere

Incidence and angle-length preservation with dualization

The spherical geometry is a projective plane with the elliptical definition of the points . In projective geometry, all sentences can be dualized, that is, the terms point and straight line are interchanged (consequently lengths and angles as in the table above). On the sphere, each straight line a can be uniquely assigned its dual point A and, conversely, each point A its dual straight line a. For a circle, the dual point pair is obtained as the intersection of the sphere with the perpendicular to the plane of the circle that runs through the center of the sphere (see figure).

With dualization, the incidence of points and straight lines is retained. The following applies: If a point A lies on a straight line b, the straight line a, which is dual to it, runs through the point B, which is dual to the straight line b. But not only the incidence is retained, but angles and lengths also merge. The dimension d of the angle between two straight lines a and b corresponds (on the unit sphere) to the dimension of the distance d between the points A and B, which are dual to the straight lines.

→ This duality is a special correlation , namely an elliptical, projective polarity . This is explained in more detail in the article Correlation (Projective Geometry) .

Coordinates

To create a coordinate system, you first arbitrarily take a great circle as the equator . Then you choose a meridian as the prime meridian and define a direction of rotation . Now you can measure the angles from the equator and the prime meridian and thus clearly define each position on the sphere. Circles of latitude are parallel to the equator, while circles of longitude go through the two poles.

Borderline rule

For calculations on the spherical surface, the principle applies that all formulas that contain the spherical radius and therefore take into account the absolute size must be converted into valid formulas for flat geometry for the borderline case . ${\displaystyle r\,}$ ${\displaystyle r\to \infty }$

See also

• Spherical trigonometry
• Orthodrome (shortest distance on the sphere)

Web links

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

• Lecture on spherical geometry