Artykuły Historical Aspects of Development of the Theory of Azimuthal Map Projections Rostislav Sossa, Pavel Korol Th e problem of adequate imaging of the Earth’s the fundamental scientifi c work that is dedicated surface on the plane of a map is not new. Over to the history of map projections is the defi nitive the past 2000 years mathematicians and cartog- book by John Parr Snyder6. Historical aspects of raphers throughout the world have been con- the theory of azimuthal map projections were not sidering various methods of fi nding the ideal considered separately. cartographic projection, but a priori the exist- Azimuthal are map projections, parallels ence of such a projection is not possible, since of which are represented by concentric circles, the spheroid surface is not deployed on the plane and the meridians – straight lines that intersect without folds or breaks. at the center of the circle (pole of projection) at Th e historical aspects of the development an angle which equal to the diff erence of lon- of mathematical cartography and the theory of gitudes of corresponding meridians. From this normal transverse oblique Fig. 1. Aspects of azimuthal projections. Source: M. Kennedy, S. Kopp, “Understanding Map Projections”, Redlands 1994–2000 map projections, as the main constituent have defi nition it follows that the azimuthal projec- been considered in scientifi c papers by F.V. Bot- tions is a particular type of conic projection. ley1, J. Brian Harley, David Woodward2, Arthur Th e point of convergence of meridians in the H. Robinson, Joel L. Morrison, A. Jon Kimerling, azimuthal projections coincides with the posi- Phillip C. Muehrcke, Stephen C. Guptill3, Eber- tion of the pole of a normal coordinate system. hard Schröder4 and Waldo R. Tobler5. However, Depending on the type of functions, proper- ties of the azimuthal projections (conformality, equivalence, equidistance) are defi ned. 1 F.V. Botley, British and American World Map Projections from 1850– 1950, Univ. of London, Master’s Thesis in Geography, 1952. Depending on the position of the tangent 2 J.B. Harley, D. Woodward, The History of Cartography, vol. 1–2, Chicago point on the Earth’s surface, azimuthal projec- 1987–1994. tions are divided into three aspects: normal 3 A.H. Robinson, J.L. Morrison, Ph.C. Muehrcke, A.J. Kimerling, S.C. Guptill, Elements of Cartography, New York 1995. (polar), if the tangent point is on the pole, 4 E. Schröder, Kartenentwürfe der Erde. Kartographische Abbildungs- transverse (equatorial), if the tangent point verfahren aus mathematischer und historischer Sicht, Frankfurt am Main 1988. 5 W.R. Tobler, Medieval Distortions. The Projections of Ancient Maps, „Annals of the Association of American Geographers”, 56 (2), 1966, 6 J.P. Snyder, Flattening the Earth. Two Thousand Years of Map Projections, pp. 351–361. Chicago 1993. Studia Geohistorica • Nr 03. 2015 187 Artykuły Rostislav Sossa, Pavel Korol is on the equator, and oblique (horizontal), if and Southern hemispheres, usually a normal as- the tangent point is in any other place on the pect is used, for maps of the Western and East- Earth surface. ern Hemisphere a transverse aspect is used, and Depending on the placement of the picture for maps of continents and parts of continents plane relative to the Earth’s surface, the azi- – oblique azimuthal projection is used. muthal projection may be tangent or secant. Some azimuthal projections that were known For azimuthal projections, which use a tangent 2500 years ago have not lost their theoretical plane, the tangent point is a point of zero distor- and practical importance in our times, although tion, and for projections using a secant planes, they are used primarily in small-scale mapping. a secant circle is a line of zero distortion. In Depending on the nature of the construct- both cases, lines of equal distortions have the ing, azimuthal projections are divided into form of concentric circles that coincide with perspective and non-perspective. According- Fig. 2. General classifi cation scheme of azimuthal map projections. Note: in blue font marked conformal, in red – equal-area, in green – equidistant, in black – other arbitrary azimuthal projections the parallels of a normal grid. Distortions of all ly, we developed a general classifi cation scheme types grow with distance from the point or line of azimuthal map projections that are shown in of zero distortion. fi gure 2. Despite the limited variability of vari- Azimuthal projections in normal, transverse ables in the equations of azimuthal projections, and oblique aspects are widely used for the map- the number of modifi cations, as opposed to cy- ping of areas with a rounded shape. In particu- lindrical and conical projections, are virtually lar, for the construction of maps of the Northern unlimited. 188 Studia Geohistorica • Nr 03. 2015 Historical Aspects of Development of the Theory of Azimuthal Map Projections Artykuły Perspective projections therefore it is used for the mapping of territories A perspective azimuthal projection is obtained which have less than one hemisphere. Projec- by projecting points of the Earth’s surface onto tion refers to the class of arbitrary perspective a picture plane by rays propagating from a cer- azimuthal projections. Th e gnomonic projec- tain constant perspective point. A picture plane tion is said to be the oldest map projection. may be secant, tangent to the Earth or it may It was developed by ancient Greek philosopher be at a certain distance from it, and a perspective Th ales of Miletus (624–546) about 580 BC. point located on the perpendicular to the picture Th e name of this projection comes from the plane passing through the center of the Earth. gnomon sundial, whose hourly shadows refl ect Depending on the position of the prospect’s on the image plane one meridian of mapping point relative to the center of mass of the Earth, grid. Th e peculiarity of this projection is that perspective azimuthal projections are divided all great circles (meridians and the equator) are into: orthographic, if the prospects point is straight lines on it in such a way that the short- removed to infi nity; external, if the prospects est distance between two points on the map point is located at some distance from the center is a straight line. In the past, a projection was of mass of the Earth; stereographic, if the used for creating celestial maps and multifac- distance from the center of mass of the Earth eted globes. It is currently used in navigation for plotting direction fi nding bearings, and in astronomy for observing meteors (Lorenzoni picturepicture plan planee nets), in seismology for mapping propagation directions of seismic waves, in multidimension- al hyperbolic, in geometry for constructing of Beltrami-Klein projective model, and in pho- tography for receiving the images from camera obscura or rectilinear lenses. Stereographic (Ptolemy planisphere) pro- jection refers to the class of conformal per- orthographic externalstereographic gnomonic spective azimuthal projections, the perspective point of which is located in a nadir point. Stere- Fig. 3. Types of perspective azimuthal projections. ographic projection (from the Greek: στέρεο – Source: Л.М. Бугаевский, „Математическая картография”, solid, γραφικός – drawing) was proposed in the Москва 1998 second century BC by an ancient Greek astron- omer Hipparchus (190–126), but for a long to a prospects point is equal to the radius of time the scope of its use was limited by maps of the Earth, i.e. prospect’s center is located at the the star sky. It is believed that one of the earliest point opposite the point of contact to picture maps of the world, established in stereographic plane; and gnomonic (central), if the prospect’s projection, is a map of the French mathemati- point is placed in the center of the Earth. cian Gualterious Lud (1507). Th e transverse Gnomonic, stereographic and orthographic aspect of stereographic projection was used by projections form a group of classic perspec- Jean Rose (1542) and Rumold Mercator (1595). tive azimuthal projections, and externals – Th e current name of the projection was pro- a group of modern perspective azimuthal posed by the French mathematician François projections. d’Aguilon (1613) in work Opticorum libri sex philosophis juxta ac mathematicis utiles. All great Classic perspective projections and small circles (including meridians and Gnomonic (central, azimuthal centrograph- parallels) on the Earth’s surface, represented ic or gnomic) projection is based on the prin- as arcs of concentric circles or straight lines, ciple of azimuthal stereographic, but a perspec- and the rhumb lines have a form of logarith- tive point is situated in the center of the sphere, mic spiral. In the mathematical sense, ste- Studia Geohistorica • Nr 03. 2015 189 Artykuły Rostislav Sossa, Pavel Korol gnomonic stereographic orthographic Fig. 4. Types of classic perspective azimuthal projections. Source: M. Kennedy, S. Kopp, “Understanding Map Projections” a b c Fig. 5. Classic perspective azimuthal projections: a) gnomonic; b) stereographic; c) orthographic. Source: D. Strebe, “Geocart 3. User’s Manual. Manual describes Geocart 3.1”, Seattle 2009 reographic projection is conformal mapping – used to display the paths of solar eclipses. Ster- it preserves angles between curves and the shape eographic projection is used for a visual display of infi nitesimal pieces, transforming circles on of point symmetry groups of crystals, as well as the plane in the circle on the sphere, and on the for displaying spherical panoramas. plane – in a circle passing through the cen- Orthographic (orthogonal or Ptolemy anal- ter of projection. It provides a homeomorphism emma – sundial latitude and longitude) projec- of complex projective line in the transition to tion refers to the class of arbitrary perspective a two-dimensional sphere, and moves its sphere azimuthal projections which has been used in generating a Moebius transformation on the the design of a system of parallel rays emanat- complex plane.