An Efficient Technique for Creating a Continuum of Equal-Area Map Projections

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Cartography and Geographic Information Science

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An efficient technique for creating a continuum of equal-area map projections

Daniel “daan” Strebe

To cite this article: Daniel “daan” Strebe (2017): An efficient technique for creating a continuum of equal-area map projections, Cartography and Geographic Information Science, DOI: 10.1080/15230406.2017.1405285 To link to this article: https://doi.org/10.1080/15230406.2017.1405285

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Download by: [4.14.242.133] Date: 05 December 2017, At: 13:13 CARTOGRAPHY AND GEOGRAPHIC INFORMATION SCIENCE, 2017 https://doi.org/10.1080/15230406.2017.1405285

ARTICLE An efficient technique for creating a continuum of equal-area map projections Daniel “daan” Strebe Mapthematics LLC, Seattle, WA, USA

ABSTRACT ARTICLE HISTORY Equivalence (the equal-area property of a map projection) is important to some categories of Received 4 July 2017 maps. However, unlike for conformal projections, completely general techniques have not been Accepted 11 November developed for creating new, computationally reasonable equal-area projections. The literature 2017 describes many specific equal-area projections and a few equal-area projections that are more or KEYWORDS less configurable, but flexibility is still sparse. This work develops a tractable technique for Map projection; dynamic generating a continuum of equal-area projections between two chosen equal-area projections. mapping; equal-area projec- The technique gives map projection designers unlimited choice in tailoring the projection to the tion; adaptable projection; need. The technique is particularly suited to maps that dynamically adapt optimally to changing animated map; area- scale and region of interest, such as required for online maps. preserving homotopy; blended projection

1. Introduction themselves, preserve area in the general case, as shown The increasing use of interactive, adaptable maps, par- below. Therefore, efforts to create such transitional ticularly in software hosted on the World Wide Web, projections tended to be bespoke because general tech- often begs for seamless transitions between map pro- niques for hybridizing equal-area projections had not jections. Compare, for example, the adaptive projection been developed. Some efforts likely never came to system devised by Jenny (2012) or the National fruition because the hybridization proved intractable. Geographic animation created by Strebe, Gamache, To hybridize two equal-area projections, we would Vessels, and Tóth (2012). Strebe (2016) developed an like to do something like equal-area, continuous hybrid between Bonne and C ¼ kB þð1 kÞA Albers projections for dynamic or animated maps as a solution to a particular case. where k is a parametric constant such that 0 k 1, If the adaptable map does not need to preserve A is the desired initial projection, B is the desired areas, then hybridization can be simple, such as a linear terminal projection, and C is the resulting hybrid. combination of the initial and the terminal projections. This would yield a smooth transition from A to B.A Downloaded by [4.14.242.133] at 13:13 05 December 2017 However, even that simplicity can be deceptive. If the continuum of maps between two projections via a two projections diverge sufficiently in planar topology, parameter in the range [0, 1] is called a homotopy in then linear combinations could result in unacceptable algebraic topology and related fields. behavior such as the intermediate projection overlap- However, usually the hybrid projection C would not ping itself. An overlap means that two or more points preserve areas. This is demonstrated next. For a sphe- from the sphere map to the same point on the plane, rical projection, the property of equivalence is detected and this happens across a region. See, for example, the by the following analog to the Cauchy–Riemann equa- “wild vines” Snyder (1985, p. 86) describes for his GS50 tions (Snyder, 1987, p. 28): projection outside its useful domain. @ @ @ @ y x y x ¼ φ When the initial and terminal projections (hereafter, @φ @λ @λ @φ s cos the limiting projections) are equal-area, and area must φ be preserved throughout intermediate projections, where s is constant throughout the map, is the λ solutions are much more difficult. This is because latitude, and is the longitude. Equivalence can be ’ linear combinations of equal-area projections do not, defined strictly or loosely; Snyder s definition is strict in that s ¼ R2 only, where R is the radius of the globe.

By this definition, any region on the globe has the same More sophisticated new equal-area projections can be area on the map. By a looser definition, s may be any created at will because any equal-area projection can finite, nonzero value. The rationale in the looser case is become an area-preserving transformation applied to that all regions on the map remain proportioned cor- any other equal-area projection, given suitable scaling. rectly to each other. The looser definition permits an This was the basis for the Strebe 1995 projection affine transformation, as any constant, nondegenerate, [Šavrič, Jenny, White, and Strebe (2015)], for example, 2 2 real-valued matrix M, to be applied to the where Strebe scales the Eckert IV projection to fit within Cartesian coordinates of an equal-area mapped func- the confines of a Mollweide projection; back-projects the tion, resulting in a new equal-area projection. In this results to the sphere; and then forward projects to the case s ¼ R2= detðÞM . When s is not relevant, we assume plane via the Hammer. However, results depend on the s ¼ 1 for a unit sphere and identity linear transforma- projections available rather than on specific needs of the tion, giving: projection designer, and also do not obviously contribute @ @ @ @ to hybridization. y x y x ¼ φ: : @φ @λ @λ @φ cos (1 1) Some techniques for generating new equal-area projections were developed to approximate desirable distor- For C ! x, y as a linear combination of the two pro- tion characteristics. Snyder (1988) gives a transformation jections A ! xA; yA and B ! xB; yB, preserving areas that can be applied to Lambert azimuthal equal-area and would require that repeatedly thereafter in order to coax the angular isocols toward desired paths. Canters (2002) gives polynomial @½kyB þð1 kÞyA @½kxB þð1 kÞxA @φ @λ transformations for the same purpose that can be applied to any equal-area map and optimized via, for example, @½ky þð1 kÞy @½kx þð1 kÞx B A B A simplex minimization against specified constraints. @λ @φ Neither technique appears obviously adaptable to gener- ¼ φ: : cos (1 2) ating homotopies. Given the amount of calculation However, simultaneously, involved, neither technique would be well suited to @ @ @ @ dynamic maps in any case. yA xA yA xA ¼ φ @φ @λ @λ @φ cos @ @ @ @ yB xB yB xB ¼ φ @φ @λ @λ @φ cos 2. Background which, substituting into Equation (1.2), implies In describing local distortion on a map projection, the ’ @ @ @ @ @ @ @ @ usual metric is Tissot s indicatrix, presented in 1859 by yB xA þ yA xB yB xA yA xB @φ @λ @φ @λ @λ @φ @λ @φ Tissot (1881). The indicatrix projects an infinitesimal circle from the manifold to the plane. On smooth por- ¼ 2 cos φ: (1:3) tions of the map, the circle deforms into an ellipse with

Downloaded by [4.14.242.133] at 13:13 05 December 2017 This equality places stringent constraints beyond just semimajor axis a and semiminor axis b.Ifa b is equivalence upon choices for A and B. Therefore, linear constant throughout the indicatrices of the map, then combinations of arbitrary choices for A and B will not the map is equal-area. If a ¼ b throughout the map, then themselves be equivalent. the map is conformal. New equal-area projections can be generated via a As noted by Laskowski (1989), Tissot’s indicatrix of a variety of simple transformations applied to known projected point can be described via the singular value equal-area projections. As mentioned above, any non- decomposition of the point’s column-scaled Jacobian degenerate, affine transformation to an equal-area pro- matrix. Hereafter the Jacobian will be denoted J.Inthe jection yields an equal-area projection. Or, all mapped case of a sphere-to-plane projection, the affine transforma- points sharing the same y value can deform into the arc tion representing the complete Tissot ellipse T is given by of a circle, with neighboring horizontals becoming sec φ 0 concentric arcs. Applied to the sinusoidal projection, T ¼ J : (2:1) this technique results in the Bonne projection. Other 01 methods appear in the literature, such as Umbeziffern In particular, the areal inflation or deflation (or (renumbering), first used by Hammer to develop the generically flation, as per Battersby, Strebe, and Finn Hammer from the Lambert azimuthal equal-area, and (2016)) can be calculated as then named and elaborated on by Wagner (1949) for ¼ : : several of his projections. And so on. s det T (2 2) CARTOGRAPHY AND GEOGRAPHIC INFORMATION SCIENCE 3

The term isocol appears in Section 1. The term is then C will be merely k A as k ! 0. This is true specific to cartographic maps, and denotes a level curve despite having been projected by B because B of distortion. In the case of a conformal map, it is the has no distortion locally at B and therefore does level curve of constant scale factor. In the case of an not contribute to projection. equal-area map, it is the level curve of constant angular (6) If k is chosen to be 1, then the deprojection 0 deformation, which also implies Tissot ellipses of con- procedure giving Ak results in full coverage of stant dimensions. For maps that are neither conformal the original manifold, undistorted, and therefore nor equal-area, the term is not defined, although iso- C will be merely B. cols of angular deformation and isocols of flation both (7) In a “reasonable” projection B, distortion apply and generally do not coincide. increases away from P in most directions if P The term standard parallel appears in the text as an has no distortion. attribute of a conic projection. This is the cartographic term for the geographic parallel along which the conic These observations lay the groundwork for the techni- projection is undistorted. Conceptually, it is the circle que. While the observations above, and the technique of tangency of the cone on the sphere. When there are to follow, are applicable to any sufficiently smooth two standard parallels, they are, conceptually, the manifold, the remainder of this monograph discusses secant circles – that is, where the cone cuts through the sphere specifically, but without loss of generality. the sphere. “Conceptually,” because generally the projections involved are not literal perspectives. 3.2. Synthesis

Let k represent the weight of B desired in the blended 3. Development projection, such thatðÞ 1 k shall be the weight of A. 3.1. Observations By Observations (3) and (5), when k ¼ 0, we have k A=k ¼ A. By Observation (6), when k ¼ 1, we have B. (1) By the loose definition of equal-area transforma- For k inðÞ 0; 1 , when k is small, the contribution of B tion, an equal-area projection A from manifold in the description of C is small. This is because the (such as sphere) to plane, then scaled by s, and distortion of B is low in the neighborhood of P and thence deprojecting back to the manifold via the therefore, by Observations (5) and (7), its distinguish- 0 – 0 ¼ 0ðÞ – inverse A that is, As A s A results in ing characteristics as a projection are small in that 0 the area-preserving transformation As from the region. Meanwhile, the heavy reduction in A’s scale manifold to itself. This must be true because due to small k places all of A into that region of low each step preserves relative areas. In general, distortion in B. That leaves A’s distortion characteris- the result covers only part of the manifold, and tics to dominate. indeed s ought to be chosen such that s 1. Conversely, as k increases, the portion of B’s range

Downloaded by [4.14.242.133] at 13:13 05 December 2017 Hereafter we refer to this parametric s as k that A fills increases, and this increases the influence of such that ½0 k 1. B’s unique characteristics. Simultaneously, A’s charac- 0 (2) Projecting Ak to the plane via some other area- teristics diminish because the sphere-to-sphere map- preserving transformation B yields an equal-area ping from Observation (2) places points closer and projection Ck. closer to their original location as k increases. = (3) Scaling Ck by 1 k yields C having the same We thereby have a continuum of equal-area map nominal scale as A: that is, any region in A projections. These features fulfill the requirements for will have the same area measure in C, insofar an area-preserving homotopy. The technique is illu- as B’s area change is unity. strated in Figure 1. (4) As k ! 0, the fraction of projection B’s range Expressing this synthesis using notation introduced that is devoted to the transformation shrinks heretofore, and presuming no distortion at P ¼ BPðÞ0 , toward a single point, and so C’s distortion ¼ ð 0½Þ = : : characteristics approach being described by a C B A k A k (3 1) single Tissot indicatrix. In practice, our choice of anchor point P might vary as 0 “ ” 0 (5) If Ak is contrived such that an anchor point P k varies, so that when k is 0, P has no distortion, but as on the manifold remains undistorted by k increases, P moves toward some center common to A0ðÞk A =k, and contrived such that the Tissot both projections, such as a point of bilateral symmetry. indicatrix at P ¼ BPðÞ0 =k shows no distortion, This might be our choice if that common center is 4 D. STREBE Downloaded by [4.14.242.133] at 13:13 05 December 2017

k 1= Figure 1. How the homotopy works, 2. CARTOGRAPHY AND GEOGRAPHIC INFORMATION SCIENCE 5

1ð Þ distorted in B. Or, perhaps we keep P constant but noting that it is not the inverse TA P involved, but apply an affine transformation MB to B for large k in TAðPÞ itself because the goal is to apply the original order to eliminate the distortion at the common center distortion, not to undo it. P, with MB approaching identity as k ! 0. We express As seen in the examples in Appendix Section A.2, MA 0 this more general latter case as and MB are not inevitable even when P or P is distorted, and, in fact, the inverse transformation from k A back ¼ ð 0½Þ = : C MB B A k A k (3 2) to the sphere need not even use A0: Any equal-area ¼ where, for example, projection that limits to A when k 0 could potentially be used. MA and MB are just general rote devices for ¼ þ ðÞ 1ðÞ NB k I 1 k TB P reliably achieving a homotopy. pffiffiffiffiffiffiffiffiffiffiffiffiffiNB MB ¼ (3:3) det NB 4 Characteristics with I as the identity matrix and TB as given in 4.1. Computational cost Equation 2.1 for projection B. This yields a constant determinant of 1 across the parameterization, and The computational cost of this technique is the cost of 0 therefore preserves area. Obviously other ways of con- A,ofA ,ofB, and a little overhead for scaling and any structing MB are possible. other adjustments the projection designer might want. Other ways of selecting the anchor point and parame- Obviously, this is far cheaper than a direct interpreta- terizing the projections involved might present themselves, tion of the partial derivatives of the projections in play, depending upon the limiting projections. The example in which would imply computing a double integral for Appendix Section A.2 demonstrates such a case. every point. Despite the cheap cost, the result is still Note a qualification in Observation (5): P0 must superior to the naïve double integral because this tech- 0 remain undistorted in Ak. This is crucial to the techni- nique will not result in overlaps. que because the pre-image of k A has to retain the distortion characteristics of k A as k ! 0. If it does 0 4.2. Asymmetry not, then BAðÞk =s will not result in A as k ! 0. Therefore, the scaling by k on the plane must be This technique is sensitive to which of the two chosen arranged such that P ¼ APðÞ0 anchors the projection, projections gets assigned to be A and which to B. That with the rest shrunk toward it radially. What is impor- is, the homotopy from A to B differs from the homo- tant is that P0 round-trips to its original position: That topy from B to A. Therefore, the homotopy is direc- is, P0 ¼ A0ðÞk AP½0 . tionally asymmetric. It may be that APðÞ0 does not remain undistorted for Another asymmetry arises due to the use of linear para- the desired P0. For example, if A and B are equatorial meter k as a scale factor in both dimensions when con- 0 pseudocylindric projections, and A is distorted at the structing As. A linear progressionpffiffiffi in k generally yields a center as a vertical elongation, and the desired homo- weight that favors A.Using k in place of k throughout the Downloaded by [4.14.242.133] at 13:13 05 December 2017 topy would consist of pseudocylindric projections formulations yields a subjectively more linear homotopy. throughout, then more needs to be done. A0 yields an undistorted P0 because it undoes the distortion A 0 4.3. Distortion enacted. When then projected by B,ifBPðÞremains undistorted, then C will be squashed compared to A as Distortion characteristics of homotopies between arbi- ! k 0. The stretch could be reintroduced after B. The trary pairs of limiting projections are difficult to speak ¼ stretch after B would be linear against k, such that k of in generalities. They tend to be highly contingent ¼ 0 gives the full compression/stretch, and k 1 gives upon the homotopy. However, if the two limiting pro- none. In the general case this can be expressed as jections are chosen such that both have low distortion 0 in the region in which they anchor to each other, we C ¼ MA MB BðA ½Þk A =k (3:4) can expect the homotopy to have low distortion with MB as from Equation (3.3) and MA being the throughout the same region. In those cases, regions of analogous correction for A: unexpected or wildly varying distortion tend to occur only toward the boundary of the hybrid projection, if at N ¼ k I þ ðÞ1 k T ðÞP A A all. When boundaries are topologically similar (such as pffiffiffiffiffiffiffiffiffiffiffiffiffiNA for two pseudocylindric projections), the resulting pat- MA ¼ ; (3:5) det NA terns of distortion can be intuited readily. 6 D. STREBE

4.4. Outer boundary Section 3.2 are present only to correct for non-conformal behavior in equal-area maps, they could have no Projections whose planar topologies are the same retain purpose in a conformal mapping anyway. the common topology under the homotopy. Hence, the Likewise, this homotopy can be developed between continuum between two equatorial pseudocylindric aphylactic (neither conformal nor equal-area) projec- projections with pole-lines, for example, continues to tions, with little expectation of preserving identifiable have the outer meridian, duplicated left and right, as its properties other than topological integrity. Or, for outer boundary, along with a line for the pole. If they example, it can be used to synthesize a progression have pointed poles, then so will the continuum. If they between some conformal map and some equal-area have flat poles, then so will the continuum. If the initial map. Such a progression has didactic value and, in projection is pointed and the other not, then the con- dynamic maps, perhaps even practical value when the tinuum will be pointed except at the flat extreme, but thematic focus of the map changes or when zooming the angle of incidence may be imperceptibly slight out from large scale to medium and small scales. approaching the flat pole extreme. If the initial projection has a flat pole but the terminal is pointed, then the continuum will be flat throughout, with width shrink- ing to a point at the terminal extreme. 5. Conclusion When the limiting projections’ topologies differ, By combining several observations about equal-area little can be said generally other than that the solution maps in a novel way, this research produced a tract- becomes specific to the projections. The description of able technique for hybridizing any two equal-area the boundary is a tractable calculation, but not neces- projections. Using such transitions, a map designer sarily simple or convenient. The Lambert azimuthal-to- might find combinations that yield distortion char- Albers continuum of Appendix Section A.2 is an exam- acteristics more favorable to the region being ple of a nontrivial topological evolution. mapped than are otherwise available. Because of the generality of the technique, and the fact that its 4.5. Applicability to conformal and other maps parameterization operates on the topology of the sphere (or ellipsoid or other manifold), the technique In cartography, one might need a homotopy between two can hybridize even projections of otherwise incom- conformal projections. Due to the fact that any analytic patible planar topologies, such as an azimuthal and a function acting on a conformal map results in another conic projection. The problem of tailored projections conformal map, a wealth of functions can be drawn upon has been acute for equal-area needs, since no simple, and composed in order to do this. However, when acting general system existed. This new technique fills the purely on the plane, nothing guarantees a priori that a void. given mapping will behave well, with the potential for the The same ability to parameterize for new projections map failing bijection criteria being an acute concern. While between known projections makes the technique well it is common for cartographic maps not to be invertible at suited for dynamic mapping and animations. For the Downloaded by [4.14.242.133] at 13:13 05 December 2017 singularities or along some boundaries, overlapping of first time, equal-area projections can dynamically adapt regionsisnotacceptableforrangesthatmustbepreserved optimally for the entire region brought into view when in the mapping. Naïve blending techniques, such as linear panning and zooming. combinations of the two projected spaces, hazard such The technique is computationally cheap (that is, on overlapping. the order of the limiting projections involved); com- Fortunately, nothing about this homotopy is specific pletely general; and not limited to equal-area maps. to equal-area projections. As a general topological pro- Because the technique is built upon the topology of cedure, it can be used for any sufficiently smooth the surface being projected, rather than acting purely in mapping from any sufficiently smooth manifold. It Cartesian space, overlapping can be avoided with mini- can be used equally successfully for conformal maps, mal precautions, unlike a simple linear combination of and indeed, it preserves conformality in the intermedi- the limiting projections. Therefore, the technique ate maps. The only caveat is that the affine transforma- applies equally well to conformal projections, projections MA and MB must not contain shear components, tions that are neither conformal nor equal-area, and but instead only scale isotropically and/or rotate. Since even hybridizing conformal projections with equal-area the shear components in MA and MB as given in projections. CARTOGRAPHY AND GEOGRAPHIC INFORMATION SCIENCE 7

Acknowledgments Tissot, N. A. (1881). Memoire sur la représentation des sur- faces et les projections des cartes géographiques. Paris: This author thanks Mary Solbrig for recognizing this techni- Gauthier-Villars. que to be a method for generating homotopies. Thanks also Wagner, K. (1949). Kartographische Netzentwürfe. Leipzig: go to Bernhard Jenny for pushing to solve his specific pro- Bibliographisches Institut. blem; without that push and the intractability of a specific solution, who knows how long I would have let the search for a general solution simmer. Appendix. Examples

A.1. Sinusoidal and cylindrical equal-area

Disclosure statement The cylindrical equal-area and the sinusoidal projec- No potential conflict of interest was reported by the author. tions are two of the simplest equal-area projections from sphere to plane. The cylindrical equal-area projection is defined as ORCID x ¼ λ cos φ “ ” ¼ φ φ1 Daniel daan Strebe http://orcid.org/0000-0001-6410- y sec 1 sin (A1) 6259 φ with 1 being the latitude along which scale is correct and conformality preserved. It has Jacobian "# @x ¼ cos φ @x ¼ 0 @λ 1 @φ : References @y ¼ @y ¼ φ φ (A2) @λ 0 @φ sec 1 cos Battersby, S. E., Strebe, D., & Finn, M. P. (2016). Shapes on a The projection’s inverse is plane: Evaluating the impact of projection distortion on ÀÁ spatial binning. Cartography and Geographic Information φ ¼ arcsin y cos φ – λ ¼ φ : 1 Science, 44, 410 421. doi:10.1080/15230406.2016.1180263 x sec 1 (A3) Canters, F. (2002). Small-scale map projection design. London: Taylor & Francis. The sinusoidal is defined as Jenny, B. (2012). Adaptive composite map projections. IEEE x ¼ λ cos φ Transactions on Visualization & Computer Graphics, 18, y ¼ φ: (A4) 2575–2582. doi:10.1109/TVCG.2012.192 Š č If we let A be the cylindrical equal-area and B the Jenny, B., & avri ,B.(2017). Enhancing adaptive composite 0 map projections: Wagner transformation between the sinusoidal, then A is given by the inverse of k A lambert azimuthal and the transverse cylindrical equal- like so: area projections. Cartography and Geographic φ0 ¼ ðÞφ 0arcsin k sin Information Science. doi:10.1080/15230406.2017.1379036 λ ¼ kλ: (A5) Laskowski, P. H. (1989). The traditional and modern look at φ 0 Tissot’s indicatrix. The American Cartographer, 16, 123– Notice that 1 does not appear in A ;regardlessof φ 0 0 133. doi:10.1559/152304089783875497 choice for 1,theyallresultinthesameA .AnyA Nell, A. (1890). Äquivalente kartenprojektionen. Petermanns will undo the distortion enacted by A at P0,butin Downloaded by [4.14.242.133] at 13:13 05 December 2017 Mitteilungen, 36, 93–98. φ this case, the influence of 1 gets removed not only Šavrič, B., Jenny, B., White, D., & Strebe, D. R. (2015). User 0 from P but throughout the projection because the preferences for world map projections. Cartography and φ Geographic Information Science, 42, 398–409. doi:10.1080/ projection parameterized by any 1 is just an affine 15230406.2015.1014425 scaling of the same underlying projection. This Snyder, J. P. (1985). Computer-assisted map projection research. removal will be counteracted by MA,computedas (US Geological Survey Bulletin 1629). Reston, VA: USGS. Snyder, J. P. (1987). Map projections – a working manual. ¼ þ ðÞ ðÞ (US Geological Survey Professional Paper 1395). NA kI 1 k TA P Washington: US Government Printing Office. þ ðÞ φ ¼ k 1 k cos 1 0 Snyder, J. P. (1988). New equal-area map projections for þ ðÞ φ 0 k 1 k sec 1 noncircular regions. The American Cartographer, 15, 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 – þðÞ φ 341 356. doi:10.1559/152304088783886784 k 1 k cos 1 N þðÞ φ 0 Strebe, D. (2016). An adaptable equal-area pseudoconic map A 4 k 1 k sec 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 MA ¼ ¼ kþðÞ1k sec φ projection. Cartography and Geographic Information det NA 1 0 kþðÞ1k cos φ Science, 43, 338 –345. doi:10.1080/15230406.2015.1088800 1 Strebe, D., Gamache, M., Vessels, J., & Tóth, T. (2012). Mapping the oceans [video]. National Geographic by Equations (2.1), (3.4), and (A2), given Magazine, Digital Edition,9.https://www.youtube.com/ P0 ¼ ðÞ0E; 0N . No correction is required for B watch?v=OQCoWAbOKfg because sinusoidal has no distortion at P,soMB ¼ 8 D. STREBE

I and can be ignored. BAðÞ0 =k is given by applying the pole, a homotopy for the two is trivial because Equation (A4) to Equation (A5): Lambert is a limiting form of Albers, with both standard parallels being the pole. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi When Lambert needs to be centered elsewhere, how- ¼ λ 2 2φ x 1 k sin ever, no obvious solution makes itself known. Jenny ¼ ðÞφ = : y arcsin k sin k (A6) (2012) proposed a transformation from Lambert azi- By Equations (3.4) and (A6), then, muthal to transverse Lambert cylindrical equal-area pro- 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 þðÞ φ "#pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jection through Albers by using the limiting-form k 1 k cos 1 6 kþðÞ1k sec φ 0 7 λ 1 k2sin2φ C ¼ 4 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 relationship of Lambert to Albers, with other adjust- kþðÞ1k sec φ 1 arcsinðÞk sin φ =k ments. However, in order to keep the region of interest 0 kþðÞ1k cos φ 1 away from the split that appears as soon as the homo- (A7) topy’s parameter leaves 0, the Lambert azimuthal must be and thereby the homotopy from cylindrical equal-area rotated, pushing the region of interest into higher-distor- to sinusoidal is tion portions of the projection. This is not satisfactory sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k þ ðÞ1 k cos φ because the region of interest is where low distortion is x ¼ λ 1 k2sin2φ 1 Š č k þ ðÞ1 k sec φ most desired. Jenny and avri (2017) acknowledge this 1 shortcoming and propose an improved transition via a sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi transverse Wagner. However, applicability of the Wagner arcsinðÞk sin φ k þ ðÞ1 k sec φ y ¼ 1 : (A8) route is limited to “portrait format” maps. Jenny and k k þ ðÞ1 k cos φ 1 Šavrič (2017,p.6)write: Parameterized with k ¼ 0:226 and φ ¼ 0(suchthat 1 … Equation (A6) suffices), this formulation results in a A conic transformation remains in the adaptive composite projection for landscape-format maps for projection that is practically indistinguishable from the transitioning between the azimuthal (for maps at con- pseudocylindric limiting form of the equal-area pseudo- tinental scales) and the conic (for larger scales) projec- conic projection devised by Nell (1890). Nell’sisthefirst tions . . .. However, distortion caused by the conic knownequal-areapseudocylindricwithapole-line,andis transformation is comparatively large . . .. Improving a compromise between the sinusoidal and the equal-area this transformation between the Lambert azimuthal φ ¼ ’ and the Albers conic projections is an open challenge. cylindric with 1 0(Lamberts). Unlike Nell, no itera- tion is required in computing Equation (A8), illustrating Here we meet this challenge. the technique’s characteristic computational efficiency. The oblique Lambert azimuthal equal-area from the To illustrate the asymmetry noted in Section 4.2,we sphere is formulated as give the reverse homotopy from sinusoidal to cylind- pffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ = þ φ φ þ φ φ λ rical equal-area, omitting the intermediate calculations: z 2 1 sin 3 sin cos 3 cos cos sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x ¼ zÀÁcos φ sin λ λ cos φ cos φ k þ ðÞ1 k sec φ ¼ φ φ φ φ λ ¼ 1 1 y z cos 3 sin sin 3 cos cos (A10) x ðÞφ þ ðÞ φ cos k skffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 k cos 1

Downloaded by [4.14.242.133] at 13:13 05 December 2017 after Snyder (1987), with its inverse being sinðÞkφ sec φ k þ ðÞ1 k cos φ y ¼ 1 1: (A9) pffiffiffiffiffiffiffiffiffiffiffiffiffiffi k k þ ðÞ1 k sec φ ρ ¼ x2 þ y2 1 ρ ’ c ¼ 2 arcsin This projection is the same as Kavraiskiy sfifth, ÀÁ2 φ ¼ φ þ ρ1 φ matching his recommended parameterization at arcsin cos c sin 3 y sin c cos 3 ¼ : φ ¼ : k 0 738340936 and 1 29 8924267 . ÀÁ λ ¼ ; ρ φ φ arctan x sin c cos 3 cos c y sin 3 sin c (A11) A.2 Lambert azimuthal equal-area and Albers conic and arctan being the typical two-argument form ½π; πÞ φ Albers and Lambert azimuthal equal-area have two yielding the full range . 3 is the latitude at highly divergent topologies. The perimeter of Albers which no distortion is wanted, achieved at the cen- is the same as a pseudocylindric projection’s: pole lines tral meridian only. and the 180th meridian replicated left and right. The standard Albers from the sphere is formu- Lambert, on the other hand, projects the center’s anti- lated as podal point as a circle of radius 2, given unit sphere. sin φ þ sin φ n ¼ 1 2 With the constraint that the Lambert be centered on 2 CARTOGRAPHY AND GEOGRAPHIC INFORMATION SCIENCE 9

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiÀÁ 2φ þ φ φ (A11), and B as Equation (A12), the homotopy follows cos 1 2n sin 1 sin ρ ¼ immediately from Equation (3.1). The configurable n φ φ φ φ parameters 1 and 2 are taken as 1k and 2k for θ ¼ nλ x ¼ ρ sin θ each k. Delineated, using definitions from Equations y ¼ρ cos θ: (A12) (A10) and (A12) unless replaced here:

Its inverse is not needed here because Albers serves as xL ¼ kzÀÁcos φ sin λ ¼ φ φ φ φ λ B, for which only forward is used. qyffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL kz cos 3 sin sin 3 cos cos ρ ¼ 2 þ 2 Let Lambert azimuthal equal-area be A and Albers be B. L xL yL φ ρ Lambert is undistorted at its point of obliquity ( 3,0). c ¼ 2 arcsin L L ÀÁ2 Albers is undistorted all along selectable constant latitudes φ ¼ φ þ ρ1 φ L arcsinÀÁ cos cL sin 3 yL L sin cL cos 3 φ and φ .TheangularextentoftheAlberswedgeis λ ¼ ; ρ φ φ 1 2 L qarctanffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffixL sin cL L cos 3 cos cL yL sin 3 sin cL determined by n.Letusassumethatsinφ is chosen to ÀÁ 3 ρ ¼ cos2φ þ 2n sin φ sin φ be n to place it about halfway between the standard paral- A 1k 1k L θ ¼ nλL lels. Albers would be distorted at φ – in fact, it reaches a ρ sin θ 3 x ¼ A local maximum there – unless φ ¼ φ ¼ φ .Hence, k 1 2 3 ρ cos θ something must be done for small k so that the area y ¼ A : (A13) k around φ has low distortion then. We could address this 3 The results are as shown in Figure 2, with distortion by means of MB asdescribedinEquation(3.2). Another method presents itself based on the char- diagrams as Figure 3. We do not address the matter of acteristics of Albers. We are free to vary the standard the perimeter here; its description is complicated and not parallels insofar as the sum of their sines is constant so instructive for other homotopies. Most practical uses of that we do not vary the angular extent of the cone. this particular homotopy would bound the region well ¼ φ ¼ φ ¼ φ short of the topological perimeter by some easily described Hence, if we start at k 0 with 1k 2k 3, and gradually move φ and φ apart as k ! 1, we achieve meanssuchasarectangleorasnorth/west/south/east 1k 2k extents, mapped. the needed effect. So, ÀÁ In a reverse homotopy going from Albers to Lambert, φ ¼ φ þ ðÞ φ φ sin 1k sin 1 1 k ÀÁsin 3 sin 1 we might again choose to situate P about halfway sin φ ¼ sin φ þ ðÞ1 k sin φ sin φ : 2k 2 3 2 between the two standard parallels so that we converge This frees us to use the simplest form of the transfor- most directly from Albers to Lambert azimuthal. To 0 mation. Given A as Equation (A10), A as Equation achieve that, we could use MA as described in Equation Downloaded by [4.14.242.133] at 13:13 05 December 2017

pffiffiffi Figure 2. Homotopy from Lambert azimuthal equal-area to Albers, k in 0.2 increments. 10 D. STREBE

Figure 3. Patterns of maximum angular deformation, 10 increments, deeper color signifying greater distortion.

(3.2). This would correct for the fact that P reaches a local maximum of distortion on the Albers, whereas if left uncorrected, P would be the point of no distortion and the standard parallels would gain distortion. However, just as in the forward homotopy, a better method suggests itself. Ideally, we would want the benefits of Albers to dominate for small k.ThebenefitofAlbersisin the two standard parallels having no distortion. We can protract those benefits beyond the vicinity of k ¼ 0, at least on the Albers side of the transformation. We can do this by using a different Albers parameterization for the inverse, onewhoseforwardwewillcallA^. Its parameterization is contrived so that A^ has standard parallels that coincide in Downloaded by [4.14.242.133] at 13:13 05 December 2017 Cartesian space with the standard parallels of k A.(See Figure 4.) This works because as k ! 0, A^ ! A.Because the paths of no distortion on A^ coincide with those on k A, no correctional affine transformation is needed and the homotopy retains more “Albers-ness” over the low Figure 4. A^ with standard parallels (green dashes) coincident to parametric space as compared to a rote approach those of k A (solid green). using MA. Without getting into computational details, the result of this method from Albers to Lambert, and the round- trip return via Equation (A13), can be seen in the video Lambert azimuthal round-trip homotopy distortion” in “Albers-Lambert azimuthal round-trip homotopy” in the Supplementary Material. Lower-quality versions can the Supplementary Material. The corresponding anima- also be found at https://youtu.be/D1CuUPi2yA0 and tion showing distortion can be seen in the video “Albers- https://youtu.be/AcTFAXPLReE.