Cartesian coordinates

A Cartesian coordinate system consists of an origin, a pair of axes that intersect at a perpendicular angle at the origin, and a fixed unit of distance. The axes are labeled X and Y. Values are positive to the right of the origin on the X axis, and above the origin on the Y axis. The location of any point can be specified by an X,Y coordinate pair.

Point locations in the four quadrants of a Cartesian coordinate system.

Cartesian coordinates divide a plane into four quadrants as shown in the graphic. Commonly, the origin of the coordinate system is made to coincide with the intersection of the central meridian and central parallel of the map. This is a natural place for it, but it also conflicts with the natural desire of mapmakers to keep all their map coordinates positive (in effect, to keep the map within the first quadrant).

False easting and false northing
This conflict can be resolved with false easting and false northing, which you learned about in the previous module. Adding a number to the Y axis origin (false easting) and another number to the X axis origin (false northing) is equivalent to moving the origin of the system to the southwest. The easting and northing values are arbitrary—they just have to be large enough to ensure that every point on the map has positive coordinates.

Question: In this graphic, if a false easting of 10 and a false northing of 10 were added to the coordinate system, what would the coordinates of the point (-8,-2) become?

(2,8)

(8,2)

(10,10)

(18,12)

Calculating distance and direction
The main value of Cartesian coordinates is for making measurements on maps, as the French realized in World War I. The increasing range of artillery made it possible to train guns on targets that were out of sight—but on the battlefield, the distance and direction of fire had to be calculated quickly. Formulas for converting latitude and longitude were too cumbersome, but Cartesian coordinates offered a satisfactory solution.

Distance and direction formulas

Look at the two points labeled A and B. Since the distance between them (green line) can be regarded as the hypotenuse of a right triangle, you can use the Pythagorean formula to solve it.

To solve for distance, where d_ab is the distance between points A and B:

d_ab = {(x₁ - x₂)² + (y₁ - y₂)²}^1/2

To solve for direction, where a_ab is the direction between points A and B:

tan a_ab = {(y₁ - y₂) / (x₁ - x₂)}

The corresponding angle (angle BAC on the graphic) is given by the inverse tangent function on a calculator: 30.96 degrees. We'll round it to 31.

Since you want the complementary angle (the angle between grid north and line AB), you subtract from 90.

So if you are at point A and your target is at point B, you need to fire your gun a distance of 5.83 units at an angle of 59 degrees.