Conclusion

In the course of this project I have introduced in detail the concept behind perspective projections onto each class of surface. Analysis of projections has been shown in terms of orthogonal scale factors, to show area and shape properties.

Non-perspective have been introduced as a way to satisfy desired conditions on the final projection. These were defined and found in terms of orthogonal scale factors. Other non-perspective methods have been briefly introduced, notably the interrupted projection technique.

A cartographic map projection is a systematic transformation (also called mapping) from a round surface to a plane. There are many

different projections, since there are several interesting or useful properties to fulfill. For instance, it would be desirable to keep shape,

distance and area relationships exactly as in the original surface. Unfortunately, it can be proved that there is not and there will never

be such a perfect projection: every one is bound to distort at least part of the mapped region.

Therefore, cartography is an art and science of trade-offs and

guidelines for designing and choosing the least inappropriate projection for each purpose.

Basic Definitions and Concepts

The Shape of the Earth

Since a map is a representation, the original shape of the represented subject must first be defined. An important branch of cartography, geodesy studies the Earth shape and how it is related to its surface's features.

Spheres, spheroids and geoids

Geoid, from the Greek for "Earth-shaped", is the common definition of our world's shape. This recursive description is necessary because no simple geometric shape matches the Earth:

· like in all space bodies above a certain mass, the Earth's materials aggregated in a spherical shape, which minimizes gravity and potential energy

· however, quick rotation around its axis caused a bulging at the middle (Equator) and a flattening at the poles; the resulting shape is called an spheroid or oblate ellipsoid. The equatorial

diameter is nearly 1/300 longer than polar diameter · on average, the surface rests perpendicular to the gravitational

force at each point, which also influences land level. However, mass concentration is not uniform, due to irregular crust density and land distribution. In addition to the rotation bulge, some

researchers concluded that the southern hemisphere is expanded and its pole depressed, while the other half is

compressed with a raised pole (the resulting shape resembles a pear, but average distortion of curvature does not exceed 50

meters). Others have the opinion that the Equator itself is elliptical

· finally, the surface is not smooth, further complicating the shape

Taken in account, those factors greatly complicate the cartographer's job but, depending on the task, some irregularities can be ignored. For

instance, although important locally, terrain levels are minuscule in planetary scale: the tallest land peak stands less than 9km above sea level, or nearly 1/1440 of Earth diameter; the depth of the most profound sea abyss is roughly 1/1150 diameter.

For maps covering very large areas, especially worldwide, the Earth may be assumed perfectly spherical, since any shape imprecision is dwarfed by unavoidable errors in data and media resolution. This assumption holds for most of this document. Conversely, for very small areas terrain features dominate and measurements can be based on a flat Earth.

The Datum

For highly precise maps of smaller regions, the basic ellipsoidal shape can not be ignored. A geodetic datum is a set of parameters (including axis lengths and offset from true center of the Earth)

defining a reference ellipsoid. For each mapped region, a different datum can be carefully chosen so that it best matches average sea

level, therefore terrain features. Thus, data acquisition for a map involves surveying, or measuring heights and distances of reference points as deviations from a specific datum (a delicate task: due to mentioned irregularities, gravity - and therefore plumb bobs and levers - is not always aligned towards the center of the Earth).

Several standard datums were adopted for regional or national maps. International datums do exist, but can not fit any particular area as well as a local one.

Coordinate Systems

Latitudes and Longitudes

Although the Earth is a three-dimensional object, when supposed spherical its surface has a constant radius, so any point on it is

uniquely identified using a polar two-coordinate system.

Wooden sphere with an octant removed for clarity; copper arrows define the coordinate system's origins. The white "point" is located by two angles or coordinates: its latitude and longitude. Every point has a counterpart directly on the opposite side, called its antipode (not shown here).

Selected parallels (in red) and meridians (in blue), here spaced 15° apart, comprise a spherical graticule. The number of possible parallels and meridians if infinite; how many should be presented depend on the globe's (or map's) purpose and size.

Given a polar axis (around which the planet rotates daily), an orthogonal plane which divides the globe in halves (i.e., an equatorial plane) and an arbitrary reference axis on it, any surface point

determines a latitude, or the smallest angle, measured from the center of the Earth, from it towards the equatorial plane, and a longitude, or the smallest angle from the arbitrary axis to the projection of the point on the Equator determined by the latitude.

A graticule is a spherical grid of coordinate lines over the planetary surface, comprising circles on planes normal - i.e., perpendicular to the north-south axis - called parallels, and semicircular arcs with that axis as chord, called meridians. True to their name, no parallels ever cross one another, while all meridians meet at each geographic pole.

Every parallel crosses every meridian at an angle of 90°. This and other properties help assessing map distortion.

Both sets of parallels and meridians are infinite, but of course only a

subset can be included in any map. A point's latitude and longitude, both usually measured in degrees, define the crossing of a parallel and

a meridian, respectively. So, latitudes mean north-to-south angles from the equatorial plane, while longitudes express west-to-east angles from a particular meridian defined by the reference axis.

Latitudes conventionally range from 90° South to 90° North, while longitudes range from 180° West to 180° East. reference axis.

Parallels and Their Properties

A natural reference, the longest parallel divides the Earth in two equal hemispheres, north and south; thus its name, Equator. Four other important parallels are defined by astronomical constraints. The

geographical north-south axis is actually tilted slightly less than 23.5° from the plane of the Earth's orbit around the sun. This accounts for

the different seasons and different lengths of day and night periods throughout the year.

Schematic cross section of Earth's orbit. AT is the axial tilt, about 23.5°

Every year about December 21st, the solar rays fall vertically upon a parallel near 23.5°S. That is the longest day in the southern hemisphere (notice how most of it is exposed to the sun, so that date is called the southern summer solstice), but the shortest day in the northern hemisphere (therefore winter solstice); not only shorter daylight periods but a shallower angle of incidence of solar rays explain the lower temperatures north of Equator.

Near June 21st, a similar phenomenon happens along the parallel opposite North. By definition, these two parallels encircle the torrid or

tropical zone; they are named after the zodiacal constellations where the sun is at those dates, thus Tropic of Capricorn (south) and

Tropic of Cancer (north). In regions south of the Tropic of Capricorn the sun around noon appears to run always north of the observer; at the same hour, in places north of the Tropic of Cancer the sun runs

always south of the observer, while in tropical regions the sun appears sometimes south, sometimes north, depending on the season.

Subtracting the axial tilt from 90° we get the latitudes of the Arctic

(about 66.5°N) and Antarctic (about 66.5°S) polar circles. Around December 21 the sun does not set at the Antarctic circle for a full day.

Going south, we get even longer consecutive daylight periods, up to six months at the pole. There are correspondingly long nights at the

Antarctic winter. Of course, the same occurs at the northern latitudes, with a six-month offset.

Points on the same parallel suffer similar rates of exposure to the sun,

therefore are prone to similar climates (disregarding other factors like altitude, wind/sea conditions and terrain).

A point's latitude can be inferred from the sun's angle above the horizon at noon (the moment when the sun appears highest at the sky and a vertical stake projects its shortest shadow); Sailors use

instruments like the sextant for measuring it.

East-west distances between

points separated by one minute of

solar time at different latitudes. Distance is zero at poles, where one "sees" every moment of the

day at any time.

Meridians and Their Properties

All points on a meridian have the same solar, or local, time. Due to different day lengths throughout the year, correction formulas are

applied to convert it to a local mean time. Since it would be impractical having nearby regions with different time reckonings (one

nautical mile, approximately 1853 meters, corresponds to an angle of 0°1" along the Equator, or a temporal difference of 4 seconds), the world is divided in 24 fuses, or time zones, each 15° wide. For everyday purposes, every point inside a zone is considered having the same standard time (actually, a few countries still use solar time).

In practice, the time-jumping boundaries seldom follow the meridians, bending (usually at national or regional borderlines) to keep related

places conveniently synchronized.

Unlike the Equator, there's no easily defined prime or "main" meridian, which was fixed (mainly by political consensus) in 1884 over the Royal

Observatory in Greenwich, near London, UK. This choice's only obvious advantage is setting the opposite meridian (near or at the left

or right edges of many world maps) away from most inhabited areas. That opposite meridian is the base of the international date line, which separates world halves in two different days. Again, this line is

somewhat irregular in order to keep national territories (mostly Pacific islands) in a single timezone.

Compared to finding a point's latitude, getting its longitude is a much more involved procedure, usually comparing the time separating the noon at the reference meridian and at the point in question.

Basic Definitions and Concepts

"To project" means transferring features from Earth to a suitable surface, like a tangent plane, a secant cylinder or a secant truncated cone. Regions nearer the surface are usually better presented.

From ellipsoid or sphere (left) to flat map (right). A conceptual intermediary surface (center) may be useful for either actual construction or mere visualization, but the darker conversion paths unavoidably incur in distortion.

An orange's peel provides a classic demonstration of distortion in maps: it cannot be completely flattened unless compressed, stretched or torn apart.

Maps, Globes and Projections

Any study in geography requires a reduced model of the Earth, like a globe or map. Neither is perfect: a globe is seldom practical, and flat maps are never free from errors. Selecting or creating a good map

involves interesting choices and trade-offs.

What's a Projection?

A cartographical map projection is a formal process which converts (mathematically speaking, maps) features between a spherical or ellipsoidal surface and a projection surface, often flat. Although many projections have been designed, just a few are currently in widespread use. Some were once historically important but were superseded by better options, several are useful only in very specialized contexts, while others are little more than fanciful curiosities.

Projection Surfaces

The map's support, the projection surface is usually created, i.e., developed, conceptually touching the mapped sphere in one (the

surface is tangent to the sphere) or more (the surface is secant) regions. Intuitively, portions of the surface nearer the touching regions

depart less from the original spherical shell; therefore, the corresponding portions on the map are more faithfully reproduced.

Some projections are actually composites, fitting separate surfaces to different regions of the map: overall error is reduced at the cost of

greater complexity.

Sometimes a conceptual auxiliary surface like a cone, open cylinder, ellipsoid or torus is employed: the sphere's features are (often by

perspective construction) transferred to that surface, which is then flattened. Many projections are classified as "cylindrical" or "conic"; however, for most of them, the naming is just an analogy or didactic

device, since they aren't actually developed on an intermediary surface; rather, the resulting map can be rolled onto a tube or a cone.

The Unavoidable Distortion

Two quasi-orthographic views of a sphere (plus polar axis) divided in eight equal sections, surrounded by four maps at the same scale using, clockwise from top right: azimuthal

equidistant, Lambert's equal-area cylindrical, Maurer's equal-area star with four lobes, Winkel Tripel projections.

A few selected projections illustrate how the same spherical data can be stretched, compressed, twisted and otherwise distorted in different ways. The azimuthal equidistant map has interesting properties regarding directions and distance from the central pole, but the outer hemisphere is greatly stretched: its pole becomes a circle. Both poles become lines in the equal-area cylindrical map, but it covers the same area as the original sphere; also, all octants have identical shape. This particular star projection has unequal octants and marked loss of continuity; however, it also preserves area. In the Winkel Tripel map, octants have different shapes, area is changed and poles are linear, but overall distortion is subjectively smaller. Finally, the orthographic views, projections themselves, show only part of the sphere. All projections suffer from some distortion; none is "best" for all purposes. Octants would assume even stranger shapes in oblique aspects.

No matter how sophisticated the projection process, the original surface's features can never be perfectly converted to a flat map: distortion, great or small, is always present in at least one region of planar maps of a sphere. Distortion is a false presentation of angles, shapes, distances and areas, in any degree or combination.

Every map projection has a characteristic distortion pattern. An important part of the cartographic process is understanding distortion and choosing the best combination of projection, mapped area and coordinate origin minimizing it for each job.

Cones and cylinders are developable surfaces with zero Gaussian curvature (in a nutshell, at every point passes a straight line wholly contained in the surface). Distortion always occur when mapping a sphere onto a cone or cylinder, but their reprojection onto a plane incurs in no further errors.

The Choice of Coordinate Origin

Another key feature of any map is the orientation (relative to the sphere) of the conceptual projection surface.

A particular projection may be employed in several aspects, roughly defined by the graticule lay-out and the sphere's region nearest the conceptual projection surface, commonly the center of a whole-world map (not necessarily the actual center, due to cropping or recentering

):

· a polar map aligns the Earth axis with the projection system's, thus one of its poles lies at the map's conceptual center;

· an equatorial map is centered on the Equator, which is set across one of the map's major axes (mostly horizontally);

· an oblique map has neither the polar axis nor the equatorial plane aligned with the projection system.

Also (orthogonally),

· the most "natural" aspect of a projection, called normal, conventional, direct or regular, is ordinarily determined by geometric constraints; it often demands the simplest calculation and produces the most straightforward graticule. The polar aspect is the normal one for the azimuthal and conic groups of

projections, while the equatorial is the normal for cylindrical and pseudocylindrical groups. The normal graticule for azimuthal and

conic projections comprises exclusively straight lines and circular arcs; normal cylindrical have altogether straight graticules

· the transverse aspect frequently resembles the normal one, except by a simple rotation of 90°.

Normal Transverse Oblique

(polar) (equatorial)

(equatorial)

(polar) (seldom used) (seldom used)

Three (normal, transverse and oblique) aspects applied to three (azimuthal equal-area, Gall's stereographic cylindrical and Albers's conic) projections with different tangent projection surfaces in blue (just a few of infinitely many possible oblique maps are presented). Some projections like Gall's stereographic may actually be derived via perspective geometry; for most, however, surfaces are only illustrative: the map may be laid on a developable surface, but is not calculated from it.

Distinctive graticules of some projection groups (radially symmetric meridians in azimuthal and conic maps, rectangular grid in cylindrical maps) are only realized using their simple, normal aspects. Trivial rotations of the finished map, like turning it sideways or upside-down, leave both aspect and projection unchanged. On the other hand, modifying the aspect does not affect either represented area or the shape of the whole map.

Some authors consider a different definition of "aspect": it determines whether the projection surface is secant or tangent to the globe (this is of course a more limited meaning, since many projections are not defined via an auxiliary surface). Still others reserve "aspect" for one meaning and "case" for the other, or even for distinguishing equations devised for an ellipsoid versus the simpler sphere.

Theoretically, especially supposing a spherical Earth, any projection may be applied in any aspect: after all, the parallel/meridian system is a convention which might have origins anywhere, although it is hard imagining others more useful than the poles. However, many projections are almost always used in particular aspects:

· their properties may be less useful otherwise. E.g., many factors like temperature, disease prevalence and biodiversity depend on climate, thus roughly on latitude; for projections with constant parallel spacing, on equatorial aspects latitude is directly

converted to vertical distance, easing comparisons. · several projections whose graticules in normal aspects are

comprised of simple curves were originally defined by geometric construction. Since many non-normal aspects involve complex curves, they were not systematically feasible before the

computer age (indeed, mapping was an important motivation for calculation shortcuts like logarithms).

Even though oblique aspects are frequently useful, in general

calculations for the actual ellipsoid are fairly complicated and are not developed for every projection.

Useful Map Properties

Matching Projection to Job

No map projection is perfect for every task. One must carefully weigh pros and cons and how they affect the intended map's purpose before choosing its projection. The next sections outline desirable properties in a map, mentioning how projections can be used or misused.

Like in the real Earth, this globe's north-south axis is tilted at 23°27'30" from the orbital

plane.

For any map, the most important parameters of accuracy can be

expressed as:

· can distances be accurately measured? · how easy is obtaining the shortest path between two points?

· are directions preserved? · are shapes preserved?

· are area ratios preserved? · which regions suffer the most, and which kind of, distortion?

Globe Properties

Unfortunately, only a globe offers such properties for any points and regions. Since crafting a globe is only a matter of reducing dimension (no projection is involved), every surface feature can be reproduced with precision limited only by practical size, with no loss of shape or distance ratios. As a bonus, a globe is a truly three-dimensional body whose surface can be embossed in order to present major terrain features. But globes suffer from many disadvantages, being:

· bulky and fragile, clumsy to transport and store; · expensive to produce, especially at larger sizes, thus impractical

for showing fine details; · difficult to look straight at every point, therefore · cumbersome for taking measurements or setting directions;

· able to show a single hemisphere at a time; · completely unfeasible for widespread reproduction by printed or

electronic media

Changing minor map properties

Several maps using the Mercator projection show how fundamental projection properties are not affected by changes in minor features like scale, aspect, and choice of mapped area. What is constant in all Mercator maps is how scale quickly changes the farther one gets from a reference line (which may or may not be horizontal), but not in a direction parallel to that line.

1 The Mercator projection is frequently (and usually inappropriately) employed in world maps. Any Mercator map must be arbitrarily clipped, in this normal equatorial aspect at its top and bottom; a complete map would be infinitely tall. The cropped, transverse and oblique maps on the right are much more legitimate uses for this projection.

2 Translating the central meridian is a common and trivial modification

4 Scaling - in this case by 100% - and cropping the map are common editorial decisions which change no projection properties.

5 Still a Mercator map, but in a transverse aspect scaled

strongly dependent on the map's theme. Here a translation is used to avoid interrupting the Pacific Ocean. The distortion pattern remains the same as above, with areas drastically enlarged at the top and bottom.

3 Departing from tradition, some maps half-jokingly show Australia and other meridional nations "on top of the world". Here, a bit more of Antarctica was arbitrarily cropped off. Of course, neither projection nor aspect differ from the above maps. Mercator's projection is often naïvely accused of privileging the northern hemisphere and thus First World nations, but an uncropped equatorial Mercator map is perfectly symmetrical around the Equator; it's the map editor who decides whether a large portion of Antarctic is or not relevant

up 25% and rotated 90°: there's a different graticule and the poles can be visible, however the distortion pattern remains: scale is constant vertically, but increases left and right. The central meridian, instead of the Equator, is perfectly represented, while Africa, bent strongly enough to seem split in two, is barely recognizable.

6 This oblique Mercator map (scaled up 25% and rotated) lays more of the Americas in the central band of reduced scale distortion. The Equator is no longer straight.

Exploiting Map Properties

So, flat map projections are usually more important and useful than globes, despite their shortcomings. In particular, no flat map can be simultaneously conformal (shape-preserving) and equal-area (area-preserving) in every point.

However, a reasonably small spherical patch can be approximated by a flat sheet with acceptable distortion. In most projections, at least one specific region (usually the center of the map) suffers little or no distortion. If the represented region is small enough (and if necessary

suitably translated in an oblique map), the projection choice may be of little importance.

On the other hand, the fact that no projection can faithfully portray the whole Earth should not lead to a pessimistic view, since distorting the planet on purpose makes possible - unlike a globe - uncovering important facts and presenting at a glance relationships normally obscure. Skillfully used, distortion is a powerful visual tool; this becomes explicit in a kind of pseudoprojection called cartogram, where the place a point is drawn depends not only on its location on Earth, but also on attributes of the mapped region, like a county's population or a country's economic output. E.Raisz presented cartograms using simple rectangles with proportional areas but no relation to the original shapes (1934). W.Tobler developed the modern concept of cartogram, which instead uses distorted but actually recognizable shapes; electronic computers are an indispensable tool.

No projection is intrinsically good or bad, and a projection suitable for

a particular problem might well be useless or misleading if applied elsewhere.

Major and Minor Properties

For any projection, the "major" properties - concerning whether and how much distances, areas and angles are preserved - are largely independent of changes in scale, aspect and the choice of mapped

area (this last detail is significantly determined by the aspect, selection of central meridian, and any post-projection rotation and cropping), even though the graticule and other shapes may appear radically

different. Therefore, the same projection may be the source of many maps, often only superficially unrelated.

Sometimes, for historical or convenience reasons, particular uses of a

single projection are known by different names. E.g., the Gauss-Krüger projection is a transverse case of the ellipsoidal Mercator

projection; the Briesemeister and Nordic projections are oblique (with or without rescaling) aspects of the Hammer projection; and many rescaled versions of Lambert's equal-area cylindrical projection were proposed.

Identically marked

graticules in orthographic (representing the original globe) and

van der Grinten III projections

at same scale.

The projection known as Van der Grinten's third violates all five graticule properties:

1. purple lines are very stretched near the poles; green lines are longer farther from the vertical axis (the constant vertical spacing between parallels is deceptive)

2. red lines are also slightly stretched closer to the map edges

3. the blue cells are also enlarged near the edges; this is more obvious at high latitudes

4. of all meridians, only the central remains straight

5. parallels and meridians cross at right angles only at the Equator and central meridian; elsewhere, there's shearing, symmetrical around the vertical axis: angles are compressed in opposite directions east and west of the central meridian

The Graticule as a Guide to Distortion

Especially for a map in the normal aspect, a quick visual inspection of its graticule provides obvious clues of whether its projection preserves features. For instance, if the coordinate grid is uniformly laid (say, one line every ten degrees),

1. along any meridian, the distance on the map between parallels should be constant

2. along any single parallel, the distance on the map between meridians should be constant; for different parallels, should decrease to zero towards the poles

3. therefore, any two grid "cells" bounded by the same two parallels should enclose the same area

Also,

4. the Equator and all meridians should be straight unbroken lines, since they don't change direction on the Earth's surface

5. any meridian should cross all parallels at right angles

Again, for any particular projection, violation of any or all these properties doesn't necessarily make it poorly designed or useless; rather, it suggests (and constrains) both the range of applications for which it is suitable and, for each application, regions on the map where distortion has significance.

Onther one

1. Azimuthal Projections These azimuthal projections are the earliest types, devised by the 'Ancient Greeks'. In each case, the globe touches the projection plane at a point: the simplest cases are the polar azimuthal examples

Graphics: projections overview - azimuthal

a. Gnomonic

· Point of origin is at the centre of the GG (Generating globe). · Scale increases rapidly from the centre, so the equator can't be shown. · The advantage of this projection is that it is the ONLY one where all great

circles are straight lines.

b. Stereographic

· Point of origin is at far pole. · Scale increases from centre, but not as much as gnomonic. · Can show one hemisphere, but with increasing distortion. · It is a conformal projection.

c. Orthographic

· Point of origin at infinity (like a projector), or distant planet. · Scale decreases from centre, can only show 1 hemisphere. · It shows the perspective as seen from space, e.g. the earth from the

moon.

d. Equidistant

· Origin is located between the stereographic and orthographic (opposite pole and infinity)

· All meridians are standard lines, so equatorial areas are exaggerated (SF along equator is 1.6).

· An advantage is that the parallels are equally spaced (polar case)

e. Equal-Area

· Origin between orthographic and equidistant to compensate for stretching, but it is equal-area.

Each of the five has equatorial and oblique versions, which can be centered on any chosen location,. e.g. St. John's NL

The main use for azimuthal projections is to show the polar regions

2. Cylindrical Projections The cylindrical projections date from the 16th century. The globe is 'wrapped' by a cylinder, which is 'unwrapped' at a convenient line (180 degrees E/W most commonly). The equator becomes the standard line. All projections have a rectangular grid, with parallels of equal length. Meridians are at right angles, equally spaced. examples

Plate Carree (geographic) projection

· equidistant spacing of parallels (1 degree = 1 degree anywhere on the map) graphic

Mercator Projection (1569) : 'the navigator's friend'

Cylindrical projections are especially well know as a result of the widespread use of Mercator's projection: graphic

· Lines of constant compass bearing (rhumb lines or loxodromes) are straight lines, crucial for early navigation.

· N-S stretching to match E-W, hence shapes are 'maintained'.

· Scale Factor is the same in two directions at any point, and hence is conformal.

· It exaggerates polar areas greatly: x4 at 60, x30 at 80, infinite at poles (which cannot be shown)

Transverse Mercator graphic

· Standard line is on a selected line of longitude (meridian) graphic · This projection is the basis for the (Universal) UTM system in many

countries - Australia, Canada, US, UK etc. · Each UTM zone is 6 degrees of longitude wide, each Central Meridian is a

standard line. · The UTM system consists of 60 UTM projections · Polar areas use the azimuthal stereographic projection

Gall's stereographic Projection (1880) / and other cylindrical projections

· Details are projected 'stereographically' from the other side of globe.

Gall's orthographic projection, now is well known as Peters (1972)

· E-W stretching is matched by N-S compression, for an Equal-Area projection.

· Used recently by UNESCO and religious organizations to emphasize the importance of the third world compared to the Mercator's projection, but while areas are preserved, shapes are (grotesquely) distorted.

Used for World (historical) maps - fill a rectangular shape

3. Conic Projections Conic projections date from the 18th century. The cone opens out to form a 'pie' shape with concentric parallels, and meridians as radial spokes. examples The standard line can be located at a chosen latitude: there are no oblique and transform cases, since the ability to select standard parallels makes them redundant. Cylindrical and azimuthal projections could be considered as the extreme cases of the conic (highly acute or highly flattened cones respectively).

Simple Perspective graphic

One standard parallel

· Scale increases away from the standard parallel.

Two standard parallels

· Scale increases outside standard parallels. · Scale decreases inside the standard parallels.

Common conic projections include Albers and Lambert's equal area projections (Albers is used in BC with standard parallels at 50 and 58.5 north) BC Albers

Used for Mid-latitude regions, not world maps

4. Conventional Projections These are also called 'pseudo cylindrical' and are geometrically constructed.

The parallels are generally equally spaced as in the cylindrical projections, but are made proportional to their real length to minimize distortion. Graphic

a. Sinusoidal graphic

· Central area has least distortion (as opposed to the band around equator in cylindricals). · Meridians are not at right angles. Equal area is preserved at the cost of shape. · Parallels equally spaced, correct length.

b. Mollweide / Homolographic (1805)

· Equal area but less distortion than sinusoidal. · More 'pleasing shape' particularly in high latitudes.

· Parallels are not true to scale, too long above 40 degrees. · The 90 E/W line forms a circle; meridians equally spaced.

· Parallels spaced closer together from equator to maintain equal-area.

c. Goode's Homolosinal (1923)

· it combines sinusoidal below 40 degrees latitude and Mollweide above 40 degrees; they join at 40 degrees latitude, where they are the same length. Equal-area; often used in modern atlases, more aesthetically pleasing

These world projections are often seen in interrupted form: saving distortion by cutting out ocean areas.

d. Robinson's graphic

· Adopted by National Geographic in 1988; The poles are shown as lines to preserve shape in extremes.

Used for global distribution maps

5. Summary of Projection Basics

· Projections are needed to transform the earth from a sphere to a flat map. · Distortions are unavoidable in this process. · Scale Factor (SF) describes this distortion in scale.

Projections can maintain some, but not all of these:

· Shapes Conformal: SF is the same N-S and E-W at a point (Therefore SF1/SF2 = 1).

· Areas Equal-area: product of SFs, N-S, and E-W at a point = 1. (so SF1*SF2 = 1)

· Distances Equidistant in any direction from a point (zenithal) or in one direction from a line.

· Azimuthal Preserves all directions from a point (zenithal).

6. Choosing a Map Projection

a. Ease of construction and visualisation

Manually azimuthal polar, normal cylindrical, conic, conventional, are easier than oblique projections, but construction is less of a concern with computer software; however projections that would be harder to construct tend to be also harder to visualise. e.g. many oblique projections

b. Size & location of area

World: conventional

Polar: polar azimuthal

Equatorial: normal cylindrical

Mid latitudes: conic

c. Shape of area

E-W extent: conic or cylindrical (equatorial)

N-S extent: transverse cylindrical

Square or Circular: oblique or equatorial azimuthal

d. Map purpose

Navigation: Conformal

Mapping distributions: equal area

World distributions: conventional equal area

Specific locations: azimuthal oblique

a. Azimuthal Equidistant

b.

Orthographic

c. Transverse

Mercator

d. Robinson

e. Mercator

f. Sinusoidal Equal Area

g. Gnomonic

h.

Stereographic

Next

Introduction

A map can be simply defined as a graphic representation of the real world. This representation is always an abstraction of reality. Because of the infinite nature of our Universe it is impossible to capture all of the complexity found in the real world. For example, topographic maps abstract the three-dimensional real world at a reduced scale on a two-dimensional plane of paper.

Maps are used to display both cultural and physical features of the environment. Standard topographic maps show a variety of information including roads, land-use classification, elevation, rivers and other water bodies, political boundaries, and the identification of houses and other types of buildings. Some maps are created with very specific goals in mind. Figure 2a-1 displays a weather map showing the location of low and high pressure centers and fronts over most of North America. The intended purpose of this map is considerably more specialized than a topographic map.

Figure 2a-1: The following specialized weather map displays the surface location of pressure centers and fronts for Saturday, November 27, 1999 over a portion of North America.

The art of map construction is called cartography. People who work in this field of knowledge are called cartographers. The construction and use of maps has a long history. Some academics believe that the earliest maps date back to the fifth or sixth century BC. Even in these early maps, the main goal of this

tool was to communicate information. Early maps were quite subjective in their presentation of spatial information. Maps became more objective with the dawn of Western science. The application ofscientific method into cartography made maps more ordered and accurate. Today, the art of map making is quite a sophisticated science employing methods from cartography, engineering, computer science, mathematics, and psychology.

Cartographers classify maps into two broad categories: reference maps and thematic maps. Reference maps normally show natural and human-made objects from the geographical environment with an emphasis on location. Examples of general reference maps include maps found in atlases and topographic maps. Thematic maps are used to display the geographical distribution of one phenomenon or the spatial associations that occur between a number of phenomena.

Map Projection

The shape of the Earth's surface can be described as an ellipsoid. An ellipsoid is a three-dimensional shape that departs slightly from a purely spherical form. The Earth takes this form because rotationcauses the region near the equator to bulge outward to space. The angular motion caused by the Earth spinning on its axis also forces the polar regions on the globe to be somewhat flattened.

Representing the true shape of the Earth's surface on a map creates some problems, especially when this depiction is illustrated on a two-dimensional surface. To overcome these problems, cartographers have developed a number of standardized transformation processes for the creation of two-dimensional maps. All of these transformation processes create some type of distortion artifact. The nature of this distortion is related to how the transformation process modifies specific geographic properties of the map. Some of the geographic properties affected by projection distortion include: distance; area; straight line direction between points on the Earth; and the bearing of cardinal points from locations on our planet.

The illustrations below show some of the common map projections used today. The first two-dimensional projection shows the Earth's surface as viewed from space (Figure 2a-2). Thisorthographic projection distorts distance, shape, and the size of areas. Another serious limitation of this projection is that only a portion of the Earth's surface can be viewed at any one time.

The second illustration displays a Mercator projection of the Earth (Figure 2a-3). On a Mercator projection, the north-south scale increases from the equator at the same rate as the corresponding east-west scale. As a result of this feature, angles drawn on this type of map are correct. Distortion on a Mercator map increases at an increasing rate as one moves toward higher latitudes. Mercator maps are used in navigation because a line drawn between two points of the Earth has true direction. However, this line may not represent the shortest distance between these points.

Figure 2a-2: Earth as observed from a vantage point in space. This orthographic projection of the Earth's surface creates a two-dimensional representation of a three-dimensional surface. The orthographic projection distorts distance, shape, and the size of areas.

Figure 2a-3: Mercator map projection. The Mercator projection is one of the most common systems in use today. It was specifically designed for nautical navigation.

The Gall-Peters projection was developed to correct some of the distortion found in the Mercator system (Figure 2a-4). The Mercator projection causes area to be gradually distorted from the equator to the poles. This distortion makes middle and high latitude countries to be bigger than they are in reality. The Gall-Peters projection corrects this distortion making the area occupied by the world's nations more comparable.

Figure 2a-4: Gall-Peters projection. The Gall-Peters projection corrects the distortion of area common in Mercator maps. As a result, it removes the bias in Mercator maps that draws low latitude countries as being smaller than nations in middle and high latitudes. This projection has been officially adopted by a number of United Nations organizations.

The Miller Cylindrical projection is another common two-dimensional map used to represent the entire Earth in a rectangular area (Figure 2a-5). In this project, the Earth is mathematically projected onto a cylinder tangent at the equator. This projection in then unrolled to produce a flat two-dimensional representation of the Earth's surface. This projection reduces some of the scale exaggeration present in the Mercator map. However, the Miller Cylindrical projection describes shapes and areas with considerable distortion and directions are true only along the equator.

Figure 2a-5: The Miller Cylindrical projection.

Figure 2a-6 displays the Robinson projection. This projection was developed to show the entire Earth with less distortion of area. However, this feature requires a tradeoff in terms of inaccurate map direction and distance.

Figure 2a-6: Robinson's projection. This projection is common in maps that require somewhat accurate representation of area. This map projection was originally developed for Rand McNally and Company in 1961.

The Mollweide projection improves on the Robinson projection and has less area distortion (Figure 2a-7). The final projection presented presents areas on a map that are proportional to the same areas on the actual surface of the Earth (Figure 2a-8). However, this Sinusoidal Equal-Area projectionsuffers from distance, shape, and direction distortions.

Figure 2a-7: Mollweide projection. On this projection the only parallels (line of latitude) drawn of true length are 40° 40' North and South. From the equator to 40° 40' North and South the east-west scale is illustrated too small. From the poles to 40° 40' North and South the east-west scale is illustrated too large.

Figure 2a-8: Sinusoidal Equal-Area projection.

Map Scale

Maps are rarely drawn at the same scale as the real world. Most maps are made at a scale that is much smaller than the area of the actual surface being depicted. The amount of reduction that has taken place is normally identified somewhere on the map. This measurement is commonly referred to as themap scale. Conceptually, we can think of map scale as the ratio between the distance between any two points on the map compared to the actual ground distance represented. This concept can also be expressed mathematically as:

On most maps, the map scale is represented by a simple fraction or ratio. This type of description of a map's scale is called a representative fraction. For example, a map where one unit (centimeter, meter, inch, kilometer, etc.) on the illustration represents 1,000,000 of these same units on the actual surface of the Earth would have a representative fraction of 1/1,000,000 (fraction) or 1:1,000,000 (ratio). Of these mathematical representations of scale, the ratio form is most commonly found on maps.

Scale can also be described on a map by a verbal statement. For example, 1:1,000,000 could be verbally described as "1 centimeter on the map equals 10 kilometers on the Earth's surface" or "1 inch represents approximately 16 miles".

Most maps also use graphic scale to describe the distance relationships between the map and the real world. In a graphic scale, an illustration is used to depict distances on the map in common units of measurement (Figure 2a-9). Graphic scales are quite useful because they can be used to measure distances on a map quickly.

Figure 2a-9: The following graphic scale was drawn for

map with a scale of 1:250,000. In the illustration distances in miles and kilometers are graphically shown.

Maps are often described, in a relative sense, as being either small scale or large scale. Figure 2a-10helps to explain this concept. In Figure 2a-10, we have maps representing an area of the world at scales of 1:100,000, 1:50,000, and 1:25,000. Of this group, the map drawn at 1:100,000 has the smallest scale relative to the other two maps. The map with the largest scale is map C which is drawn at a scale of 1:25,000.

Figure 2a-10: The following three illustrations describe the relationship between map scale and the size of the ground area shown at three different map scales. The map on the far left has the smallest scale, while the map on the far right has the largest scale. Note what happens to the amount of area represented on the maps when the scale is changed. A doubling of the scale (1:100,000 to 1:50,000 and 1:50,000 to 1:25,000) causes the area shown on the map to be reduced to 25% or one-quarter.

2. Cartography as Communication One of the most useful approaches to the study of cartography is to view maps as a form of visual communication--a special-purpose language for describing spatial relationships. Although it is perhaps unwise to draw a direct analogy between cartography and language, concepts such as "grammar" and "syntax" help to explain, at least metaphorically, the sorts of decisions cartographers make as they compose maps. Cartographers seek to make use of visual resources such as color, shape and pattern to communicate information about spatial relationships. The analogy with language also helps explain why training in principles of effective cartography is so important--it allows us to communicate more effectively. Without a knowledge of some of these basic principles, the beginning cartographer is likely to be misunderstood or cause confusion.

2.1 Cartography is closely related to graphical communication

Cartography is related to, but different from other forms of visual communication. Cartographers must pay special attention to coordinate systems, map projections, and issues of scale and direction that are in most cases of relatively little concern to other graphic designers or artists. But, because cartography is a type of graphical communication, some insights to the demands of cartography can be gleaned from the literature of graphical communication and statistical graphics. Often cartographers are faced with some of the same challenges faced by graphical designers and can learn much from their insights. As you begin to study cartographic design, you may find it useful to consult some of the standard works on graphical communication. You will find the following books particularly interesting, and maps are often the focus of discussion.