Gnomonic projection

A gnomonic projection is a central projection, in which the center of projection is the center of the imaged body. The term gnomonic projection is taken from the sundial, in which the celestial sphere is mapped with the sun in central projection using the gnomon tip on the dial. The type of projection is arbitrary. Ancient example is the inner surface of a sphere at one of the oldest sundials, the Skaphe. Often is projected onto a plane, such as the most in ( mapping ).

The projection is used in the mapping of the center of projection in the center of the earth to map network designs and in crystallography of the center of projection at the center of the crystal. Depending on the geometry of the imaging surface the picture to all or each edge is towards highly distorted. For the gnomonic projection topographic maps is therefore not suitable. However, the practical use of gnomonic Azimutalprojektionen in which the projection surface is a plane. Gnomonic Azimutalprojektionen are " just true ", that is, all great circles on the surface and thus all orthodromic are represented as straight lines. If one wishes to navigate from one point to another, so it is therefore possible in a card with gnomonischer Azimuthal determine the path by connecting the points with a straight line. Therefore find these cards along with conformal maps in the navigation at sea and in the air as well as in the radio navigation application. The crystallography uses the fact that Pole tautozonaler faces lie in gnomonischer Azimuthal on straight.

The gnomonic Azimuthal allows the mapping of the beginning of the projection center open half- space in which the picture plane. A lying in the boundary of the half space great circle is projected into infinity. Due to the strongly increasing with increasing distance from the center of the projection, the projection distortion is usually limited to an angular range of 60 ° around the central axis.

Gnomonic projections in cartography

Normal ( polar ) Location

If the projection plane of a gnomonic Azimuthal in normal position, ie it touches the earth at the pole, so the parallels are shown as concentric circles around the pole and the meridians as radial lines. The card network design can be carried out with the help of ruler and compass. The imaging equations in this position are as follows for the polar coordinates of the map, azimuth and radius, with longitude and Breitenkomplement (90 ° - latitude )

With a scale factor.

Transversal location

The contact point is on the equator. The meridians are shown as parallel lines, the parallels as hyperbolas. The equator becomes a straight line which intersects perpendicularly the meridians.

Leaning location

The card touches any point on the earth's surface (in this case Japan). It forms the meridians also as a line bundle from the parallels are to conic sections.

Transversal location: Equator ( 0 ° geog. Length)

Leaning Location: Japan

Illustration

If a globe is illuminated from the inside ( with almost point-like light source in the center of the planet ), so that the surface of the globe is projected onto a flat wall, so this image corresponds to the wall of the gnomonic Azimuthal.

Historical

The name derives from Gnomon ( γνόμον ), the Greek word for a gnomon used as an astronomical instrument, from.

With the tip of the gnomon shadow- throwing center of projection of the position of the sun has been mapped into a sundial and used to display the time of day, for example, already in antiquity. A design specification for a composition containing a gnomon sundial is preserved as the analemma of Vitruvius.