In the first two in this series of four articles, we will examine the four different types of coordinate systems. These are:

1. Two Dimensional Cartesian
2. Three Dimensional Cartesian
3. Two Dimensional Spherical
4. Three Dimensional Spherical

Coordinate derives from co meaning jointly, implying that there is more than one, and ordinatus which means to order or arrange. Thus a coordinate is one of two or more items used to order or arrange things. A coordinate system is a system which uses two or more coordinates to order or arrange things. Our interest in such things is to order, or arrange, things spatially such that the spatial relationship (and possibly also their relative size and orientation), is appropriately shown relative to other things in our data set.

Cartesian coordinate systems get their name from René Descartés. While Descartes did not actually invent these things, he was the first to analyze them and formulate rules and procedures concerning their use. Descartes defined a coordinate system (two dimensional for now, we deal with three dimensions below) as one being defined by two axes, i.e. two lines. This is what distinguishes a cartesian system from other types; that the system is defined by lines.

In so doing, Descartes also established the convention of x being the unknown quantity, and a being a known quantity. More generally, the letters at the end of the alphabet, x, y, and z for example, were to be used for unknown quantities, and the letters at the beginning of the alphabet, e.g. a, b, c, . . . , were to be used for known quantities. Since coordinate systems were often used to obtain solutions to problems, the axes got the labels X and Y.

In the "general" theory of things, these axes did not have to be orthogonal (i.e. perpendicular), merely non parallel. Given such axes, lines drawn parallel to the axes can be used to uniquely identify a particular location in the domain of the coordinate system. However, if the axes are not orthogonal (perpendicular), we loose much of the convenience of the coordinate system. For example, if the X and Y axes are not orthogonal, what is the formula for the distance between two points. Does the old formula still work? Probably not. Neither will most of the traditional formulas we frequently use such as plane geometry and coordinate manipulation. Thus, while not a requirement for a cartesian coordinate system, all of the cartesian systems Casual Cartographers use are orthogonal, i.e. the X axis is perpendicular to the Y axis. If this were not the case, our repertoire of formulas would not work and for the same reason, our CAD/GIS systems would not work either.

Another issue is that of units. In order to get our simple formulas to work, we assume that the units of each axis are the same. For example, if the units of our X axis were meters, and the Y axis feet, our basic distance formula would require us to add meters and feet. Like adding apples and oranges, this does not work very well. So again, for our convenience as Casual Cartographers, we make another very important requirement of any cartesian coordinate system we use to make maps: the units of both axes must be the same.

In high school, we learned a lot about calculating stuff using the X and Y coordinates. Essentially, we learned most of what we know about trigonometry and geometry in this environment. Thus, for the practical solution of many problems, we prefer to use this knowledge. Thus, for solving many of these types of problems we prefer to use the cartesian coordinates which work in a flat plane.

While the earth is round, we find it most desirable to represent it on a plane. Yes, we all know that this introduces some distortion. However, how do you compute the distance between two points if you’re given the latitude and longitude of the two points? What is the angle between two lines if the coordinates of the lines are given in latitude and longitude? What is the area of a rectangular plot if the corners of the plot are given in latitude and longitude?

Moving our cartesian coordinate system to three dimensions requires that we change the defining entities to three dimensional objects. So, the two dimensional lines which we used to define the axes of the two dimensional cartesian coordinate system must now become planes; and of course there are three of them. Like the two dimensional lines in the 2D case, and for many of the same reasons the planes in the 3D case are orthogonal. That is, each plane is orthogonal to the other two.

Coordinates in the three dimensional case are the perpendicular distance from each of the three planes. The origin, i.e. the point which has 0:0:0 as its coordinates, is that point where the three defining planes intersect. (Your homework assignment is to prove that three orthogonal planes have to intersect somewhere:>)

Almost always, such systems are set up so that they are right handed. To understand this, separate the fingers of your right hand so that the index finger, the middle finger, and your thumb are all approximately at right angles to each other. If you adjust the orientation of your hand so that the index finger points along the X axis in the positive direction and the middle finger points along the Y axis in the positive direction, your thumb will point in the direction of the positive Z axis. (You may now call you chiropractor to get straightened out again.) Again, this is a convention which is assumed if you expect the formulas to all work.

In cartography/geodesy, the three dimensional cartesian coordinate system is almost always used with the origin at the center of mass of the earth. Now, this point is somewhat subjective since nobody other than Jules Vern has ever been there. However, there is no doubt that such a point exists. By definition, it is also the point at which the axis of rotation intersects the plane of the equator. The resulting x, y, and z coordinates are often called geocentric coordinates. In this case, the positive Z axis is defined proceed from the origin to the north pole; and the positive X axis to proceed from the origin to the point where the prime meridian (Greenwich) intersects the equator. The Y axis completes the right handed system. Just in case you don’t want to go back the chiropractor, that means the Y axis proceeds from the origin to a point in the Indian Ocean about halfway between Sri Lanka and Indonesia (i.e. 0 degrees latitude and 90 degrees east longitude).

As indicated above, the linear nature of the cartesian system allows us to use rather simplified formulas for many calculations. For example, the distance between two points is , not terribly different from the two dimensional formula. Several texts in the literature provide the means of computing arc length on the ellipsoid by converting the points to geocentric coordinates, computing the straight line (i.e. chord) length, and then making adjustments for the curvature of the ellipse. As we have seen in previous issues of The Casual Cartographer, geocentric coordinates are also very useful in calculating approximate datum shifts.