Choose Map Proj

No Slide TitleChoosing Map Projections
Of the theoretically infinite number of map projections possible, less than 50 are in common use
Within the common application areas there is little choice in the types of projections used; this is due to:
Specialized needs with respect to accuracy, use, features, etc.
The need to interoperate with other organizations through standards
Atlas maps allow more flexibility in the selection of projections for relatively small scale maps:
countries, continents, hemispheres, world
Choosing Map Projections
LIS for legal, administrative, and economic decision-making
The ability to integrate these files or layers which represent different themes of information from different sources (Figure 11.01)
utilities
Display results of analysis (soft or hard copy)
Transformation / registration of multiple layers into a single frame of reference
Quantitative measurements validation
density / gradient calculations
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survey, cartographic process, map projection, cartometric operations
projection distortion is minimal in large-scale mapping for some applications
approximately .2 mm is the size of the finest point visible to the naked eye
Generalization is inevitable on maps
some features are exaggerated for legibility
Threshold of perception and threshold of separation (Figure 11.02)
separation required to distinguish 2 separate objects on the ground by 2 symbols on the map
Generalization involves exaggeration, selection, priority of features, deselection, among other techniques
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The zero dimension varies according to the original source materials
Computers have increased the range of information that is extractable from source and displayable
The larger the scale the smaller the limiting ground distance of the zero dimension
This is particularly true of survey-quality aerial photography and other remotely sensing imagery systems
Digital processing makes what once seemed to be very small errors in the cartographic production process very large
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Converting multiple data sets which are each based on different sources and projections into one single projection system is a common and necessary operation in a GIS
Selecting an internal projection system for the GIS is a common procedure
Often this is the UTM projection system for relatively large-scale mapping
For smaller scale mapping, geographic coordinates are typically used
Even here, decisions on internal formats must be decided:
radians, decimal degrees, degrees-minutes-seconds, precision, etc.
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Factors in choosing a suitable projection
Since all projections have some degree of distortion and generally distortion increases away from the point or line(s) of zero distortion toward the map edges, the main objective is “to select a projection in which the extreme distortions are smaller than would occur in any other projection used to map the same area”
Deliberate distortion is sometimes employed to depict certain spatial relationships, e.g., Hagerstrand’s logarithmic azimuthal projection to illustrate population migration
Intended use is therefore a key factor
Location, size, and shape of the area to be mapped are also key components in selecting a map projection
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Purpose and intended use usually dictate the required special property: conformality, equal-area, equidistance
Selecting the specific projection, though, requires an evaluation of the purpose against the amount and distribution of distortion
For a conformal projection we look at the behavior of area scale near the boundaries
For an equal-area map we look at angular deformation
A minimum-error projection may be appropriate if neither special property is required and is nevertheless a good starting point in selecting the appropriate projection
The expected quantitative uses of a map or chart usually place the most stringent requirements in map projection selection
Angles for navigation charts and surveying
Areas for statistical mapping
Distance for route planning
+/- 1 to 2% precision for area and distance calculations
Up to 1o accuracy for most non-precision applications
Maps of small areas or countries
Choice of projection here is of little significance since on practically any projection the distortion will not be visually apparent
The distortion in the vicinity of the point or line(s) of zero distortion is less than the zero dimension
Typically the base topographic map of the country is appropriate
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Maps of large areas or countries
For countries such as Canada or Russia distortion levels of 3% or greater, or angular deformation of 3o or more must be tolerated
Larger areas will have greater distortion
Equal-area maps of Asia may exhibit angular deformation of 15o near the edges; of Africa or North America 6 - 8o (Figures 12.01 and 12.02)
Equal-area maps of the world may exhibit 30o angular distortion, and world maps in general usually exhibit singular points of 180o distortion, and indeterminate areal exaggeration
Purpose and shape of the area to be mapped
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Modified Projections
Redistribution of particular scales and introducing more than one line of zero distortion
Introduce special boundary conditions on the map edges
Recentered or interrupted projections
Time involved in compiling, plotting, and redrawing graticules and features, although this consideration is practically obsolete in our current digital age
Availability of process cameras to perform scale change and optical rectification
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Rectifications are limited by graticule projection
Transformation of a rectilinear graticule to one of curves is not possible
For example, normal aspect cylinderical to normal aspect conical
Specialized optical equipment has been used in certain situations but is expensive
Historically, use of previous materials was preferred over compiling a new graticule, master grid, and transferring coordinates due to the time involved
Digital techniques have made obsolete many of these considerations but data capture still remains as an obstacle
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Obstacles
In the digital realm, the concept is based on scale-free databases which originate from the largest scale source available
Smaller scale products are made from the master database
The concept is sound but data coverage while reasonable at relatively small scales is fairly sparce at larger scales for much of the world
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General rules:
principal scale preserved along the equator
Conical projections for countries in the temperate zones
principal scale preserved along a parallel of latitude
Azimuthal projections for the polar regions
principal scale preserved at the pole
Other general rules
Locate the point or line of zero distortion in the center of the area
Orient the lines of zero distortion through the longer axis of the area
Orientation effects aspect
Shape effects class
These rules have been regarded as one of the classical foundations of cartographic design in the selection of map projections and production of atlas maps since the 16th century
However, the rules are limiting with respect to today’s technology
Consideration of other map projection classes was not practised
Use of other aspects has not been common
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Selecting the point of origin
Calculate the center of gravity of the land mass to be mapped
Select this exact point or the nearest graticule intersection
This is no longer an issue of simplifying calculations but may be desirable from an aesthetic standpoint
Table 11.01 gives suggested origins for maps of the continents
Selecting the line(s) of zero distortion
Orient the line of zero distortion along the major axis through the country
Use two standard parallels where appropriate
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Generally, select azimuthal for circular countries and cylindrical or conical for asymmetrical countries
The rule is based on the measurements z and d of the country to be mapped (Figure 11.04)
z is the maximum angular distance from the center of the country to the most distant boundary
d is the minimum distance between two bounding parallel arcs of small circles
shortest distance between arcs
independent of orientation
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Ginzberg and Salmanova suggest three critical values related to special properties:
Conformal z/d = 1.41
Equidistant z/d = 1.73
Equal-area z/d = 2.00
Chile (z/d = 2.3) - conical or cylindrical; Australia (z/d = .63) - azimuthal
Selecting special property
Conformal and equal-area projections are mutually exclusive and are considered at the opposite ends of the spectrum of choice
Conformal projections have large areal exaggeration
Equal-area projections have large angular distortion
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Selecting special property
Figures 11.05, 11.06, and 11.07 show areal exaggeration (p) and angular deformation (w) against z and d for the projection classes
Note areal exaggeration and angular distortion of conformal and equal-area properties versus the equidistant property
Equidistant projections provide a reasonable compromise
Table 11.02 summarizes a distortion continuum by special property and use
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Minimum-error representation
Based on premise that the sums of the squares of scale errors throughout the mapped area are minimized (Eq. 6.33)
Not an exclusive property
e.g., minimum-error conformal retains special property
While minimum-error is a good choice for relatively small-scale mapping it has been seldom used
Four examples: Airy, Clarke, Hinks, Sears
Mathematically difficult
Not widely known
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Redistribution of particular scales
Introduce a standard circle or 2 standard parallels (Figures 5.08, 5.09, 5.10)
Two standard parallels have the following characteristics:
Principal scale is preserved on both standard parallels
Between the standard parallels and map edges, the maximum and minimum particular scales behave similar to the unmodified projection
Between the standard parallels, the directions of the maximum and minimum scales are reversed
No effect on special property
Reduces deformation toward edges (Tables 10.02, 10.03)
No effect on singular points
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Modification of cylindrical projections
Ratio of the length of the equator and a meridian is altered
For conformal projections, since a = b a single scale factor may be applied to coordinates to effect the change
The scale factor value is the scale of the line of zero distortion on the unmodified projection
This value results in scales of 1 on the 2 standard parallels
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Choice of standard parallels
de l’Isle projection standard parallels are located halfway between the central and bounding parallels of the map area
However, not all map areas are symmetrical
Kavraisky K constant is a shape measure (Figure 11.08)
K = 7: short latitude and large longitude extent
K = 5: long latitude and short longitude extent
K = 4: circular
K = 3: square
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Normal aspect Cylindrical equal-area projection has singlular point poles equal in length to the equator: unrealistic
Normal aspect Sinusoidal projection shows poles as point: extreme distortion near edges
Use of a pole-line can reduce distortion
Implemented by a constant which creates a singular point at the poles
Length is determined by the constant value
One half the equator is common
Eckert VI pseudocylindrical (Figure 13.05) is referred to as a truncated or flat polar projection
Pole-lines may also be curved (Aitoff-Wagner in Figure 1.05)
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Choosing Map Projections (II) - Graphical and Analytical Methods
Graphical methods of selection by visual comparison of overlays
Based on transparencies or overlays of different layers of distortion isograms
Just like any standard registration process; requires the same scale, origin, and axes
Parallels and meridians are typically not shown
Showing points/lines of zero distortion are helpful
Overlays are placed over a rough sketch of the area to be mapped
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Comparison of Bonne’s projection and the Azimuthal equal-area (Figure 12.01)
Both are equal-area; Bonne’s is pseudoconical
Same origin
From graphical analysis, the Azimuthal projection seems preferable
Studying extreme values graphically is only one criteria; analysis of distortion values in other areas of the map is also important (e.g., at the map center - covered in Chapter 13)
Overlays against rough sketches provide estimates only, and just for that particular origin and aspect
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Combined graphical and analytical methods
Summary of the technique used that led to the choice of the Bipolar oblique conformal conical projection of Latin America by Miller, 1941 (Figure 11.03)
First, specify limiting values of distortion
e.g., for a conformal map we might specify .95 < p < 1.05 which equates to an aread distortion that never exceeds +/- 5%
or
we might specify the limiting values in terms of linear distortion
.965 < m < 1.035 or not to exceed +/- 3.5%
Figures 12.03, 12.04, and 12.05 show the area to be mapped - S. America and C. America
Preliminary origin is j0 = 00, l0 = 72oW
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Miller analyzed the following projections
Transverse Stereographic, Normal Mercator, Transverse Mercator; m = 1.035 or +/- 3.5% (Figure 12.03)
Modified versions of the above with .965 < m < 1.035 or +/-3.5%; m = .965 at the origin (Figure 12.04)
A much larger part of the area to be mapped is within the distortion constraints in the modified versions making the modified versions preferable
Miller sampled other points within the mapping area to evaluate distortion at other than the extreme portions
Table 12.01 shows the average and maximum scale distortion percentages for the modified projections based on a sample of 49 points
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Selection of the Oblique aspect Conformal Conical projection (Figures 12.05 and 11.03)
Location of the pole at j 20o S and l 110o W
Centerline z0; z3 and z4 defining the limits; and z1 and z2 the standard parallels
K = 7 initially selected; K = 8 was the final choice
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Selection of a projection for internal storage within a GIS for Europe
Projections compared
Albers’ conical equal-area; Figure 12.06 and Table 12.02
Table 12.03 reflects an equal distribution of the standard parallels between the limiting parallels with a slight improvement in distortion characteristics
Murdoch’s third projection; minimum-error conical; Table 12.04; scale factors k, p and w are comparable to Alber’s
Alber’s oblique
Pole in northern Russia (55oN, 43oE)
Table 12.05 shows particular scales for the extremes of the map
An improvement over the normal aspect Alber’s
Azimuthal equal-area
Table 12.06 gives distortion characteristics based on the center point 48oN and 9oE
Best choice
Automatic methods
Automatic analysis
Registration and overlay against map background
Work by Bugaevskii (82); and Jankowski and Nyerges (89), the latter implemented in a system titled Map Projection Knowledge-Based System
MPKBS asks a series of questions from the general to the specific (page 264)
Geographical area category

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Choose Map Proj