Gnomonic projection

Gnomonic projection: great circles become straight lines

A gnomonic projection is a central projection in which the projection center is in the center of the body to be depicted. The term gnomonic projection is taken from the gnomon (shadow stick) of the sundial , in which the celestial sphere with the sun is depicted centrally from the gnomon tip ( nodus ) on the dial. The type of projection surface is arbitrary. An ancient example is the inner surface of a sphere in one of the oldest sundials, the scaphe . Often it is projected onto a plane, as for example mostly in cartography .

Construction of a gnomonic azimuthal projection in normal position, projected elements are marked with a line

The projection is used in cartography with a projection center in the center of the earth for map network designs and in crystallography with a projection center in the center of the crystal. Depending on the geometry of the image surface , the image is heavily distorted towards all or individual edges. The gnomonic projection is therefore not suitable for topographic maps. Nevertheless, the gnomonic azimuthal projections , in which the projection surface is a plane, are of practical use .

Projection true to the great circle

Azimuthal gnomonic projections are "true to the line"; H. all great circles on the earth's surface and thus all orthodromes are shown as straight lines. If you want to navigate from one point to another, you can therefore determine the route in a map with a gnomonic azimuthal projection by connecting the points with a straight line. That is why these maps are used together with conformal maps in navigation at sea and in the air as well as in radio navigation. The crystallography utilizes the fact that poles tautozonaler surfaces lie in gnomonischer Azimuthal on straight lines.

The gnomonic azimuthal projection allows the image of the open half-space beginning at the projection center , in which the image plane lies. A great circle lying in the boundaries of the half-space is projected into infinity. Because of the increasing distortion with increasing distance from the center of the projection, the projection is usually limited to an angular range of at most 60 ° around the central axis.

Gnomonic projections in cartography

Normal (polar) position

If the projection plane of a gnomonic azimuthal projection is in the normal position , ie it touches the earth in the pole, the parallels are shown as concentric circles around the pole and the meridians as radial straight lines. The map network design can be carried out with the help of compasses and ruler. The mapping equations in this position are for the polar coordinates of the map, azimuth and radius , with longitude and latitude complement (90 ° - geographical latitude) ${\ displaystyle \ alpha}$ ${\ displaystyle m}$ ${\ displaystyle \ lambda}$ ${\ displaystyle \ delta}$

{\ displaystyle \ alpha = \ lambda}

,

{\ displaystyle m = r \ cdot \ tan \ delta}

,

where is a scale factor. ${\ displaystyle r}$

Transversal position

The point of contact is on the equator. The meridians are shown as parallel straight lines, the circles of latitude as hyperbolas. The equator becomes a straight line that intersects the meridians perpendicularly.

Inclined position

The map touches any point on the earth's surface (here Japan). It also shows the meridians as a bundle of straight lines , the circles of latitude become conic sections .

Normal location:
North Pole
Transversal position:
Equator (0 ° geogr. Longitude)
Leaning position:
Japan

illustration

If you illuminate a globe from the inside (with an almost point-like light source in the center of the globe) so that the globe surface is projected onto a flat wall or the ceiling, this image on the wall corresponds to the gnomonic azimuthal projection.

If the axis of the globe is perpendicular to the ceiling, all parallel circles are mapped as concentric circles, at an inclined position as ellipses and hyperbolas . The meridians, however, remain straight.

Historical

The name is derived from Gnomon (γνόμον), the Greek word for a shadow stick used as an astronomical instrument.

With the tip of the gnomon as a shadow-casting projection center, for example, the position of the sun was depicted in a sundial and used to display the time of day as early as ancient times . A design specification for a sundial containing a gnomon has been handed down as The Analemma of Vitruvius .

Web links

Institute for Discrete Mathematics and Geometry at TU Wien: Gnomonic azimuthal projection of the earth - interactive
Institute for Discrete Mathematics and Geometry at the Vienna University of Technology: Gnomonic cone projection of the earth - interactive
Böhm hiking maps: Gnomonic azimuthal projection - u. a. in different positions