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Journal of Applied 2016; aop

Research Article

Ebrahim Ghaderpour* Some Equal-, Conformal and Conventional Projections: A Tutorial Review

DOI 10.1515/jag-2015-0033 There are a number of techniques for map , received December 22, 2015; accepted April 23, 2016. yet in all of them occurs in length, angle, , area or in a combination of these. Abstract: Map projections have been widely used in many showed that a ’s cannot be represented on such as , oceanography, meteorology, a map without distortion (cf., [8]). geology, geodesy, photogrammetry and global positioning A terrestrial is a three dimensional model systems. Understanding different types of map projections of the that does not distort the real shape and the is very crucial in these areas. This paper presents a tutorial real size of large futures of the Earth. The term globe is used review of various types of current map projections such as for those objects that are approximately spherical. The equal-area, conformal and conventional. We present these equation for spherical model of the Earth with radius R is map projections from a model of the Earth to a flat sheet of paper or map and derive the plotting equations for them in x 2 y 2 z 2 ++=1. (2) detail. The first fundamental form and the Gaussian fun- 2 2 2 R R R damental quantities are defined and applied to obtain the plotting equations and in length, shape and An oblate or is a quadratic surface size for some of these map projections. obtained by rotating an ellipse about its minor axis (the axis that passes through the pole and the south Keywords: Conformal, Distortion, Equal-area, Gaussian pole). The shape of the Earth is appeared to be an oblate Fundamental Form, Geographical Information System, ellipsoid (mean ), and the geodetic , Tissot’s indicatrix and of positions on the surface of the Earth coming from satellite observations are on this ellipsoid. The equation for spheroidal model of the Earth is 1 Introduction x 2 y 2 z 2 ++=1, (3) 2 2 2 A map projection is a systematic transformation of the a a b latitudes and longitudes of positions on the surface of where a is the semi-major axis, and b is the semi-minor the Earth to a flat sheet of paper, a map. More precisely, axis of the spheroid of revolution. a map projection requires a transformation from a set of The spherical representation of the Earth (terrestrial two independent coordinates on a model of the Earth (the globe) must be modified to maintain accurate representa- f and l) to a set of two independent tion of either shape or size of the spheroidal representation coordinates on the map (e. g., the Cartesian coordinates of the Earth. We discuss about these two representations x and y), i. e., a transformation matrix T such that in Section 3.  x   φ  There are three major types of map projections: =.T (1)  y   λ  1. Equal-area projections. These projections preserve the area (the size) between the map and the model However, since we deal with partial derivative and fun- of the Earth. In other words, every section of the map damental quantities (to be defined in Section 2), it is keeps a constant ratio to the area of the Earth that rep- impossible to find such a transformation explicitly. resents. Some of these projections are Albers with one or two standard parallels [6] (the conical equal-area),

*Corresponding author: Ebrahim Ghaderpour, Department of Earth the Bonne [8], the azimuthal and Lambert cylindrical and Space Science and Engineering, York University, Toronto, ON, equal-area [6, 8] which are best applied to a local area Canada M3J1P3, e-mail: [email protected] of the Earth, and some of them are world such

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as the sinusoidal [8], the Mollweide [8, 12], the para- in length for the and in length and area bolic [13], the Hammer-Aitoff [8, 12], the Boggs eumor- for the are calculated [8, 9, 15]. phic [13], and Eckert IV [11]. As an application of equal-area projections, we can mention for instance Lambert azimuthal projection [6, 8] which is often used by researchers in structural geology to plot crys- 2 Mathematical Fundamentals tallographic axes and faces, lineation and foliation of Map Projections in rocks, slickensides in faults, and other linear and planar features. In this section, we derive the first fundamental form for 2. Conformal projections. These projections, while a general surface that completely describes the metric not so useful for portraying large areas, are very properties of the surface, and it is a key in map projec- important in surveying and mapping, as angles are tion. Furthermore, we discuss about Tissot’s indicatrices truly preserved. These projections include the Mer- (Tissot’s ellipses) that usually depict on the maps to show cator, the Lambert conformal with one standard the various types of distortions. This section is mainly parallel, and the stereographic (e. g., [6, 8]. These based on [5, 6, 8, 9, 12, 15]. projections are only applicable to limited areas on the model of the Earth for any one map. Conformal projections have applications in , certain 2.1 Differential Geometry of a General kinds of navigation and geographical information Surface and First Fundamental Form system (GIS). Lambert conformal conic and Merca- tor conformal cylindrical are commonly used for GIS. The parametric representation of a surface requires two Google and Bing Maps use a variant of the Mercator parameters. Let a and b be these two parameters. For conformal projection called Web Mercator or Google instance, for a particular surface such as the model of the Web Mercator. This for instance shows Greenland Earth, these two parameters are the latitude and longitude.   as large as Australia, however, in reality Australia is The vector at any point P on a surface is given by r = r (a, b). more than three and half times larger than Greenland If either of parameters a or b is held constant and the other (e. g., [4]). one is varied, a space curve results (cf., Figure 1). 3. Conventional projections. These projections are The vectors to a-curve and b-curve at point P neither equal-area nor conformal, and they are are respectively as follows: designed based on some particular applications, for ∂r ∂r instance in GIS and on web pages. Some examples are a =, b =. (4) the simple conic [6], the gnomonic [8], the azimuthal ∂αβ∂ equidistant [9], the Miller [12], the polyconic [13], the Robinson [8], and the Plate Carree projections [8, 9]. The Plate Carree was commonly used in the past since it has a very simple calculation before computers allowed for complex calculations.

In this paper, we only show the derivation of plotting equations on a map for the Mercator cylindrical confor- mal and Lambert cylindrical equal-area for a spherical model of the Earth (Section 4), the Albers with one stan- dard parallel and the azimuthal for a spherical model of the Earth and the Lambert conformal with one standard parallel for a spheroidal model of the Earth (Section 5), the sinusoidal (Section 6), the simple conic and the plate carree projections (Section 7). The methods of obtaining other projections are similar to these projections, and we refer the reader to [3, 6, 8, 9]. In Section 8, the equations for distortions of length, area and angle are derived. As an example, the distortions Figure 1: Geometry for parametric curves.

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 The total differential of r is Example 1 The first fundamental form for a planar surface 1. in the Cartesian coordinate system (a cylindrical (5) dr =+adαβbd . surface) is (ds)2 = (dx)2 + (dy)2, where E = G = 1, 2. in the (a conical surface) is The first fundamental form (e. g., [8]) is defined as the dot (ds)2 = (dr)2 + r2(dθ)2, where E = 1 and G = r2, product of Equation (5) with itself: 3. in the spherical model of the Earth, Equation (2), 2 is (ds)2 = R2(df)2 + R2 cos2 f(dl)2, where E = R2 and ds ==dr ⋅⋅dr adαβ++bd adαβbd 2 2 () ()() G = R cos f, and 22 4. in the spheroidal model of the Earth, Equation (3), =+Edαα2+Fd dGββd , (6) () () is (ds)2 = M2(df)2 + N 2 cos2 f(dl)2, where E = M 2 and       G = N2 cos2 f in which M is the radius of curvature in where E = a . a, F = a . b and G = b . b are known as the and N is the radius of curvature in prime ver- Gaussian fundamental quantities. tical which are both functions of f: –– From Equation 6, the between two arbitrary

points P1 and P2 on the surface can be calculated: 2 ae1– ab a M = (),=N , P 22 1.5 0.5 2 22 22 sE=+∫ dFαα2+ddββGd 1–eab sin φφ1–eab sin P1 () () ()() 22 2 2 ab– eab = . (11) P2 dβ dβ =+∫ EF2+G dα. (7) a2 P1 dα dα See Figure 2, and [8, 15] for the derivations of M and N.   –– The angle between a and b is simply given by ab⋅ F 2.2 Transformation Equations of Map θ cos= =. (8) Projections ab⋅ EG Suppose that a = f and b = l are the parameters of the –– Incremental area is the magnitude of the cross product model of the Earth with the fundamental quantities e, f   of ada and bdb, i. e., and g given by e = R2, f = 0, g = R2 cos2 f for the spherical model of the Earth, or given by e = M2, f = 0 and g = N 2 cos2 f dA ==adαβ×⋅bd adαβbd sinθ for the spheroidal model of the Earth (cf., Example 1). =sab⋅ inθαddβ

=1EG –cos2 θαddβ EG– F 2 = EG ddαβfrom Equation (8) EG =–EG Fd2 αβd . (9)

Since we are dealing with latitudes and longitudes on a  spherical or spheroidal model of the Earth, the vectors a and b are orthogonal (meridians are normal to equator parallels). Also, in maps, we are dealing with the polar and Cartesian coordinate systems in which their axes are perpendicular. Thus, from Equation (8), because 90° = 0, one obtains F = 0. Therefore, the first fundamental form Equation (6) in map projection will be deduced to the following form: x 2 y2 z 2 222 Figure 2: Geometry for the spheroidal model of the Earth, ++=1. ()ds =+Ed()αβGd(). (10) a2 a2 b2

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Consider a two-dimensional projection with para- Equations (14), (15) and (16), the following relation can be metric curves defined by the parameters u and v. For derived (see [8, Chapter 2]): instance, for the polar or conical coordinates, we have   22    22  u = r and υ = θ. Let E¢, F¢ and G¢ be its fundamental quan- ∂u ∂v ∂u ∂v   ′   ′   ′   ′ (17) tities. Also, assume that on the plotting surface a second E =+  E   GG,+  E   G . ∂φφ ∂  ∂λλ ∂  set of parameters, x and y, with the fundamental quanti- ties E, F and G. From Equation (9), a mapping from the Earth to the plot- The relationship between the two sets of parameters ting surface preserves area if (e. g., [8]): on the is given by eg =.EG (18) xx==()uv,, yy()uv,. (12) From Equations (15), (16), (17) and using F = F¢ = 0, one As an example, x = r cos θ and y = r sin θ for the polar and obtains Cartesian coordinates. The relationship between the parametric curves f, l, ∂uu∂ u and υ is 2 ∂φλ∂ EG =,JE⋅ ′′GJ=,(19) ∂vv∂ uu=,()φλ,=vv()φλ,. (13) ∂φλ∂

Equation (13) must be unique and reversible, i. e., a point where J is the Jacobian determinant of the transformation on the Earth must represent only one point on the map from the coordinate set f and l to the coordinate set u and υ. and vice versa. From Equations (12) and (13), we have By a theorem of (e. g., [8]), a mapping for the orthogonal curves is conformal if and xx=,uvφλ,,φλ ,=yyuvφλ,,φλ,.(14) ()()() ()()() only if

From the definition of the Gaussian first fundamental E G =. (20) quantities, we have e g

      22  ∂xy∂ ∂xy∂ ∂xy∂   ⋅      Ea==⋅a  ,,   =+    , ∂φφ∂  ∂φφ∂  ∂φφ ∂  2.3 Tissot’s Indicatrix     ∂xy∂ ∂xy∂ ∂xx∂ ∂yy∂   ⋅  Fa==⋅b  ,,   =+, Suppose that a terrestrial globe is covered with infinites- ∂φφ∂  ∂λλ∂  ∂φλ∂ ∂φλ∂ imal circles. In order to show distortions in a map projec-       22  tion, one may look at the projection of these circles in a ∂xy∂ ∂xy∂ ∂xy∂   ⋅      Gb==⋅b  ,,   =+    . (15) map which are ellipses whose axes are the two principal ∂λλ∂  ∂λλ∂  ∂λλ ∂  directions along which scale is maximal and minimal at that point on the map. This mathematical contrivance is Note that in here a and b in Equation (4) are replaced by f called Tissot’s indicatrix (e. g., [6, 12]). and l, respectively. Similarly, we have Let a and b respectively be the semi-major and       22  semi-minor axes of the projected ellipse on a map with ∂x ∂y ∂x ∂y ∂x ∂y ′   ⋅      coordinates x ad y. It is shown in [12] that E =,   ,=   +,  ∂u ∂u ∂u ∂u ∂u ∂u       22 22 ∂x ∂y ∂x ∂y ∂x ∂x ∂y ∂y ah=0.5  ++kh2skhin ϑϑ++kh–2 k sin, (21) ′   ⋅    F =,   ,= +, ∂u ∂u ∂v ∂v ∂u ∂v ∂u ∂v 22           22 22 ∂x ∂y ∂x ∂y ∂x ∂y bh=0.5  ++kh2skhin ϑϑ–+kh–2 k sin, (22) ′   ⋅        G =,   ,=   +.  (16) ∂v ∂v  ∂v ∂v  ∂v  ∂v  where h and k represent scales along the meridian and As we mentioned earlier, since we are dealing with parallel for a given point respectively, and υ is the angular orthogonal curves, f = F = F¢ = 0. Using this fact and deformation:

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2 2 2 2 2 2 have e = M , g = N cos f, E¢ = R A and G¢ = R A cos fA. By   22  1 ∂xy ∂  Equations (18) and (19), h =   +,  (23) R ∂φφ∂    2

∂φAA∂φ

22 ∂φ ∂λ    22 242  MN cos=φφRAAcos.(27) 1 ∂xy ∂  ∂λ ∂λ k =   +,  (24) AA Rcos φλ∂  ∂λ  ∂φ ∂λ

  1 ∂yx∂ ∂xy∂ In the transformation from the ellipsoid to the authalic sin=ϑ  –, (25) 2   sphere, longitude is invariant, i. e., l = lA. Moreover, fA is Rhk cos φφ∂ ∂λφ∂ ∂λ  independent of lA and so l. Thus Equation (27) reduces to where f, l, R are latitude, longitude and the radius of 2 ∂φA the globe, respectively. Moreover, the maximum angular 22 242 0 MN cos=φφRAAcos ∂φ . (28) distortion denoted by ω can be calculated as (cf., [12]): 01   ab– ω   =2arcsin  . (26) Substitute the values of M and N given by Equation (11) ab+  into Equation (28) to obtain

Usually Tissot’s indicatrices are placed across a map along 22 ae1– ab the intersections of meridians and parallels to the equator, () 2 cos=φφdRAAcos.φφd A 2 (29) and they provide a good tool to calculate the magnitude of 22 1–eab sin φ distortions at those points (the intersections). In an equal- () area projection, Tissot’s indicatrices (Tissot’s ellipses) Integrating the left hand side of Equation (29) from 0 to f change shape, whereas their areas remain the same. In (using binary expansion), and the right hand side from 0 conformal projection, the shape of Tissot’s indicatrices to fA, one obtains preserves (i. e., a = b in Equation (26), and so ω = 0), but their area varies. In conventional projection, both shape  2 and area of Tissot’s indicatrices vary. 22φφ22 3 φ RaAAsin= 1–eeab sin+ ab sin () 3  3 4 45φφ67… + eeab sin+ ab sin+. (30) 3 Projection from an Ellipsoid 5 7  to a Sphere Assuming fA = 90° when f = 90°, Equation (30) gives:

In this section, we describe how much the latitudes   2 3 4 Ra22=1–1ee22 + + ee46+ +.… and longitudes of a spheroidal model of the Earth will Aa()ba babab  (31) be affected once they are transformed to a spherical  3 5 7  model, i. e., how much distortion in shape and size Substituting Equation (31) into Equation (30), one obtains happens when one projects a spheroidal model of the Earth to a spherical model [3, 8, 9, 14]. We distinguish    2 223 444 66  two cases, equal-area transformation and conformal 1+ eeab sin+φφab sin+eab sin+φ  3 5 7  transformation. sin=φφA sin  . Case 1. A spherical model of the Earth that has the  2 243 4 6   1+ eeab + ab + eab +  same surface area as that of the is  3 5 7  called the authalic sphere [8]. This sphere may be used as (32) an intermediate step in the transformation from the ellip- soid to the mapping surface. Since the eccentricity eab is a small number, the above

Let RA, fA and lA be the authalic radius, latitude and series are convergent. The relation between authalic and longitude, respectively. Also, let f and l be the geodetic geodetic latitudes is equal at latitudes 0° and 90°, and latitude and longitude, respectively. From Example 1, we the difference between them at other latitudes is about

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0°.1 for the WGS-84 spheroid (see [8] for the definitions of 4 Mercator and Lambert Cylindrical the WGS-84 and WGS-72 ). Projections Example 2 In this section, by an elementary method, we show the 1. For the WGS-72 spheroid with a ≈ 6,378,135 m and cylindrical method that Mercator used to map from a spher- eab ≈ 0.081818, the radius of the authalic sphere is ical model of the Earth to a flat sheet of paper and derive   its plotting equations. Also, we give the plotting equations 2 3 Ra≈ 22 4  ≈6,371,004 m. (33) for the Lambert cylindrical equal-area projection. In the fol- A 1–eeab 1+ ab + eab  () 3 5  lowing sections, however, we apply the mathematical equa- tions of map projections described in Section 2 to obtain the 2. For the I. U. G. G spheroid (cf., [8]) with f = (a–b) / plotting equations. This section is based mostly on [7]. 2 a ≈ 1 / 298.275, we have eab = 2 f – f ≈ 0.0066944, and Let S be the globe, and C be a circular tangent from Equation (32), for geodetic latitude f = 45°, we to S along the equator, see Figure 3. Projecting S along the

have sin fA ≈ 0.70552 which gives fA ≈ 44°.8713. rays passing through the center of S onto C, and unrolling the cylinder onto a vertical strip in a plane is called central Case 2. A conformal sphere is an sphere defined for con- cylindrical projection. Clearly, each meridian on the sphere formal transformation from an ellipsoid, and similar to is mapped to a vertical line to the equator, and each parallel the authalic sphere may be used as an intermediate step of the equator is mapped onto a circle on the cylinder and so in the transformation from the reference ellipsoid to a a line parallel to the equator on the map. All methods dis- mapping surface. cussed in this section and other sections are about central

Let Rc, fc and lc be the conformal radius, latitude and projection, i. e., rays pass through the center of the Earth to longitude for the conformal sphere, respectively. Let e a or cylinder. Methods for those projections that are and g be the same fundamental quantities as Case 1, and not central are similar to central projections (see [8, 9]). 2 2 2 E¢ = R c and G¢ = R c cos fc. Also, let fc = fc(f) and lc = l. Let w be the width of the map. The scale of the map Thus, from Equation (17), along the equator is s = w / (2πR) that is the ratio of size of objects drawn in the map to actual size of the object it 222 represents. The scale of the map usually is shown by three ∂φ  ∂λ  ∂φ   cc ′   ′  c  ′ EE=+    GE=,  methods: arithmetical (e. g. 1:6,000,000), verbal (e. g. 100  ∂φ   ∂φ   ∂φ  miles to the inch) or geometrical. 22  At latitude f, the parallel to the equator is a circle ∂φ  ∂λ  cc ′   ′′ GE=+    GG=. (34) with circumference 2πR cos f, so the scale of the map at ∂λ ∂λ     this latitude is

w Combining Equations (20) and (34), one obtains sh = =ss ec φ, (37) 2cπR os φ   2 ∂φ where the subscript h stands for horizontal.  c  R2   c 22  ∂φ  Rcccos φ = , (35) M 2 N 22cos φ that after integrating and simplifying with the condition fc = 0 for f = 0, it gives

eab φφ    φ  2  caππ   1–e b sin  tan  + =tan  +    . (36)  2 4 2 4 1+eab sinφ 

One can calculate fc from Equation (36) which is a func- tion of geodetic latitude f. Also, it can be shown that

RMc = N for a given latitude f which in this case f = π / 2. We refer the reader to [8, Chapter 5] for the derivation. Figure 3: Geometry for the cylindrical projection.

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Assume that f and l are in radians, and the origin in Figure 4 shows the Mercator projection with Tissot’s the Cartesian coordinate system corresponds to the inter- indicatrices that do not change their (all of them section of the Greenwich meridian (l = 0) and the equator are circles indicating a conformal projection) while (f = 0). Then every cylindrical projection is given explicitly their areas increase toward the poles. More precisely, it by the following equations can be easily verified from Equations (23), (24), (25) and (44) that h = k, sin υ = 1, and so a = b = w / (2πR cos f) = wλ x = ,=yf()φ . (38) s / cos f (cf., Equations (21) and (22)), and so ω in Equa- 2π tion (26) will be zero. Now if the goal is preserving size rather than shape, For instance, it can be seen from Figure 3 that a central then we would make the horizontal and vertical scaling cylindrical projection is given by reciprocal, so the stretching in one direction will match wλ shrinking in the other. Thus, from Equations (37) and (41), φ (39) x = ,=yrtan, ¢ f f 2π we obtain f ( )sec = c or where for a map of width w, a globe of radius r = w / (2π) fc′()φφ=cos , (45) is chosen.

In a globe, the arc length between latitudes of f and f1 where c is a constant. From Equations (42) and (45), we can (in radians) along a meridian is choose c in such away that for a given latitude, the map also preserves the shape in that area. For instance if f = 0,

φφ1 – then we choose c = w / (2π), and so the map near equator 2πR ⋅−=,R φφ1 (40) 2π () is conformal too. Hence, the equations for the cylindrical equal-area projection (one of Lambert’s maps) are and the image on the map has the length f (f1) – f (f). So the overall scale factor of this arc along the meridian when wλ w x = ,=y sin.φ (46) f1 gets closer and closer to f is 2π 2π

φφ Figure 5 shows the Lambert projection with Tissot’s 1 1 ff1 – () () (41) sv = f ′ φ = lim , indicatrices that have the same areas (indicating an () φφ→ R R 1 φφ– 1 equal-area projection) while their shapes change toward the poles. where the subscript υ stands for vertical. The goal of Mercator was to equate the horizontal scale with vertical scale at latitude f, i. e., sh = sυ. Thus, from Equations (37) and (41),

w f ′ φφ= sec. (42) () 2π

Mercator was not be able to solve Equation (42) precisely because logarithms were not invented! But now, we know that the following is the solution to Equation (42) (use f (0) = 0 to make the constant coming out from the integra- tion equal to zero),

w yf==φφln sec+tan.φ (43) () 2π

Thus, the equations for the Mercator conformal projection (central cylindrical conformal mapping) are

wλ w x = ,=y ln sec+φφtan. (44) 2π 2π Figure 4: The Mercator .

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Figure 5: The Lambert equal-area map.

5 Albers and Lambert, One Standard Therefore, the first polar coordinate, θ, is a linear Parallel function of l, i. e., (47) θλ=∆ sin.φ0 In this section, we describe the Albers one standard paral- lel (equal-area conic projection) and Lambert one standard The second polar coordinate, r, is a function of f, i. e., parallel (conformal conic projection) at latitude f0 which give good maps around that latitude (cf., [2, 8, 9, 14]). rr=.()φ (48) We start with some geometric properties in a cone tangent to a spherical model of the Earth at latitude f0. The constant of the cone, denoted ϱ, is defined from the In Figure 6, ACN and BDN are two meridians sepa- relation between lengths on the developed cone on the rated by a longitude difference of ∆l, and CD is an arc of Earth. Let the total angle on the cone, θT, corresponding to the circle parallel to the equator. We have CD = DO¢∆l and 2π on the Earth be θT = d / r0, where d = 2πR cos f0 is the cir-

DN¢ sin f0 = DO¢ and approximately θ . DN¢ = CD. cumference of the parallel circle to the equator at latitude

f0, and r0 = R cot f0. Thus θT = 2π sin f0, and the constant of

the cone is defined as ϱ = sin f0. Case 1. The Albers projection. Consider a spherical model of the Earth. From Example 1, we know that the first fundamental quantities for the sphere are e = R2 and g = R2 cos2 f and for a cone (the polar coordinate system) are E¢ = 1 and G¢ = r 2. Hence, from Equations (18) and (19),

2 ∂rr∂ ∂φ ∂λ Rr42cos=φ 2 . (49) ∂θ ∂θ ∂φ ∂λ

Using Equations (47) and (48), Equation (49) becomes

2 ∂r 42 2 0 Rrcos=φ ∂φ . (50)

0sin φ0

Solving Equation (50) by knowing the fact that an increase Figure 6: Geometry for angular convergence of the meridians. in f corresponds to a decrease in r, one gets

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–2R2 sin φ r 2 = +.c (51) sin φ0

Imposing the boundary condition r0 = R cot f0 into Equa- 2 2 2 tion (51), c = 2R + R cot f0, and so after some simplifica- tions, Equation (51) becomes

R 2 r = 1+sin–φφ002sin sin.φ (52) sin φ0

The Cartesian plotting equations for a conical projection are defined as follows: Figure 8: The Albers equal-area azimuthal map.

xs=sryinθθ,=sr–cr os , (53) ()0   22    2 ∂r ∂θ ∂r   ′   ′   E =+  EG  =,  where s is the scale factor, θ and r are given respectively ∂φ  ∂φφ ∂  by Equations (47) and (52), and r0 = R cot f0. The origin of 22     ∂r ∂θ the projection has the coordinates l0 (the longitude of   ′   ′ 2 φ 2 (55) G =+  EG  =sin 0r . central meridian) and f0. Figure 7 shows the Albers pro- ∂λ  ∂λ  jection with one standard parallel. If we let f0 = 90°, then Equations (47) and (52) reduce to Substituting these values in Equation (20), integrating, sim- plifying and noting that r increases as f decreases, one gets θλ=∆ ,=rR2(1–sin)φ , (54) sin φ0  eab /2       π φ 1+eab sin φ  that are the polar coordinates for the azimuthal equal- tan –          area projection, a special case of the Albers projection,   4 2  1–eab sin φ   rr= 0   , (56) see Figure 8. The shape of Tissot’s indicatrices in Figures 7 eab /2       π φ 1+eab sin φ0 and 8 changes when the latitude changes, but they all tan – 0           have the same area.   4 2  1–eab sin φ0   Case 2. The Lambert projection. In this case, we con- sider a spheroidal model of the Earth. From Example 1, where the fundamental quantities for this model are e = M2 and acot φ0 2 2 rN00=cφ ot φ = . (57) g = N cos f, and the fundamental quantities for a cone 0 0.5 2 1–e22sin φ are E¢ = 1 and G¢ = r . Again using Equations (47) and (48), ()ab 0 Equation (17) becomes The Cartesian equations are the same as Equation (53)

with these new r0 and r. We show the Lambert projection with one standard parallel in Figure 9. Tissot’s indicatrices in this figure have different areas for different latitudes, but they all have the same shape (circle).

Figure 7: The Albers equal-area map with standard parallel 45°N. Figure 9: The Lambert conformal map with one standard parallel.

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6 indicatrices in Figure 10 change their shapes (the ellipses with different eccentricities indicating angular distortion) In this section, we only discuss about the sinusoidal toward the poles while having the same areas. equal-area projection that is a projection of the entire The inverse transformation from the Cartesian to geo- model of the Earth onto a single map, and it gives an ade- graphic coordinates is simply calculated from Equation (59): quate whole world coverage [3, 8]. y x Consider a spherical model of the Earth with the φλ=, ∆= . (60) fundamental quantities e = R2 and g = R2 cos2 f. The first sR sR cos φ fundamental quantities on a planar mapping surface is E¢ = G¢ = 1. Substituting these fundamental quantities into Equation (19) (using Equation (18)), one gets 7 Some Conventional Projections 2 ∂xx∂ In this section, we give the plotting equations for two ∂φ ∂λ R42cos=φ , conventional projections, the simple conic projection ∂ ∂ yy (one standard parallel) and the plate carree projection ∂φ ∂λ (cf., [3, 8, 13]). As we mentioned earlier, these projections neither preserve the shape nor do they preserve the size, which by imposing the conditions y = Rf and x = x(f,l) and they are usually used for simple portrayals of the reduces to world or regions with minimal geographic data such as 2 index maps. ∂xx∂   2 42 2 ∂x  1. The simple conic projection is a projection that the R cos=φ ∂φ ∂λ =.R   (58) ∂λ  along every meridian are true scale. Suppose R 0 that the conic is tangent to the spherical model of the

Earth at latitude f0, see Figure 11. In this figure, we have Taking the positive square root of Equation (58) and r0 = R cot f0. We want to have DE = DE¢, but DE¢ = R(f – f0). using the fact that l and f are independent, one obtains Thus the polar coordinates for this projection are dx = R cos fdl, and so by integrating x = lR cos f + c.

Using the boundary condition x = 0 when l = l0, one gets rr=–R φφ–, θλ=∆ sin.φ (61) 00() 0 c = –l0R cos f, and so the plotting equations for the sinu- soidal projection become as follow (f and l in radians): Replacing these values into Equation (53) gives its xs=∆Ryλφcos, = sRφ, (59) Cartesian coordinates. where s is the scale factor. Figure 10 shows a normalized plot for the sinusoidal projection. In this map, the merid- ians are sinusoidal curves except the central meridian which is a vertical line and they all meet each other in the poles. This is why this map is known as the sinusoi- dal map. The x axis is also along the equator. Tissot’s

Figure 10: The sinusoidal equal-area projection. Figure 11: Geometry for the simple conic projection.

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2. The distortion in area is defined as the ratio of an area on a map to the true area on a model of the Earth. From Equation (9) ( f = F = 0), the area on the map is

AEM = G, and the corresponding area on the model

of the Earth is AeE = g . Thus, the distortion in area is

AM EG K A == =.KKme (66) AE eg

In equal-area map projections, from Equation (18),

Figure 12: The plate carree map, 10° graticule. KA = Km Ke = 1. 3. The distortion in angle is defined as (in percentage): 2. The plate carree, the equirectangular projection, is a αβ– conventional cylindrical projection that divides the Kα =100⋅ , (67) meridians equally the same way as on the sphere. α Also, it divides the equator and its parallels equally. where a is the angle on a model of the Earth (the The plate carree plotting equations are very simple: ), and b is the projected angle on a map (the xs=∆Ryλφ,=sR , (62) azimuth a on the map, cf., Figure 13).

where f and y are in radians. Figure 12 shows the In order to obtain b as a function of the fundamental plate carree map with Tissot’s indicatrices which are quantities and a, we first calculate sin (b ± a). From changing their shapes and areas when moving toward Figure 13, we have the poles indicating that this map is neither equal- area nor conformal. sin=()βα±±sincβαos cossβαin         dλφd dφλd    ⋅  ±⋅   ⋅  = G ⋅   e   E   g  8 Theory of Distortion  dS   ds   dS   ds  dφλd ± (68) In this section, we discuss about three types of distor- =.KKem eg ()dS dS tions from differential geometry approach: distortions in length, area and angle, and we present them in term of the Hence, Gaussian fundamental quantities (cf., [8, 9, 15]). KK– 1. The distortion in length, also known as the scale factor sin–()βα= emsin+()βα. (69) in the surveying and mapping world, is defined as the KKem+ ratio of a length of a line on a map to the length of the Define true line on a model of the Earth. More precisely, KK– 2 22 f ()ββ=sin ()––αβemsin+()α . (70) ds Edφλ+Gd KKem+ 2 ()M () () KL == . (63) 2 22 ds edφλ+ gd ()E () ()

From Equation (63), the distortion along the meridians (dl = 0) is E K =, (64) m e and along the lines parallel to the equator (df = 0) is

G Ke=. (65) g Figure 13: Geometry for differential parallelograms.

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Now the goal is to find the roots of f. This can be done by Example 4 In this example, we first use the first funda- Newton’s iteration as follows: mental form to obtain the plotting equations for the Mer- cator projection, and then we show its length and area f β ()n distortions. From Example 1, the first fundamental form for ββ=– , (71) nn+1 the cylindrical surface (the Cartesian coordinate system) is f ′ β ()n 222 ds =+dy dx . (78) where ()M () () KK– f ′ ββ=cos ––αβemcos+α . (72) ()nn() ()n Taking the derivative of Equation (38) and substituting in KKem+ Equation (78), one finds

The iteration is rapidly convergent by letting b0 = a. In 2 2   2 222 dy 22 conformal mapping, from Equation (20), Ke = Km, and so   ()ds =+  ()dsφλRd()=+Ed()φλGd(), (79) the function f will have a unique solution (b = a). M dφ 

Example 3 In this example, we show the distortions in where s is the scale of the map along the equator, length in the Albers projection with one standard parallel. E = (dy / df)2 and G = s2R2. The first fundamental quan- From Example 1, the first fundamental form for the map is tities for the spherical model of the Earth are e = R2 and g = R2 cos2 f. Substituting these fundamental quantities in 22 222 ds =+dr rd2 θθ=+Ed′′rGd , (73) Equation (20) and simplifying, one obtains ()M () () () () sRdφ dy = . (80) and the first fundamental form for the spherical model of cos φ the Earth is It is easy to see that integrating the above differential 2 2 22 2 ds =+Rd2 φφRd22cos=λφed +.gdλ (74) equation and applying the boundary condition y(0) = 0, ()E () () () () Equation (43) follows. By Equation (80), E = (dy / df)2 = s2R2 / cos2 f. Therefore, substituting Equations (74) and Taking the derivatives of Equations (47) and (50), one obtains (79) in Equation (63), the length distortion will be φφ –cRdos s ddθφ=sin 0 λ,=dr , (75) 2 KL = . (81) 1+sin–φφ002sin sin φ cos φ

respectively. One may substitute Equation (75) in Equation It can be seen that KL = Km = Ke, and so from Equation (66), (73) to get the distortion in area for the Mercator projection is

2 2 22φ 2 s R cos KKAm==Ke . (82) ds = dφ 2 ()M 2 () cos φ 1+sin–φφ002sin sin φ 222 Hence, in the Mercator projection both length and area +srd22in φλ=+EdφλGd . (76) 0 () () () distortions are functions of f not l.

Substituting Equations (74) and (76) in Equation (63) gives the total length distortion. Also, 9 Discussions and Conclusions E cos φ Km == , There are a number of map projections used for different e 2 φφφ 1+sin–002sin sin purposes, and we discussed about three major classes of them, equal-area, conformal, and conventional. Users 2 G 1+sin–φφ002sin sin φ may also create their own map based on their projects by Ke == , (77) g cos φ starting with a base map of known projection and scale (e. g., [10]). In certain applications such as global web which are functions of f. Clearly, KmKe = 1. map visualization, some of the map projections can be

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also combined to minimize the distortion in both shape [3] Deetz, C. H., Adams, O. S., Elements of map projection, Spec. and area [1, 4]. Publ. 68, coast and geodetic survey, U. S. Gov’t. Printing office, In this paper, in cylindrical projections, we assume Washington, D. C., 1944. [4] Girin, I., Development of a multi-projection approach for that the cylinder is tangent to the equator. Making the cyl- global web map visualization, Thesis (MSc), York University, inder tangent to other closed curves on the Earth results Canada, 2014. good maps in areas close to the tangency. This is also [5] Goetz, A., Introduction to differential geometry, Addison-Wesley, applied for conical and azimuthal projections. Reading, MA, 1958. nd In all projections from a 3-D surface to a 2-D surface, [6] Grafarend, E. W., You, R. J., Syffus, R., Map Projections. 2 ed., Springer, Berlin, 2014. there are distortions in length, shape or size that some of [7] Osserman, R., Mathematical Mapping from Mercator to the them can be removed (not all) or minimized from the map Millennium, Mathematical Sciences Research Institute, 2004. based on some specific applications. In Section 3, we also [8] Pearson, F., Map projections, theory and applications, Boca showed that projecting a spheroidal model of the Earth to Raton, Florida, 1999. a spherical model of the Earth will distort length, shape [9] Richardus, P., Adler, R. K., Map projections for geodesists, cartographers, and geographers, North Holland, Amsterdam, and / or angle. 1972. Intelligent map users should have knowledge about [10] Savric, B., Jenny, B., White, D., Strebe, D. R., User preferences the theory of distortion in order to compare and distin- for projections, and Geographic guish their maps with the true surface on the Earth that Information Science, 42–5 (2015), 398–409. they are studying. [11] Snyder, J. P., Voxland, P. M., An album of map projections, Professional Paper 1453, p. 249, 1989. [12] Snyder, J. P., Map projections used by the U. S. geological survey, U. S. Geol. Surv. Bull., U. S. Gov’t, Washington, D. C., 1532, 1983. References [13] Steers, J. A., An Introduction to the study of map projection, University of London, 1962. [1] Battersby, S. E., The Effect of Global-Scale Map-Projection [14] Thomas, P. D., Conformal projection in geodesy and cartography, Knowledge on Perceived Land Area, Cartographica: The Spec. Publ. 68, coast and geodetic survey, U. S. Gov’t. Printing International Journal for Geographic Information and office, Washington, D. C., 1952. , 44 (2009), 33–44. [15] Vanicek, P., Krakiwsky, E. J., Geodesy the concepts, pp 697. [2] Davies, R. E., Foote, F. S., Kelly, J. E., Surveying, Theory and Amsterdam The Netherlands, University of New Brunswick, practice, McGraw-Hill, New York, 1966. Canada, 1986.

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