Projection parameters

When you choose a map projection, you mean to apply it either to the whole world or to some part of the world—a continent, a strip of land, or an important point like Redlands, California. In any case, you want the map to be just right for your area of interest. “Just right” usually means having as little distortion as possible, at least for the spatial property that you most want to preserve. And sometimes it also means making sure that all the coordinates that mark locations on the map have positive values.

You make the map just right by setting projection parameters. Not all projections have exactly the same set of parameters, though they tend to be similar. And there are parameters that can be changed in some projections but not in others. But, what it all comes down to is that parameters let you customize a projection for your area of interest.

On the round surface of the earth, locations are described in terms of latitude and longitude. Some projection parameters, called angular parameters, are set with these latitude-longitude values. Once the earth's back has been broken with a projection, locations are described in terms of constant units like meters or feet. Some projection parameters, called linear parameters, use these constant units (or they use ratios, such as 0.5 or 0.9996).

Top: Round data is described with meridians, parallels, and latitude-longitude values. Bottom: Flat data is described with x,y units. Projection parameters use both kinds of descriptions. The projection at bottom is Plate Carrée.

Concepts

Angular parameters

·         Central meridian

·         Latitude of origin

·         Standard parallel

·         Latitude of center

·         Central parallel

Every projection has a central meridian, which is the middle longitude of the projection. In most projections, it runs down the middle of the map and the map is symmetrical on either side of it. It may or may not be a line of true scale. (True scale means no distance distortion.)

In ArcGIS, you can change the central meridian of any projection. (Occasionally, it's the only angular parameter you can change.)

The central meridian is also called the longitude of origin or the longitude of center. Its intersection with the latitude of origin (see below) defines the starting point of the projected (x,y) map coordinates.

Every projection also has a latitude of origin. The intersection of this line with the central meridian is the starting point of the projected coordinates. In ArcGIS, you can put the latitude of origin wherever you want for most conic and transverse cylindrical projections. (In many world projections, on the other hand, it is defined to be the equator and can't be changed.) The latitude of origin may or may not be the middle latitude of the projection and may or may not be a line of true scale.

More about the latitude and longitude of origin

The important thing to remember about the latitude and longitude of origin is that they don't affect the distortion pattern of the map. All they do is define where the map's x,y units will originate.

When data is unprojected, it doesn't have x,y units. Locations are measured in latitude and longitude, as you know from the previous module. But when you set a projection and flatten everything out, you also start using a new way to measure location. This new way is in terms of constant distance units (like meters or feet) measured along a horizontal x-axis and a vertical y-axis. A location like x = 500,000, y = 100,000 would refer to a point 500,000 meters (or whatever units of measure you are using) along the x-axis and 100,000 meters along the y-axis. The place where the axes cross is the coordinate origin, or 0,0 point. Commonly, this is in the middle of the map but it doesn't have to be.

In the top graphic below, the intersection of the central meridian (longitude of origin) and the latitude of origin is marked with a cross. This point becomes the origin of the x,y coordinates.

Top: The point of intersection of the central meridian and latitude of origin becomes the origin of the x,y coordinates. Bottom: Red lines represent the x and y axes.

The bottom graphic shows the grid (normally invisible) on which the x,y coordinates are located. The heavy lines are the x- and y-axes, which divide the grid into four quadrants. Coordinates are positive in one direction and negative in the other for each axis.

In essence, a map projection is a method for taking locations on a sphere, as defined by the intersection of a meridian and a parallel, and assigning them to locations on a grid, as defined by the intersection of an x-axis and a y-axis.

Often, you don't care where the x,y coordinates originate, but sometimes, on published maps that have locational reference marks, you want all the reference marks to have positive values. You can help make this happen by setting the latitude of origin below the area of interest, ensuring that all y-coordinates on the map are positive. (The same result can be achieved with false northing, discussed in the next concept).

A standard parallel is a line of latitude that has true scale. Not all projections have standard parallels, but many common ones do. Conic projections often have two. In a few projections, like the Sinusoidal and the Polyconic, every line of latitude has true scale and is therefore a standard parallel.

In ArcGIS, you can change the standard parallel for some projections and not for others. Many world projections, for instance, have fixed standard parallels. (These do not show up as parameters when you set the projection, but you can find out what they are in the online help.)

A standard parallel may or may not coincide with the latitude of origin.

Top: The Cylindrical Equal Area projection has a single standard parallel. By default, it is the equator, but you can change it. Bottom: The Behrmann projection is the same projection, but with two standard parallels at 30° N and 30° S. These standard parallels define the projection and cannot be changed.

In some projections, you will also see parameters called the latitude of center and the central parallel. These two terms seem to have the same meaning. Like the latitude of origin, they define the starting point of the y-coordinates; unlike it, they are nearly always the middle parallel of the projection. These parameters are used mainly with projections that have single points (rather than lines) of zero distortion, such as the Gnomonic and Orthographic. The intersection of the latitude of center (or central parallel) with the central meridian defines both the origin of the x,y coordinates and the point of zero distortion for the projection.

A summary of angular parameters. "Yes" means yes; "no" only means not necessarily. For example, the standard parallel may be the y-coordinate origin and the middle latitude of the projection, but it doesn't have to be. (* In transverse cylindrical projections, the central meridian does have true scale by definition.)

Other angular parameters are used only with a few specific projections, like the Two Point Equidistant and the Hotine Oblique Mercator. For example, the Hotine Oblique Mercator has special parameters for defining an oblique line of true scale. You used these parameters in a previous exercise.

Linear parameters

·         False easting

·         False northing

·         Scale factor

Projected coordinates (that is, x,y coordinates) are positive for some map locations and negative for others, depending on where the x- and y-axes intersect. But on published maps that use x,y coordinates as reference marks, it is standard practice to have all coordinates positive.

This may happen with no effort on your part (if your area of interest is favorably located). Or you may be able to make it happen through your choice of central meridian and latitude of origin. Another way to make it happen, and a convenient one, is by the use of false easting and false northing values. These are nothing but two big numbers that are added to each x- and y-coordinate, respectively. The numbers are big enough to ensure that all coordinate values—or at least all those in your area of interest—come out positive.

Top: Projected coordinates are positive or negative, depending on their location. Bottom: A false easting value of 7,000,000 and a false northing value of 2,000,000 have been set. Every x-coordinate is now its original value plus 7,000,000. Every y-coordinate is its original value plus 2,000,000. The projection is Plate Carrée.

You might be asking yourself when you should show x,y values on a map instead of latitude-longitude values. The answer is related to scale. Small and medium-scale maps normally use latitude-longitude values, while larger scales use x,y values. It's also common to see both types of values on the same map.

Details of two maps. Lower left: An Albers Equal-Area Conic projection of the northeastern U.S. at 1:10,000,000 scale. Reference marks are lines of latitude and longitude drawn on the map at ten-degree intervals. Upper right: A Universal Transverse Mercator projection of the Corn Creek, Nevada, quadrangle at 1:24,000 scale. Reference marks are both projected coordinates (red hatch marks) and latitude-longitude (blue hatch marks). This map has a false easting of 200,000 meters and false northing of 8,000,000 meters.

Reference marks and map scale

Distance units (projected coordinate values) are typically used for reference at scales larger than 1:10,000, while latitude-longitude values are used at scales smaller than 1:1,000,000. At intermediate scales, both are frequently shown. (Maling, 1992)

Map projections in the Universal Transverse Mercator and State Plane Coordinate Systems (which you will learn about in Module 5, Geographic and Planar Coordinate Systems) have conventional false easting and false northing values, which are applied by default in ArcGIS.

A scale factor is the ratio of the true map scale to the stated map scale for a particular location. Remember that no map has true scale everywhere.

The scale text or scale bar that is printed on a map is correct only for lines of true scale. For instance, on a map with standard parallels, the stated map scale will be correct for measurements along those parallels, but not for measurements along other parallels, meridians, or oblique lines.

A map of Mexico in an Albers Equal Area Conic projection. The scale information holds true only for the standard parallels at 18 and 30 degrees. But don't panic; on this map, the scale is never wrong by more than one percent for a line drawn in any direction.

For large-scale maps and maps with good distance-preserving properties, you won't go too far wrong making measurements in any direction. But with some projections (such as the Mercator) at small scales, distance measurements can be significantly wrong if you are not on or near a line of true scale. Sometimes you will see maps with stacked scale bars that show the correct scale for different lines of latitude on the map.

A line of true scale is defined as having a scale factor of 1.0. Along this line, the actual map scale is equal to the stated scale (there is no distortion of distance). A scale factor of 2.0 means that distance measurements on the map are twice too long—if your scale bar tells you it's a hundred kilometers from A to B, it's really only fifty kilometers. A scale factor of 0.5 means that distance measurements are twice too short.

A Mercator projection with a stated map scale of 1:100,000,000. Along a line of true scale, such as the equator in this projection, the scale factor is 1.0. One map unit equals the number of ground units that the map says it does. At 60° north or south, the scale factor increases to 2.0 along the parallels. The blue double-headed arrow at the bottom of the map measures only half as much ground as the one at the top.

In ArcGIS, a scale factor can be applied to the line of true scale for transverse and oblique cylindrical projections and a few others. Changing the scale factor of this line creates two parallel lines of true scale on either side of it. (In other words, it gives the projection two secant lines instead of one tangent line.) The purpose is to balance distortion within the area of interest.

For Universal Transverse Mercator projections, the default scale factor is 0.9996. For other projections, it is 1.0.

The yellow shaded area is covered by the Universal Transverse Mercator projection for Zone11North. The central meridian has a scale factor of 0.9996. This setting creates two parallel lines of true scale on either side of the central meridian. The result is a better balancing of distortion within the zone.