Stereographic projection

The stereographic projection (also conformal azimuthal projection ) is a central projection, which is used for mapping of spherical surfaces in the plane. PZ is the projection center on the ball, the image plane is a tangent plane of the opposite point TP.

The stereographic projection is useful for mapping the sky ball on star charts and the earth's surface to map network designs. Her two advantages that obtain angular stay ( conformality ) and circles again be represented as such ( circle fidelity ), have already been discovered in the ancient world (presumably by Hipparchus about 130 BC ) used and the figure of the celestial sphere on the astrolabe.

In crystallography stereographic projection finds practical application for representing the lattice planes of the crystal and in the geology in the mapping of rock fractures.

In theoretical mathematics ... you motivate the religion or belief adverse conclusion the complex plane by a single point ... ( point infinity) with the help of their stereographic projection back to the Riemann sphere.

• 2.1 star maps
• 2.2 Map Projection
• 2.3 Geometry of plane curves
• 2.4 Crystallography

Mathematical treatment

Illustration of globe

The mathematical foundations of the stereographic image can be clear when the imaged points of the ball can be described by a coordinate system with the values ​​for the length and the width. An arbitrary point of the spherical surface is referred to as represented by the center of projection S onto the tangential projection.

The two-dimensional map coordinates and the point in the projection surface are then recovered by the following imaging equations:

( Α = 0, the map layer for each map point direction from arbitrarily specified reference meridian ) For the azimuth

And for the dimension m of the card point

Now, if the point of contact of the imaging plane and the center of projection understood as poles, so the curves

δ = const and m = const the parallels and

α = λ = const polschneidenden the meridians of a geographic coordinate system dar.

The length distortion in the direction of a meridian is on the projection of the unit sphere

The associated length distortion in the direction of the parallel of k = h due to the conformality of the mapping. The length distortion decreases with increasing angle δ increasingly to:

Therefore, the stereographic pictures remain in practice limited to δ ≤ 90 °.

Other features of the stereo Graphical projection are:

The image of the projection center lies at infinity. All circles on the sphere are represented as circles ( circle loyalty ). Thus, the key for navigation orthodrome (sections of great circles ) are not represented as a straight line, but as circles. The meridians through the center of projection are shown as circles with infinite radius and thus a straight line.

Generalization to ℝ n

The projection as described above is a special case of the general stereographic projection: in the three-dimensional space, two-dimensional spherical surface is imaged onto the plane of the card, and thus in the two-dimensional space. The general picture is as follows:

However, it is also possible to choose the archetype of this function so that its equator intersects the projection hyperplane:

This figure is for the point called the North Pole, of course not defined. Looking at the picture, the place has in the denominator, then the sphere is mapped to the south pole.

If you change the prototype of the stereographic projection in this way, we obtain by the two illustrations and an atlas of the n - sphere.

Derivation

Exemplary the stereographic projection is derived by the North Pole. For the projection by the South Pole, the same derivation can be used. The stereographic projection through the north pole is designed to represent a point on the sphere to its image point in the hyperplane, so that the image point lies on the line through the North Pole and.

This straight line can be parameterized by

This straight line intersects the plane where

It follows that the coordinates of the intersection of and

Are given. Turning now to the level, you get the stereographic projection through the North Pole.

Inverse functions

The stereographic projections through north or south pole exist continuous inverse functions

Therefore, and homeomorphisms. It has been with the projection from the north pole and the south pole of the found a possible Atlas. Thus we have shown that the sphere is a -dimensional topological manifold.

Circle Loyalty

Be a hyperplane in. If and, then it follows from the plane equation and by the Cauchy- Schwarz inequality:

The image of the points of the plane by satisfying the equation

This is a sphere equation. Therefore, all forms of sections and any hyperplane thus particularly spheres, which are in this hyperplane, on spheres in from.

Compactification of the complex plane

The stereographic projection can be used among other things to the compactification of the complex plane. Is extended by an additional point, which is here designated. The set is called Einpunktkompaktifizierung of or Riemann sphere.

The illustration is by the imaging

Continued. Now is called open if and only if is open. This induces a topology.

Chordal metric

This same topology is defined by the chordal metric by

Induced.

Application Examples

Star Maps

Since the mapping is conformal and maps circles again as such, the stereographic projection can be used for star charts. A historical application is the imaging of the stars Heaven on a astrolabe. The astrolabe shown contains the northern sky. The stars (including Zodiac, Rete ) are rotatable about the image of the north celestial ( Polaris ). The current location of the stars can be read on the fixed base ( tympanum ). On it is engraved the Safiha that from the zenith concentric circles height ( Almukantarate ), the horizon and at right angles to the azimuth Keisen there.

Map projection

In the polar stereographic projection ( the point of contact of the imaging plane is located in the North or South Pole ), the meridians of the geographical coordinate system of the earth as a straight line through the Erdpol shown ( see picture below left). The navigation elements Longitude and Latitude are therefore reproduced vividly by this projection for navigation purposes at the poles. The mapping of the Earth's poles during the International map of the world is also done via the Stereographic projection.

If the contact point of the imaging plane placed in any point of the earth, for example, in a port city, so the longitude and latitude lines are represented as arbitrary arcs. However, the direction can be added to any destination port as a straight line. This on the conformal -based property together with the lightweight drawing manufacturability of the card network design have been recognized in ancient times and used for maps for navigation as well as star maps.

In geophysics maps of the distribution of forces or line structures on the globe on a stereographic grid design to be established. In general, therefore, we restrict the mapping to a maximum of one hemisphere.

Geometry of plane curves

Consider an arbitrary curve in the plane, in an explicit polar coordinate representation. Now you put the projection sphere with radius to the coordinate origin, the tangent point TP. Through the center of projection - the opposite point on the spherical surface PZ - If you put a second plane which is parallel to the original plane (ie perpendicular to Selbiger formed by parallel displacement of the first level to ). Then the given curve will by stereographic projection in PZ, projected into the first layer to the sphere. By the original point TP is used as a new center of projection is projected through another stereographic projection, the curve on the sphere to the second level - it is described in polar coordinates by there. Then we have. The given curve is thus inverted by this double stereographic projection of the circle with radius in the (parallel) image plane.

Crystallography

Common applications for the stereographic projection in the crystallography showing the lattice planes of a crystal, for example, that of a diamond.