- 1 table of living bear species
- 2 Construction of a map projection
- 3 Using a projection surface
- 4 Aspects of the projection
- 5 Scale
- 6 Choosing a model for the shape of the Earth
- 7 Classification
- 8 Projections by surface
table of living bear species
|Binomial Name||Common Name||Size||Distribution||IUCN
|Ursus maritimus||Polar Bear||M:350–700kg||Arctic||
|Ursus arctos||Brown bear
(subsp: Grizzly bear, Kodiak bear)
|100-700kg||Northern Eurasia, North America||
|Ursus americanus||American black bear||M:57–250kg
|Ursus thibetanus||Asian black bear
(Moon bear, White-chested bear)
|NE Asia, China, Himalayas||
|Melursus ursinus||Sloth bear (Stickney bear, Labiated bear)||M:80-192kg
||Termites, bees, fruits|
|Helarctos malayanus||Sun bear||27-65kg||South-east Asia||
||Bees/honey, insects, fruits|
|Tremarctos ornatus||Spectacled bear (Andean bear)||M:100–200kg
||Vegetation, some meat|
|Ailuropoda melanoleuca||Giant Panda||M: 100-160kg
Construction of a map projection
|Pseudocylindrical||none||several Sinusoidal Mollweide||(sinusoidal)||Robinson|
Some of the simplest map projections are literally projections, and are obtained by placing a light source at some definite point relative to the globe and projecting its features onto a specified surface. This is not the case for many projections, which are defined only in terms of mathematical formulae that have no direct geometric interpretation.
A surface that can be unfolded or unrolled into a plane or sheet without stretching, tearing or shrinking is called a developable surface. The cylinder, cone and of course the plane are all developable surfaces. The sphere and ellipsoid do not have developable surfaces, so any projection of them onto a plane will have to distort the image. (To compare, one cannot flatten an orange peel without tearing and warping it.) One way of describing a projection is first to project from the Earth's surface to a developable surface such as a cylinder or cone, and then to unroll the surface into a plane. While the first step inevitably distorts some properties of the globe, the developable surface can then be unfolded without further distortion. The three developable surfaces (plane, cylinder, cone) provide useful models for understanding, describing, and developing many map projections. However, these models are limited in two fundamental ways. For one thing, most world projections in actual use do not fall into any of those categories. For another thing, even most projections that do fall into those categories are not naturally attainable through physical projection.
Using a projection surface
The creation of a map projection involves two steps
As L.P. Lee notes,
- Selection of a model for the shape of the Earth or planetary body (usually choosing between a sphere or ellipsoid). Because the Earth's actual shape is irregular, information is lost in this step.
- Transformation of geographic coordinates (longitude and latitude) to Cartesian (x,y) or polar plane coordinates. Cartesian coordinates normally have a simple relation to eastings and northings defined on a grid superimposed on the projection.
No reference has been made in the above definitions to cylinders, cones or planes. The projections are termed cylindric or conic because they can be regarded as developed on a cylinder or a cone, as the case may be, but it is as well to dispense with picturing cylinders and cones, since they have given rise to much misunderstanding. Particularly is this so with regard to the conic projections with two standard parallels: they may be regarded as developed on cones, but they are cones which bear no simple relationship to the sphere. In reality, cylinders and cones provide us with convenient descriptive terms, but little else.
Lee's objection refers to the way the terms cylindrical, conic, and planar (azimuthal) have been abstracted in the field of map projections. If maps were projected as in light shining through a globe onto a developable surface, then the spacing of parallels would follow a very limited set of possibilities. Such a cylindrical projection (for example) is one which:
- Is rectangular;
- Has straight vertical meridians, spaced evenly;
- Has straight parallels symmetrically placed about the equator;
- Has parallels constrained to where they fall when light shines through the globe onto the cylinder, with the light source someplace along the line formed by the intersection of the prime meridian with the equator, and the center of the sphere.
(If you rotate the globe before projecting then the parallels and meridians will not necessarily still be straight lines. Rotations are normally ignored for the purpose of classification.)
Where the light source emanates along the line described in this last constraint is what yields the differences between the various "natural" cylindrical projections. But the term cylindrical as used in the field of map projections relaxes the last constraint entirely. Instead the parallels can be placed according to any algorithm the designer has decided suits the needs of the map. The famous Mercator projection is one in which the placement of parallels does not arise by "projection"; instead parallels are placed how they need to be in order to satisfy the property that a course of constant bearing is always plotted as a straight line.
Aspects of the projection
Once a choice is made between projecting onto a cylinder, cone, or plane, the aspect of the shape must be specified. The aspect describes how the developable surface is placed relative to the globe: it may be normal (such that the surface's axis of symmetry coincides with the Earth's axis), transverse (at right angles to the Earth's axis) or oblique (any angle in between). The developable surface may also be either tangent or secant to the sphere or ellipsoid. Tangent means the surface touches but does not slice through the globe; secant means the surface does slice through the globe. Moving the developable surface away from contact with the globe never preserves or optimizes metric properties, so that possibility is not discussed further here.
A globe is the only way to represent the earth with constant scale throughout the entire map in all directions. A map cannot achieve that property for any area, no matter how small. It can, however, achieve constant scale along specific lines.
Some possible properties are:
- The scale depends on location, but not on direction. This is equivalent to preservation of angles, the defining characteristic of a conformal map.
- Scale is constant along any parallel in the direction of the parallel. This applies for any cylindrical or pseudocylindrical projection in normal aspect.
- Combination of the above: the scale depends on latitude only, not on longitude or direction. This applies for the Mercator projection in normal aspect.
- Scale is constant along all straight lines radiating from a particular geographic location. This is the defining characteristic of an equidistant projection such as the Azimuthal equidistant projection. There are also projections (Maurer, Close) where true distances from two points are preserved.
Choosing a model for the shape of the Earth
Projection construction is also affected by how the shape of the Earth is approximated. In the following section on projection categories, the earth is taken as a sphere in order to simplify the discussion. However, the Earth's actual shape is closer to an oblate ellipsoid. Whether spherical or ellipsoidal, the principles discussed hold without loss of generality.
Selecting a model for a shape of the Earth involves choosing between the advantages and disadvantages of a sphere versus an ellipsoid. Spherical models are useful for small-scale maps such as world atlases and globes, since the error at that scale is not usually noticeable or important enough to justify using the more complicated ellipsoid. The ellipsoidal model is commonly used to construct topographic maps and for other large- and medium-scale maps that need to accurately depict the land surface.
A third model of the shape of the Earth is the geoid, a complex and more accurate representation of the global mean sea level surface that is obtained through a combination of terrestrial and satellite gravity measurements. This model is not used for mapping because of its complexity, but rather is used for control purposes in the construction of geographic datums. (In geodesy, plural of "datum" is "datums" rather than "data".) A geoid is used to construct a datum by adding irregularities to the ellipsoid in order to better match the Earth's actual shape. It takes into account the large-scale features in the Earth's gravity field associated with mantle convection patterns, and the gravity signatures of very large geomorphic features such as mountain ranges, plateaus and plains.
Historically, datums have been based on ellipsoids that best represent the geoid within the region that the datum is intended to map. Controls (modifications) are added to the ellipsoid in order to construct the datum, which is specialized for a specific geographic region (such as the North American Datum). A few modern datums, such as WGS84 which is used in the Global Positioning System, are optimized to represent the entire earth as well as possible with a single ellipsoid, at the expense of accuracy in smaller regions.
A fundamental projection classification is based on the type of projection surface onto which the globe is conceptually projected. The projections are described in terms of placing a gigantic surface in contact with the earth, followed by an implied scaling operation. These surfaces are cylindrical (e.g. Mercator), conic (e.g., Albers), or azimuthal or plane (e.g. stereographic). Many mathematical projections, however, do not neatly fit into any of these three conceptual projection methods. Hence other peer categories have been described in the literature, such as pseudoconic, pseudocylindrical, pseudoazimuthal, retroazimuthal, and polyconic.
Another way to classify projections is according to properties of the model they preserve. Some of the more common categories are:
- Preserving direction (azimuthal), a trait possible only from one or two points to every other point
- Preserving shape locally (conformal or orthomorphic)
- Preserving area (equal-area or equiareal or equivalent or authalic)
- Preserving distance (equidistant), a trait possible only between one or two points and every other point
- Preserving shortest route, a trait preserved only by the gnomonic projection
Because the sphere is not a developable surface, it is impossible to construct a map projection that is both equal-area and conformal.
Projections by surface
Pseudocylindrical projections map parallels as straight lines and represent the central meridian as a straight line segment. Other meridians are longer than the central meridian and bow outward away from the central meridian. Along parallels, each point from the surface is mapped at a distance from the central meridian that is proportional to its difference in longitude from the central meridian, i.e. meridians are evenly spaced. Pseudocylindrical projections can be expressed mathematically as the form:
Pseudocylindrical projections are most commonly used for displaying the whole world. Many pseudocylindrical projections are equal area, others are compromise projections, but none can be conformal. On a pseudocylindrical map, north-south relationships are preserved, so any point further from the equator on the map than some other point has a higher latitude than the other point. This trait is useful when illustrating phenomena that depend on latitude, such as climate. Examples of psuedocylindrial projections include:
- Sinusoidal, which was the first pseudocylindrical projection developed. Vertical scale and horizontal scale are the same throughout, resulting in an equal-area map. On the map, as in reality, the length of each parallel is proportional to the cosine of the latitude. Thus the shape of the map for the whole earth is the region between two symmetric rotated cosine curves. The true distance between two points on the same meridian corresponds to the distance on the map between the two parallels, which is smaller than the distance between the two points on the map. The distance between two points on the same parallel is true. The area of any region is true. However there is significant distortion away from the equator and central meridian.
- The Mollweide projection is another early pseudocylindrical projection, dating from 1805. Meridians are mapped as ellipses and parallels are no longer evenly spaced, but it is equal area. The area of high distortion is generally reduced compared to the Sinusoidal, and it has a "pleasant" overall shape.
- The Robinson projection. This is an example of a compromise pseudocylindrical projection - no longer equal area, but with the aim of reducing distortion overall. It also maps the poles as straight lines - "flat polar" - a feature of a number of pseudocylindrical projections, which can help reduce shearing distortion at high latitudes.
Interrupted and Hybrid forms
Pseudocylindrical projections are particularly suitable for displaying in interrupted form: typically dividing along one of the meridians, usually from the pole to the equator, and with a number of primary meridians as a result. This significantly reduces distortion, at the expense of a more complex shape and the loss of relationship between points either side of a cut. Interruption is commonly used when the focus is primarily on the land masses (or, more rarely, primarily on the oceans).
Two pseudocylindrical (or a pseudocylindrical and cylindrical) projections can also be combined, most usually by matching scale along parallels, to form a hybrid projection. An example of both approaches is the usual presentation of Goode homolosine projection, which uses the sinusoidal projection in equatorial regions and Mollweide projection in higher latitudes. The HEALPix projection is a more recent example, combining a cylindrical equal area projection in equatorial regions with an interrupted Collignon projection at higher altitudes.
A conic projection is a projection in which meridians are mapped to equally spaced lines radiating out from the apex and circles of latitude (parallels) are mapped to circular arcs centered on the apex. However, compared to the sphere, the angular distance between meridians in reduced, by a set factor the cone constant. 
When making a conic map, the map maker selects two (or one) standard parallels. Those standard parallels may be visualized as secant lines where the cone intersects the globe — or, if the map maker chooses the same parallel twice, as the tangent line where the cone is tangent to the globe. The resulting conic map has low distortion in scale, shape, and area near those standard parallels. Conical maps (in the normal aspect) can portray large areas of temperate latitudes with low to moderate distortion making them a common choice for maps of America and Europe for example.
However, distances along the parallels to the north of both standard parallels or to the south of both standard parallels are necessarily stretched and distortion rapidly increases away from the reference parallels, becoming extreme in the opposite hemisphere. This limits their use without modification for larger areas.
The most popular conic maps either
- Albers conic
- compress north-south distance between each parallel to compensate for the east-west stretching, giving an equal-area map, or
- Equidistant conic
- keep constant distance scale along the entire meridian, typically the same or near the scale along the standard parallels, or
- Lambert conformal conic
- stretch the north-south distance between each parallel to equal the east-west stretching, giving a conformal map.
Most common conic projections are not perspective projections, including these three. A perpsective projection is the Braun stereograhic conic projection but it is rarely used.
Pseudocylindrical maps have circular concentrical parallels, but meridians other than the prime meridian are curved (in a manner analagous to pseudoscylindrical maps.) Pseudoconical maps have be known since ancient times, predating pseudocylindrical maps. Psuedoconical maps can display the whole world with moderate disotrtion but have unusual outlines.
- Collignon projection which in its most common forms represents each meridian as 2 straight line segments, one from each pole to the equator.
- Lee, L.P. (1944). "The nomenclature and classification of map projections". Empire Survey Review VII (51): 190–200. p. 193
- Snyder, John P. (1993). Flattening the earth: two thousand years of map projections. University of Chicago Press. ISBN 0-226-76746-9.
- Snyder, John P. (1997). Flattening the earth: two thousand years of map projections. University of Chicago Press. ISBN 978-0-226-76747-5.
- Weisstein, Eric W., "Sinusoidal Projection", MathWorld.
- http://www.progonos.com/furuti/MapProj/Normal/ProjPCyl/projPCyl.html retrieved 20th April 2012
- Carlos A. Furuti. "Conic Projections"