Cartography and Geographic Information Science ISSN: 1523-0406 (Print) 1545-0465 (Online) Journal homepage: http://www.tandfonline.com/loi/tcag20 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 View supplementary material Published online: 05 Dec 2017. Submit your article to this journal View related articles View Crossmark data Full Terms & Conditions of access and use can be found at http://www.tandfonline.com/action/journalInformation?journalCode=tcag20 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 À 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 À 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. CONTACT Daniel “daan” Strebe [email protected] Supplemental data for this article can be accessed here. © 2017 Daniel “daan” Strebe 2 D. STREBE 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 À y x ¼ φ: : @φ @λ @λ @φ cos (1 1) Some techniques for generating new equal-area pro- jections 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 À kÞyA @½kxB þð1 À kÞxA @φ @λ transformations for the same purpose that can be applied to any equal-area map and optimized via, for example, @½ky þð1 À kÞy @½kx þð1 À kÞx À 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 À yA xA ¼ φ @φ @λ @λ @φ cos @ @ @ @ yB xB À 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 À yB xA À 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 Á : (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 Á 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.