Useful Map Properties: Distortion Pattern
Assessing and Measuring Distortion
Every flat map includes some distortion of shape, area or
length. Some regions might be free of distortion while
others could suffer from severe error. Assessing the most
affected regions is useful in choosing an appropriate
projection.
Tissot's Indicatrix
An important tool introduced by Nicolas Tissot in the 19th
century is known today as Tissot's indicatrix.
Suppose a small circle drawn upon the original sphere.
When mapped to a flat surface, the circle could:
- preserve its shape and size, thus being free of
distortion
- get smaller or larger, thus suffering a scale distortion
- suffer from a shape deformation
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Tissot indicatrices in equatorial Hammer map
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Several very small circles set along the sphere and projected
onto a map show the latter's distortion pattern.
Hammer's
elliptical projection is not conformal and
deforms the Tissot circles, except at the very center of the
map. Shape distortion is also obvious in meridians, which
are "straight" over the terrestrial surface, but curved lines
here. However, since it is an equal-area
projection, all indicatrices cover exactly the same area.
On the other hand, the rectangular Mercator projection is
conformal: all indicatrices remain circular in shape, parallels
keep parallelism, meridians are straight lines and always
perpendicular to every parallel. Areas are not preserved,
but greatly increase towards the top and bottom of the map:
circles at the poles would be infinitely large (this is to be
expected, since meridians cross one another on a sphere but
never touch in a Mercator map. Only infinite circles on
different meridians could be all concentric as in the globe's
poles).
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Equatorial Mercator map, clipped at 85°N and
85°S, with identical indicatrices (theoretically
infinitesimaly small; the greatly oversized circles here
used for illustration slightly violate conformality)
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In a projection neither conformal nor equal-area like the azimuthal
orthographic,
Tissot indicatrices keep neither original shape nor area.
Ideally, for every circle centered at a meridian-parallel
intersection, scale should be preserved in both directions,
while the intersection angle should be 90°. Tissot
developed a formula defining the angular deformation at any
given point from the scales and angle distortion. The
maximum angular deformation can be plotted on a map
thereby presenting areas of major distortion.
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Equatorial azimuthal orthographic map
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As shown by the oblique orthographic and
Mercator maps, Tissot indicatrices present an
overall deformation pattern, not affected by graticule
rotation (in the first case, distortion depends only on radial
distance from center; in the latter, only on vertical
coordinate). These are exactly the same patterns
presented by the previous equatorial versions.
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| Oblique azimuthal orthographic map |
Oblique Mercator |
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| Equatorial cylindrical equidistant
map |
Finally, the equatorial equidistant
cylindrical map proves:
- at least the horizontal scale is always exaggerated in
polar caps in every cylindrical map
- having parallels and meridians crossing at right angles
everywhere is not enough for achieving conformality
 |  |  |  |  | | www.progonos.com/furuti May 31, 2004 |