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Useful Map Properties: Distortion Pattern

Deformation Patterns

Projections at identical scale with maximum angular deformation values represented by colors
Angular deformation scale
Equatorial Sinusoidal map with angular deformation pattern
Map in sinusoidal (Sanson-Flamsteed) projection: equal-area, but only the Equator and central meridian are free of angular deformation
Equatorial Mollweide map with angular deformation pattern
Mollweide's projection: the sinusoidal's "cross" of low deformation is replaced by two oval "islands"
Oblique Mollweide map with angular deformation pattern
Mollweide's projection, in oblique aspect minimizing distortion on Southeastern Brazil and a corresponding area on the northern hemisphere
Interrupted equatorial Mollweide map with angular deformation pattern
Mollweide's projection, interrupted with four symmetric lobes. The size of each lobe limits the distortion range, at the cost of distance discontinuities
Equatorial Bromley map with angular deformation pattern
Bromley's rescaled version of Mollweide's projection: the "islands" of low distortion move to, merge at and spread along the Equator
Equatorial Goode map with angular deformation pattern
Interrupted Goode's fused homolosine projection (in a simplified lobe arrangement with no repeated areas). Notice the discontinuity at the two boundary parallels near 40°N and S
Equatorial Boggs Eumorphic map with angular deformation pattern
Interrupted Boggs's eumorphic projection (in simplified lobe arrangement, unbroken Eurasian lobe). Distortion is continuous because the partial projections are averaged, not fused
Gnomonic Waterman map with angular deformation pattern
Polyhedral maps are a particular case of interruption. On an unfolded Waterman polyhedron, the gnomonic projection centered on faces is azimuthal, therefore the deformation pattern is radially symmetric.
Icosahedral gnomonic map with angular deformation pattern
Gnomonic projection on an unfolded icosahedron
Icosahedral Dymaxion map with angular deformation pattern
Dymaxion™ projection on an icosahedron
Equatorial Eckert I map with angular deformation pattern
Eckert's projection I (neither equal-area nor conformal), with a sharp direction break at the Equator
Equatorial Eckert II map with angular deformation pattern
Eckert's projection II. A comparison with its predecessor shows that, as frequently happens in cartography, adding a favorable property (areal preservation) brings a drawback, in this case a smaller and more irregular region of low shape distortion
Equatorial equidistant cylindrical map with angular deformation pattern Equatorial equidistant cylindrical map with angular deformation pattern
Azimuthal stereographic maps are conformal everywhere, but the point opposite the projection center cannot be shown: they are customarily limited to hemispheres
Polar azimuthal equal-area map with angular deformation pattern
Lambert's azimuthal projection: equal-area, at the cost of pronounced distance and shape distortion near the periphery
Polar Wiechel map with angular deformation pattern
Wiechel's pseudoazimuthal equal-area projection. Again, an improvement (correct scale along meridians) is offset by a reduced area of lower distortion
Equatorial equidistant cylindrical map with angular deformation pattern
Equidistant cylindrical (neither equal-area nor conformal) projection with standard parallel at Equator
Equatorial equal-area cylindrical map with angular deformation pattern
Lambert's cylindrical projection. In this particular case with standard parallels 45°N and 45°S, known as Gall's orthographic and "Peters". Equal-area, but heavily distorted except along two narrow bands
Equal-area conic map with angular deformation pattern
Albers's equal-area conic projection with standard parallels 45°N and 20°N
Angular deformation scale

Tissot's equations describe how scaling factors are affected at each point in two principal directions, along the meridian and along the parallel which intersect there. On Earth, those directions are of course orthogonal; on the projected map, the transformed scaling factors provide:

The angular deformation has an orientation, clockwise or counterclockwise, though in practice it is omitted, since the absolute magnitude is much more relevant. Determining areal and angular deformations is important for selecting a projection and, once one is picked up, minimizing effects of distortion.

For some constrained groups of projections, like the normal azimuthal, cylindrical and conic projections, the two distortions can be calculated almost directly from the projection's equations. In the general case, they must be evaluated numerically via partial differentiation. Either way, their values may be represented as colors on a map, immediately showing the places of greater and lesser distortion.

Deformation Patterns and Angular Distortion

Most projections conceptually based on a developable surface present least distortion at points or lines of tangency, which usually coincide with the center or axes of the map. Similarly, the non-perspective sinusoidal projection preserves areas everywhere but is free of angular distortion (in the equatorial aspect) only along the Equator and the central meridian. In contrast, the Mollweide projection, also equal-area, has zero angular distortion in only two points, at the intersection of the central meridian with two standard parallels (about 40°N and 40°S); even though scale is constant along the Equator and all meridians cross it orthogonally, angular deformation along it is not zero due to the exaggerated vertical scale. When choosing between these two designs, it is worth considering that while the sinusoidal shows a smaller area with maximum angular deformation below 20°, its maximum deformation is only about 115° in contrast with 180° for Mollweide's.

Once a projection is decided, several approaches are available for dealing with its limitations:

Equal-area projections are never conformal, and removing the areal equivalence constraint can improve their range of angular deformation. For instance, in Eckert's series of flat-polar proposals, three pairs of projections look much alike, but one of each pair is equal-area. Even though his projection I is not conformal, its central area of lower deformation is larger and more uniformly shaped than in the equal-area II.



HomeSite MapDistortion Pattern - Tissot's IndicatricesMap PropertiesHow Projections are Created  www.progonos.com/furuti    September  2, 2013
Copyright © 2008 Carlos A. Furuti