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Historical Aspects of Development of the Theory of Azimuthal Projections Rostislav Sossa, Pavel Korol

Th e problem of adequate imaging of the ’s the fundamental scientifi c work that is dedicated on the 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 , 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 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 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 Projections from 1850– 1950, Univ. of London, Master’s Thesis in , 1952. Depending on the position of the 2 J.B. Harley, D. Woodward, The , 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 . The Projections of Ancient , „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.

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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- mapping. a secant circle is a line of zero . 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 and non-perspective. According-

Fig. 2. General classifi cation scheme of azimuthal map projections. Note: in blue font marked conformal, in red – equal-, 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 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 with a rounded . 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 . 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 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 . 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 ( 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 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 , mic spiral. In the mathematical sense, ste-

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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 and ) 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. In general, the projection is ing from a center that are removed to infi n- used for conformal mapping of round areas. ity. Geometric justifi cation of orthographic Th e normal aspect of the stereographic projec- projection is the simplest compared to other tion is widely used to create topographic maps azimuthal projections. Orthographic projec- of the polar regions; the transverse was used to tion (from the Greek: ορθο – straight, γραφικός create maps of the Eastern and Western hemi- – drawing) was proposed in the second cen- in 17th–18th centuries, and oblique was tury BC by the ancient Greek mathematician

190 Studia Geohistorica • Nr 03. 2015 Historical Aspects of Development of the Theory of Azimuthal Map Projections Artykuły

Apollonius (262–190) and used for mapping converging at an arbitrary zenithal point on of areas that are limited by one hemisphere. a line passing through the center of the Earth About 14 BC, the Roman engineer Marcus and perpendicular to the projection plane, which Vitruvius Pollio used the orthographic projec- is usually tangent at the Earth surface. Because tion in the construction of a solar clock and it is a perspective projection, for each point on for computing its location. In 1613, François Earth the line passing through it defi nes the d’Aguilon from Antwerp off ered its present former’s projection where it intersects the plane. name for the projection. Th e earliest references Th e projection is parameterized by the distance to the use of projection are engraved terrestrial between the convergence point and the center of on wood by an unknown author (1509) the Earth; it is the general case of the azimuthal and works of Johannes Schöner (1533, 1551) and orthographic, stereographic and gnomonic pro- Apian (1524, 1551). Th e original map in or- jections. General Vertical Perspective projection thographic projection was designed by the dis- is a limiting case of the Tilted Perspective pro- tinguished engraver of the Renaissance Albre- jection, which does not require the projection cht Dürer and executed by Johannes Stabius plane to be perpendicular to the convergence (1515). Orthographic perspective projection of line and is not necessarily azimuthal. the world on the tangent plane of the perspec- Th e General Vertical Perspective projection tive point that is removed to infi nity provides is found in two varieties: “near-sided”, when the a globular form of the image as used to create convergence point is “above” the mapped sur- artistic images of the Earth, especially terrestrial face, and “far-sided”, when “below” it. Th e fi rst landscapes from space. Using the orthographic kind reproduces a view from the air or space projection in was limited mainly to maps, directly downwards, bounded by a circular ho- but at the present stage of space exploration, rizon, which is limited by the curvature of the much attention is paid to high quality images globe; the visible angular range grows to a maxi- of the and other planets from the orbit, mum of 90° (a whole hemisphere) of the zenith which has caused considerable revival of inter- at infi nity, which is the classic orthographic est in this projection. projection. In contrast, the far-sided kind nor- mally shows more than one hemisphere. Th e Modern perspective projections visible angular range shrinks with distance; When taking a picture of the Earth from the the limiting case at infi nity is again the ortho- Space, the camera fi xes images in an external graphic. And, like the stereographic case, the perspective projection. If the camera is precisely projecting lines fi rst “see” the inner face of the directed to the center of the Earth, then the globe. Although near-sided general vertical per- observers get a General Vertical Perspective spective maps are limited to mimicking views projection, and otherwise – Tilted Perspec- from space, the far-sided variant was adopted by tive projection. Th ese projections are a group several authors who chose diff erent projection of modern perspective azimuthal projections. in order to minimize global distortion General Vertical Perspective projection according to arbitrary criteria. Such perspective refers to the class of arbitrary perspective azi- projections were studied in 18th–19th centuries as muthal projections (except stereographic ver- minimum error projection, which has a geomet- sion); however, unlike orthographic, a projective ric substantiation. point is not at infi nity, but is situated at some In order to minimize the overall distortion fi xed distance from the Earth’s surface, usu- for arbitrary criteria, cartographers have used dif- ally from a few hundred to tens of thousands ferent distances from the nadir point to the per- kilometers. Some versions of this projection spective point. In particular, French astronomer were known to the Greeks and the Egyptians Philippe de La Hire (1701) proposed to place 2000 years ago. Th e General Vertical Perspec- a perspective point for a distance, which equals tive projection can be defi ned by straight lines to 1.7071 of Earth’s radius. Later, Antoine

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Parent (1702), Herman Heinrich Ernst von In 1981, American cartographer John Parr Hammer (1890) and Hans Th eodor Julius Snyder (1926–1997) proposed a modifi ed Christian Karl Maurer (1935) proposed other arbitrary perspective azimuthal projection – variants of projection with low distortion. Sir Tilted Perspective. Th e projection shows the Henry James and Alexander Russell Clark Earth geometrically when removing an airplane (1857, 1862, 1879) proposed projections with or any other point in space. If the point is above coeffi cients, respectively, 1.5, 1.368, 1.4, which the surface the projection used to create maps are calculated at the points of the projection on of individual areas (less than a hemisphere), the

a b Fig. 6. Modern perspective azimuthal projections: a) general vertical perspective; b) tilted perspective. Source: D. Strebe,“Geocart 3. User’s Manual”

the opposite side, below the projection surface terrestrial landscapes resembles pictures from and cover the continents, and are limited by space, displaying the results of aerial removing great or small circles. To refl ect the visible side and in photogrammetry. of the Moon Albert Novicky (1963) off ered to place a projective point on the distance of Non-perspective azimuthal projections. 1.53748 of the Earth’s radius. In order to pre- Original projections serve the integrity of the continents’ image, Sir Equidistant azimuthal projection used by the James and Clark used secant planes and oblique ancient Egyptians to create maps of the star sky variants of projection. In particular, Sir James are shown in some sacred books, for example, proposed to use the angle of divergence of rays in the works of a Muslim scholar and polymath – 113°30’ (90°+ 23°30’, from the zenith to the of the 11th century al-Biruni. Mathematical visible horizon), and Clark, for its most famous reasoning of the projection was made in 1581 variant “Twilight” proposed 108° (90°+18°, the by a French linguist, astronomer, theologian angle below the visible horizon which defi nes and cartographer Guillaume Postel (1510– the astronomical twilight). Th e General Ver- 1581). Postel projection is extremely simple tical Perspective projection is widely used in in construction. Th e peculiarity of this projection iconography and sacred art. Current projection is that the distance between two points along used to create a new generation of Internet- a straight line passing through the center of the -mapping applications, including Google Earth projection is accurate, so it is often used in nor- and NASA World Wind allow interactive pan- mal aspect, in order to minimize distortions often ning, zooming, and fl ight simulation. appearing for mapping hemispheres or polar re-

192 Studia Geohistorica • Nr 03. 2015 Historical Aspects of Development of the Theory of Azimuthal Map Projections Artykuły

abc Fig. 7. Original azimuthal projections: a) Postel; b) Lambert; c) Airy. Source: D. Strebe, “Geocart 3. User’s Manual” gions. In polar aspect equidistant azimuthal trary azimuthal projection – Airy Minimum- projection a fragment of the fl ag and emblem of error Azimuthal, which is based on the con- the United Nations is used with a picture of olive dition of minimizing the distortion and is branches in place of Antarctica. As an example, currently used to create medium-scale Land- we can cite the use of equidistant azimuthal pro- Survey maps of the UK. jection in the National of the United States (USGS), in large-scale mapping of Micronesia Modifi ed projections and Guam, in air navigation, in seismology, in Some azimuthal projections are obtained by radiodirection fi nding. making certain changes in the equations of fa- In 1772, the Swiss mathematician, physicist, mous map projections. Th ey are derived from philosopher and astronomer Johann Heinrich the actual azimuthal projections and form a group Lambert (1728–1777) developed equal-area of modifi ed azimuthal projection, while oth- azimuthal projection that resembles Postel equi- ers, such as star projections, which are formed by distant azimuthal projection, but to preserve nucleus of azimuthal projection hemispheres, are the scale of an area the distance between the folded or composite. Th e modifi ed azimuthal parallels are reduced. Due to the simplicity of projection is obtained by changing the met- construction, the projection is used in all as- ric space of one or more projections to obtain pects, although normal aspect was proposed by a cartographic image with desired properties. Anton Mario Lorgna (1789), after the death Th is group of azimuthal map projections is the of Lambert. Often normal aspect of projec- most numerous. tion is used for the mapping of the Northern Modifi ed azimuthal projection of the Ger- and Southern hemispheres, the Arctic and man mathematician and cartographer Adam Antarctic regions. Transverse aspect is used for Maximilian Nell (1824–1901) called the maps of Western and Eastern hemispheres, and Nell projection is an intermediate option be- oblique aspect is used for maps of continents tween the ordinary globular projection and the and oceans. Lambert equal-area azimuthal stereographic projection of hemispheres. Th e projection is used for visual analysis of maps, geometric construction principle of the projec- in pictures, and in the construction of spherical tion is as follows: meridians and parallels are panoramas. As a result of combining equal-area curves that pass between the relevant merid- azimuthal projection, Lambert and William- ians and parallels globular and stereographic Olson projections formed a modifi ed version of projections. Th e projection is very rarely used the Werner projection. for the creation of thematic maps in the atlases In 1861, a British mathematician, astrono- of the world. mer and geodesist Sir George Biddell Airy In 1892, a German navigator Friedrich Au- (1801–1892) proposed a non-perspective arbi- gust Arthur Breusing (1818–1892) developed

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a geometric variant of the azimuthal projection Orthodromic, Close, McCaw, Immler) respec- – Breusing Geometric, and in 1920, Alfred tively, and in 1919 and 1921, they also proposed Ernest Young created a harmonic variant as an arbitrary modifi ed azimuthal projection a modifi cation of geometric variant – Breusing called the Two-point Equidistant (Doubly Harmonic. Th e visual projection is virtually Equidistant), which is traditionally used in ob- identical to the Airy Minimum-error - lique aspect. Th e fi rst one is used in direction al projection. Geometric and harmonic variants fi nding, if the exact coordinates of two trans- of Breusing projection generalize properties of mitters (two central points) and the direction stereographic projection and Lambert equal- of the ship on them are known. Th e other is used area azimuthal projection, in particular dis- to determine the location of the ship at meas- tances between parallels of cartographic grid is ured distances to it from the two central points, calculated: in the fi rst case – as the geometric which set the station range for fi nding radio

a b c

d e f Fig. 8. Modifi ed azimuthal projections: a) Breusing geometric; b) Breusing Harmonic; c) Two-point Equidistant; d) Two-point Azimuthal; e) Chamberlin trimetric; f) Miller Oblated Stereographic. Source: D. Strebe,“Geocart 3. User’s Manual”

mean, and the second – as the harmonic mean navigation system. In addition, the projections of the distances between the parallels of the two are used to create maps of Asia, United States aforementioned projections. and Southern Canada, telegraph networks and In 1914 and 1922, two outstanding cartogra- sea routes of US National Geographic Society. phers of the 20th century German Hans Th eodor A combined Two-point Azimuthal and Two- Julius Christian Karl Maurer (1868–1945) -point Equidistant projection was proposed by and British Sir Charles Frederick Arden- Sir Charles Frederick Arden-Close in 1922. -Close (1865–1962), independently proposed In 1946, the American cartographer who an arbitrary modifi ed azimuthal projection – led the U.S. National Geographic Society Two-point Azimuthal (Doubly Azimuthal, from 1964 to 1971, Wellman Chamberlin

194 Studia Geohistorica • Nr 03. 2015 Historical Aspects of Development of the Theory of Azimuthal Map Projections Artykuły

(1908–1976), developed a modifi ed equidis- all other cases – carthographical grid has not tant azimuthal balance of errors projection – the axes of symmetry. Chamberlin trimetric projection. It is used A separate group of modifi ed azimuthal pro- to create maps of individual parts of conti- jections in transverse aspect includes lenticular nents in National Geographic Atlas of the Uni- azimuthal projections. In 1889, a Russian sci- ted States, including maps of Canada, Alaska entist and cartographer David Alexandrovich and Greenland (1:8 000 000, 1947), Australia Aitoff (1854–1933) announced a very simple (1:6 000 000, 1948), Europe and Middle East modifi cation of the equatorial aspect of an equi- (1:7 500 000, 1949), Africa and Arabian penin- distant azimuthal projection – Aitoff projec- sula (1:12 000 000, 1950) and others. tion. Doubling longitudinal values enabled the In 1950, a Soviet cartographer Nikolay An- whole world to fi t in the inner disc of the map; dreevich Urmayev (1895–1959) developed the horizontal scale was then doubled, creating two arbitrary azimuthal projections – Urmayev 1 a 2:1 ellipse. As a result, the map is neither azi- and Urmayev 2. Although in the projections muthal nor equidistant, except along the Equa- distortions of angles and areas are present, with tor and central meridian. Neither is it equivalent properties they are very similar to minimum- or conformal. Th e is a very in- -error azimuthal projections, such as Sir Henry teresting compromise between shape and scale James projection, but they have not received distortion. It clearly shows the Earth’s shape global recognition. with less polar shearing than Mollweide’s ellipti- In 1951, Bomford projection, which be- cal projection. However, this infl uential design longed to the class of modifi ed equal-area azi- was quickly superseded by Hammer’s work. muthal projections and is an oblique aspect of In 1892, properly crediting David Alexan- Aitoff -Hammer lenticular projection, was de- drovich Aitoff ’s previous work, German survey- signed for a new edition of the Oxford World or and cartographer Herman Heinrich Ernst Atlas by a British surveyor and cartographer Guy von Hammer (1858–1925) applied exactly the Bomford (1899–1996). Th e peculiarity of this same principle to Lambert’s azimuthal equal- projection is that when turning a grid by 45° -area projection. Th e resulting 2:1 elliptical counterclockwise, all parts of the land were equal-area design, called by the author Aitoff - visually imaged on the map. , by others at fi rst Ham- In 1953, Miller Oblated Stereographic pro- mer-Aitoff projection and then simply the jection, which belonged to the class of modifi ed -Hammer projection, soon became popular conformal azimuthal projections, was proposed and is used even today for world maps. It was by a American surveyor and cartographer Osborn itself the basis for several modifi ed projec- Maitland Miller (1897–1979). In cartographi- tions, like the oblique contribution by William cal literature to defi ne a projection sometimes A. Briesemeister. It is widely used in astronomy, other synonymic forms are used, including in creating the star maps, in microbiology, in Miller Prolated Stereographic, Oblated Ste- philately and in other spheres of human activ- reographic and Prolated Stereographic. Th e ity. Th e strong superfi cial resemblance of Aitoff ’s projection was used for mapping areas that are and Hammer’s projections led to considerable placed in an oval part of map, including Os- confusion, even in technical literature. born Maitland Miller applied it when creating In 1921, a German cartographer Oswald a maps of Africa and Europe (1953) and the Winkel (1874–1953) proposed the third and Eastern Hemisphere (1955, in combination best known hybrid projections called Winkel with several non-conformal projections) and Tripel projection (from the German for triple, Lorenzo Porter Lee – when creating a map of the possibly referring to a triple compromise of re- Pacifi c ocean (1974). When creating a map of duced shape, area and distance distortion). Like Europe and Africa is used a variant with sym- his two other proposals, it is defi ned by a sim- metry concerning the central meridian and in ple arithmetic mean including the equidistant

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a b

c d

e f

Fig. 9. Modifi ed lenticular azimuthal projections: a) Aitoff; b) Hammer; c) Eckert-Greifendorff; d) Briesemeister; e) Winkel III; f) Wagner VII. Source: D. Strebe, “Geocart 3. User’s Manual”

cylindrical projection and Aitoff projection, Friedrich Eduard Max Eckert-Greifendorff using an arbitrary value for standard parallels. (1868–1938) stretched the corresponding por- Winkel’s Tripel projection is peculiarly irregu- tion of a Hammer projection. In other words, lar: it is neither equal-area nor conformal; paral- it was exactly the same idea as Hammer’s, but lels are straight at the Equator and poles, curved with longitude compressed four times and the elsewhere; the scales are constant, but not equal, horizontal scale multiplied fourfold. Before re- only at the Equator and central meridian. Ne- scaling it uses only a narrow region near the vertheless, it manages to present a pleasant and central meridian of the original azimuthal map; balanced view of the world, which led to its as a consequence, parallels are almost straight choice by several popular atlases. In 1998, it was lines. selected by the prestigious National Geographic In 1941 and 1949, a German cartographer Society for its new reference world map, in place Karlheinz (Karl Heinrich) Wagner (1892– of the . 1962) proposed his own seventh and ninth Th e Eckert-Greifendorff projection an- equal-area azimuthal map projections – Wag- nounced in 1935 by a German cartographer ner VII (Hammer-Wagner) and Wagner IX

196 Studia Geohistorica • Nr 03. 2015 Historical Aspects of Development of the Theory of Azimuthal Map Projections Artykuły

(Aitoff -Wagner), which was a modifi cation including projections of satellite navigation, ob- of the Aitoff and Hammer projections with lique and equal-area “Magnifying-Glass” azi- changed scales. Th e projections are used to cre- muthal projections. He developed his own com- ate world maps and climatic maps of Commer- puter program to create cartographical grids for cial Department of the U.S. Government. more than 120 projections. In 1953, an American cartographer and geo- grapher William A. Briesemeister (1884– Retroazimuthal projections 1967) published quite a simple and very similar Th ey are a special group of azimuthal map pro- projection to Hammer’s. Th e oblique equal-area jections. In western literature the retroazimuthal modifi ed the azimuthal projection with a cen- projection was fi rst proposed in 1909 by a Bri- tral point 45°N and 10°E is called the Briese- tish economist, mathematician and cartographer meister projection, and its ratio of axes is 7:4. James Ireland Craig (1868–1952). In 1910, an It is used to create a world map with grouping alternate version was proposed almost immedi- of continents in the center of projection. ately thereafter by a German cartographer Her- man Heinrich Ernst von Hammer. Th e main Multi-scale projections property of these projections consists in that Th e creation of new azimuthal projections the directions () from every point of the in the 20th–21st centuries became quite chal- map to chosen central point are correct. In lenging, since their equations include a small this respect the maps diff er from the usual azi- number of variables that explain the limited va- muthal projections that render azimuths from riability of this class of projections compared the center correctly. As is the case with the nor- with cylindrical and especially pseudocylindry- mal azimuthal projections, the retroazimuthal cal projections. Variations of variables in the condition is not suffi cient to completely specify vast majority of cases led to the creation of ar- the projection, since two properties are required bitrary or conventional azimuthal projections. for most map projection. In Craig’s (1910) ver- One of them was proposed in 1957 by Swedish sion, the meridians are equally spaced, straight cartographer Stig Torsten Erik Hägerstrand parallel lines perpendicular to a horizontal (1916–2004). Using the method of “magnify- line through the origin. Hammer introduced ing glass” for multi-scale images, Hägerstrand the sensible alternate condition that all places created a projection in which the distance from should be at their proper distance from the cen- the center decreases in proportion to the loga- ter, a condition comparable to that defi ning the rithm of the actual distance. However, this azimuthal equidistant projection. Th is condi- approach cannot be used for large-scale map- tion changes both the spacing and the shape of ping, so the projection is only used for creating the meridians and parallels, because the merid- medium- and small-scale maps. ians become curved lines and the curvature of For the purpose of focusing attention on the parallels is modifi ed. Probably in the case a particular area and surrounding areas in 1987, of normal azimuthal projections the property of American cartographer John Parr Snyder de- equidistance can be substituted by the term veloped a series of azimuthal projections with of radial distance, but there is no obvious reason the eff ect of magnifying glass – “Magnifying- for doing it. Glass” projections. In two of these cases, the Jackson (1968) modifi ed the spacing of the mapping area in the middle of the circle used straight parallel meridians to obtain slight vari- Postel equidistant projection or Lambert equal- ants. When the meridians were spaced accord- -area projection, and in the other four cases – ing to the sine of the longitude diff erence, he stereographic, gnomonic or other azimuthal pro- showed that the parallels of latitude are ellipti- jections were used. Snyder was a world authority cal, and when they were spaced as the tangent of in the fi eld of mathematical cartography, and he one half of the longitudinal diff erence then the developer of a number of new map projections, parallels were parabolas. If the spacing is equal

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to the tangent of the diff erence then the parallels ing to Snyder, Gilbert of Bell Telephone Labo- become hyperbolas. He also provided a variant ratories satirically constructed a retroazimuthal with curved meridians, and a “retroazimuthal projection centered on Wall Street in New York stereographic” (but not conformal) projection. City, the Mecca of the fi nancial world. Others

a b c

Fig. 10. Retroazimuthal projections: a) Craig (Mecca projection); b) Hammer; c) Littrow. Source: D. Strebe, “Geocart 3. User’s Manual”

Th e projection devised by an Austrian as- who studied the problem of determining the tronomer Joseph Johann Edler von Littrow azimuth of the include Schoy (1917) and (1781–1840) in 1833 is not only conformal Berggren (1980). but it also has the property that all places lo- Instead of using a retroazimuthal projection, cated on the central meridian have the prop- it is possible to indicate the magnitude of the erty of being retroazimuthal (Maurer, 1919). retroazimuthal directions (retro-azimuths) on It is also possible to have a projection that is any map by means of isolines. Th e preferred retroazimuthal with respect to two spherical projection for this is a conformal one. A map locations (Maurer, 1914, 1919). None of these like this, for Rugby on the Mercator projec- projections can show the entire world without tion, was reported by Reeves (1929), and one overlap. Craig’s map, when expanded to the for Mecca, also on the , has entire sphere, seems to look like a two dimen- been prepared by Kimerling (2001). Th e confor- sional orthographic view of a three dimensio- mal property of the Mercator projection makes nal hyperbolic paraboloid. Th e overlapping the direct reading of azimuth angles quite easy. can be loosely envisioned as if coming from De Hesenler drew isolines on Miller’s cylindrical a projection of this onto the drawing plane. Of projection and Hagiwara (1984) portrayed lines course the projection is not made in this way of equal retroazimuths on an azimuthal equidis- but rather is derived mathematically. tant projection of the world centered on Mecca. Th e initial incentive for these projections was to enable Muslims to determine the qibla, that Composite projections is, the directions to the Ka’aba in Mecca. Craig On the Earth’s northern hemisphere, continen- (1910) also suggested that navy ships could use tal areas are clustered around the pole, while such a map centered on their home port to de- south of the Equator wide patches of ocean termine the direction of a radio station. A com- separate sparse mainlands; moreover, Africa and parable use was later made by Hinks (1929) the Americas narrow down towards the South. with a map centered at Rugby in the United Th is peculiar distribution of lands is the foun- Kingdom for colonials in the British Empire. dation for a group of interrupted star-shaped Additionally, de Henseler (1971) produced an map projections centered on the North Pole equidistant retroazimuthal map for the geo- with a more or less circular core (often a hemi- physical observatory in Addis Ababa. Accord- sphere) surrounded by lobes with the less impor-

198 Studia Geohistorica • Nr 03. 2015 Historical Aspects of Development of the Theory of Azimuthal Map Projections Artykuły tant Antarctica split between their ends. Some Th e Octants map was widely used in medieval star maps invert this pattern, with a southern cartography to create the appearance of oval pro- polar aspect privileging oceans. Non-polar as- jections. Th ey, like star projections, divided the pects are possible but virtually unknown, at Earth’s surface into eight equal parts, which are least with manual computation. usually limited to four lines equator and meridians. As a rule, star projections are composite de- Every spherical triangle is separately projected signs; polar azimuthal projections are natural onto an octant of approximately triangular shape,

a b Fig. 11. Composite projections: a) Maurer (Regular star); b) Berghaus star. Source: D. Strebe, „Geocart 3. User’s Manual” candidates for the core due to their circular faces of which are the arcs of the circles with the parallels. Th e projection’s creator must decide center on the opposite vertices of an equilateral whether the lobes are uniform in size and shape, triangle. Th is type of octant is called Reuleaux and how well they preserve the core’s features triangle. Eight Reuleaux triangles form azimuthal like shape and spacing of parallels. projection called Octant Leonardo da Vinci.

Fig. 12. World map – Mappa mundi in „Octant Leonardo da Vinci” projection. Source: http://odtmaps.com/behind_the_maps/amundi-map-details.asp, access: 5 November 2015

Star projections should not be mistaken for Circa 1514 an Italian physicist, mathemati- projections used for representing the celestial cian and naturalist Leonardo di ser Piero da sphere; such star maps are at least as old as geo- Vinci (1452–1519) off ered a projection which graphic charts. consists of separate segments. It did not have

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a cartographical grid and did not describe the uniform lobes, with boundary meridians at method of design. In 1551, a French mathema- 160°W, 88°W, 16°W, 56°E and 128°E. Of all tician and cartographer Oronce Finé (1494– major land masses, Australia and Antarctica are 1555) published a cartographical grid of pro- interrupted. Th is design became much more jection, but again without its detail. Based on popular than Petermann’s, appearing in atlases the Octant Leonardo da Vinci’s six detailed and in the logo of the Association of American maps of Maremma, Toscana and Umbria were Geographers. created. In this projection one of the fi rst maps Like the original, only the core is azimuthal, was created which marked a new world – the and the whole map preserves neither area nor America and Southern Polar continent. Some . Further variations with any number of critics believe that the map is not Leonardo’s lobes greater than two can easily be done and, own work, because the accuracy and skill of its although not polyhedral projections by design, execution do not meet the high quality stand- assembled into pyramids; with three symmet- ards of the master. Th e projection was used in rical lobes, the map is an equilateral triangle the creation of world maps of a French priva- foldable as a regular tetrahedron. teer, explorer and navigator Guillaume Le Testu In 1935, Hans Th eodor Julius Christian (1556) and German reformed theologian Dan- Karl Maurer presented a comprehensive cata- iel Angelocrator (1616). logue of map projections, organized by hierar- Although preceded by works by Leonardo chical criteria. His taxonomy included empty da Vinci (an octant map, 1514) and Guillaume categories, i.e., combinations of features not Le Testu (1556) which could be laid out as a cir- met by actual, existing projections; for illustra- cular arrangement of lobes, the fi rst modern star tion, he fi lled some of those gaps with designs projection was published in 1865 by Gustav of his own, including the star-shaped S231 and Jäger. In its polar aspect, the inner hemisphere is S233. Th e projection Maurer named S233 is an irregular octagon whose vertices are connect- a regular version of Jäger’s map, with six identi- ed to the pole by straight semimeridians with an cal lobes. In the polar aspect, all parallels and identical scale; all graticule lines are straight lines meridians are straight lines, parallels broken with , parallels broken at the bound- at boundary meridians and uniformly spaced ary meridians, and meridians at the Equator. along them; all meridians are broken at and uni- Each lobe in the outer hemisphere is a triangle formly spaced along the Equator. Each triangu- exactly mirroring one of the core sectors. lar lobe has an exact counterpart mirrored in the A more infl uential design was almost im- inner hemisphere. For the much more interest- mediately proposed after 1865 by August ing proposal S231, Lambert’s azimuthal equal- H. Petermann (1822–1878), a German cartog- -area projection was chosen for the inner hemi- rapher and enthusiast of exploration (especially sphere. In each lobe, only the central meridian is of polar areas), but its inner hemisphere is iden- straight, and parallels are circular arcs centered tical to the polar azimuthal equidistant’s. On on the North Pole. Th e scale along the central each lobe, parallels remain circular arcs cen- meridians is the same in both hemispheres, but tered on the pole, spaced the same as in the mirrored at the Equator; the scale is constant core but with variable scale; all semimeridians along each parallel, and the same on a lobe’s are straight lines. Sources diff er on whether the and on its counterpart in the core. Th erefore the eight lobes are uniform in size. Neither Jäger’s whole map is equal-area. Projections S231 and nor Petermann’s projections are either confor- S233 were described with six uniform lobes, but mal or equal-area, but the latter is of course azi- can be extended to any number greater than one muthal in the inner hemisphere. (S231) or two (S233). A variation of Petermann’s map, Hermann A more recent star projection based on Lam- Berghaus’s (1797–1884) star projection (1879) bert’s equal-area azimuthal was devised by Wil- reduced the number of appendages to fi ve liam William-Olsson (1968); however its core

200 Studia Geohistorica • Nr 03. 2015 Historical Aspects of Development of the Theory of Azimuthal Map Projections Artykuły is bounded by the 20°N parallel instead of the with three identical lobes using the equal- Equator. Th e entire map is equal-area, but unlike -area Werner projection maintaining the same Maurer’s S231, the four lobes are derived from parallel spacing. Again, parallel scales are not the Bonne/Werner pseudoconic projection: eve- the same in the two base projections except ry parallel is a circular arc centered on the North at the poles, thus parallels must be lengthened Pole with a constant scale, and each central me- in the lobes by about 26.6%. Since, unlike in ridian is a straight line, with the same scale as on William-Olsson’s design, parallel spacing is the core at 20°N. Unfortunately, parallel lengths not reciprocally reduced, not even the lobes do not coincide at the boundary latitude on the are equal-area. pristine Lambert and Bonne/Werner projections. Published in both northern and southern Matching the lengths at the junction requires polar aspects, this projection apparently owes increasing the scale of lobe parallels by the se- its name to a fortuitous resemblance of its lobe cant of half the boundary colatitude (the angular arrangement to a common lay-out for unfold- distance from the North Pole), for William-Ols- ed tetrahedra; it is unrelated to true polyhe- son’s choice, about 22.077%. Areas are preserved dral maps. by compressing the central meridians in lobes by Th e analysis of the construction and direc- the reciprocal amount. tions of uses of the most famous azimuthal Another hybrid azimuthal/pseudoconic star- map projections allows to determine priority -shaped proposal is the “tetrahedral” projection areas for further research in this branch and introduced in 1942 by John Bartholomew to defi ne the set of subclasses within which it (1923–2008), the fourth of a long lineage is possible to search new projections. Th ese, of map publishers with the same name and fi rst of all, should include modern perspec- surname. He is also the author of the oblique tive, composite and multi-scale modifi ed azi- Atlantis and the composite “Lotus”, “Kite” muthal map projection. Th e general classifi ca- and “Regional” maps. Th e “tetrahedral” map tion scheme azimuthal map projections do not combines a northern polar azimuthal equi- limit the possibilities of the appearance of new distant core limited by the 20°30”N parallel subclasses of such class of projections. „ (instead of the Equator adopted by Berghaus)

Bibliography d’Avezac de Castera-Macaya M.A.P., Coup d’oeil Dahlberg R.E., Evolution of Interrupted Map historique sur la projection des cartes de géogra- Projections, “International Yearbook of Car- phie, “Societe de Geographie, Bulletin. Se- tography”, 2, 1962, pp. 36–54. ries 5”, 5, 1863, (Apr–May), pp. 257–361; Finsterwalder R., Die Projektion der Weltkarte in (June), pp. 438–485. den Atlanten von Battista Agnese. Ein Beitrag Balchin W.G.V., Map Projections in History, zur Entwicklung der Kartennetze in der ersten “Impulse”, 2, 1957, pp. 9–13. Hälfte des 16. Jahrhunderts, “Acta Albertina Barbie du Bocage J.D., Lacroix S.F., Notice Ratisbonensia”, 46, 1989, pp. 19–27. historique et analytique sur la construction des Flegg G., Map Projections, in Projection in His- cartes geographiques, “Journal des Sciences tory of Mathematics, unit 5: Projection, Mil- Militaires”, 1, 1825, pp. 62–85. ton Keynes, Open University Press, 1975, Botley F.V., British and American World Map pp. 22–25. Projections from 1850–1950, Univ. of London, Grüll J., Heinrich Christian Albers (1773–1833). Master’s Th esis in Geography, 1952. Leben und Wirken des Lüneburger Kartographen,

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“Kartographische Nachrichten”, 23 (5), 1973, Schröder E., Kartenentwürfe der Erde. Kartogra- pp. 196–203. phische Abbildungsverfahren aus mathemati- Hall E.F., Gerard Mercator. His Life and Works, scher und historischer Sicht, Frankfurt am Main “Journal of American Geographical Society 1988. of New York”, 10 (4), 1878, pp. 163–196. Snyder J.P., Flattening the Earth. Two Th ousand Harley J.B., Woodward D., Th e History of Car- Years of Map Projections, Chicago 1993. tography, vol. 1–2, Chicago 1987–1994. Snyder J.P., Map Projections. A Working Manual, Herrmann A., Marinus von Tyrus, “Petermanns Washington 1987 (http://pubs.er.usgs.gov/ Mitteilungen. Erganzungsheft”, 209, 1930, publication/pp1395, access: 15 September pp. 45–54. 2015). Hinks A.R., Edward Wright and Mercator’s Pro- Strebe D., Geocart 3. User’s Manual. Manual jection, in: 12th International Geographical Describes Geocart 3.1, Seattle 2009. Congress, Cambridge, July 1928. Report of the Th rower N.J.W., Projections of Maps of Fifteenth Proceedings, Nendeln 1972, p. 109. and Sixteenth Century European Discoveries, Kennedy M., Kopp S., Understanding Map Madrid 1993, pp. 81–87. Projections, Redlands 1994–2000. Tobler W.R., Medieval Distortions. Th e Projec- Keuning J., Th e History of Geographical Map tions of Ancient Maps, “Annals of the Associa- Projections until 1600, “Imago Mundi”, 12, tion of American Geographers”, 56 (2), 1966, 1955, pp. 1–24. pp. 351–361. Mollweide C.B., Mappirungskunst des Claudius Бугаевский Л.М., Математическая карто- Ptolemaeus, ein Beitrag zur Geschichte der Land- графия, Москва 1998. karten, “Zach’s Monatliche Korrespondenz Фролов Ю., От Клавдия Птолемея до Риго- zur Beförderung der Erd- und Himmels-Kun- берта Бонна, “Вести ЛГУ. Cерия Геологии de”, 11 (1805), pp. 319–340 (Apr.), 504–514 и Географии”, 6 (1), 1963, pp. 118–125. „ (June); 12 (1805), pp. 13–22 (July). Robinson A.H. et al., Elements of Cartography, New York 1995.

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Aspekty historyczne rozwoju teorii odwzorowań azymutalnych

Streszczenie Problem odwzorowania powierzchni Ziemi na ju i zastosowaniu wyłącznie odwzorowań azy- płaszczyźnie zajmuje matematyków i karto- mutalnych (płaszczyznowych), śledząc zmiany grafów od ponad 2000 lat. Zagadnienie to ma od czasów starożytnych do współczesnych. istotne znaczenie nie tylko dla teoretyków kar- Omawiając podstawowe cechy poszczególnych tografi i, ale również dla użytkowników map, odwzorowań azymutalnych, przedstawiono ich ponieważ w rezultacie przekształceń wynika- szczegółową klasyfi kację, m.in. na typy zwią- jących z rodzaju odwzorowania następuje znie- zane z położeniem środka rzutu (np. centralny, kształcenie pewnych zależności geometrycz- stereografi czny, ortografi czny), nie mające cha- nych występujących na powierzchni Ziemi. rakteru rzutów, jak najpopularniejsze z nich od- W odróżnieniu od wielu publikacji przedsta- wzorowanie Postela (równoodległościowe) czy wiających odwzorowania kartografi czne jako Lamberta (równopolowe), a także na wszelkie jedno z zagadnień kartografi i matematycznej modyfi kacje typów podstawowych. „ w niniejszym artykule skupiono się na rozwo-

Keywords: types of azimuthal map projections, perspective and non-perspective azimuthal projections, modifi ed azimuthal projections, retroazimuthal projections, composite projections

Slowa kluczowe: typy odwzorowań kartografi cznych, odwzorowania azymutalne perspektywiczne i nieperspektywiczne, odwzorowania azymutalne modyfi kowane, odwzorowania retroazymutalne, odwzorowania łączone

Pavel Korol – docent Wschodnioeuropejskiego Narodowego Uniwersytetu im. Łesi Ukrainki w Łucku, zatrudniony w Katedrze Geodezji, Gospodarki Gruntami i Katastru (e-mail: [email protected])

Rostislav Sossa – profesor Kijowskiego Uniwersytetu Narodowego im. Tarasa Szewczenki, dyrektor Państwowego Przedsiębiorstwa Naukowo-Produkcyjnego „Kartografi a” w Kijowie (e-mail: [email protected])

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