Coordinates are sets of numbers that describe
position-- position along a line, on a surface or in space. Latitude
and longitude, or declination and right ascension, each is a system of
coordinates on the surface of a sphere--on the globe of the Earth
or the globe of the heavens.
## Coordinates on a Flat Plane
A more widely used system are The system is based on two straight lines ("axes"),
perpendicular to each other, each of them marked with
the distances from the point where they meet ("origin")--distances
to the right of the origin and above it, the origin being taken as
positive and on the other sides as negative (see
The distance on one axis is named "x" and on the other
axis "y". Given then a point P, one draws from it lines
Graphs use this system, as do some maps. This works well on a flat sheet of paper, but the real world is 3-dimensional and sometimes it is necessary to label points in 3-dimensional space. The cartesian (x,y) labeling can be extended to 3 dimensions by adding a third coordinate z. If (x,y) is a point on the sheet, then the point (x,y,z) in space is reached by moving to (x,y) and then rising a distance z above the paper (points below it have negative z). Very simple and clear, once a decision is made What follows uses the trigonometric functions sine and cosine; if these are not familiar to you, either skip to the next heading, or go learn about them. ## Polar CoordinatesCartesian c oordinates (x,y) are not the only way
of labeling a point P on a flat plane by a pair of numbers.
Other ways exist, and they can be more useful in special
situations.
One system (" The two representations are closely related. From the definitions of the sine and cosine x = r cos f That allows (x,y) to be derived from polar coordinates.
To go in the opposite direction and derive (r,f) from (x,y), note that from the
above equations (or from the r Once r is known, the rest is easy cos f = x These relations fail only at the origin, where x = y = r = 0. At that point, f is undefined and one can choose for it whatever one pleases. In r All the points with the same value of ## Azimuth and Elevation
The surveyor's telescope is designed to measure two such angles.
The angle f is measured in a horizontal plane,
is known as
The angle l is called Again, one needs to decide from what direction is the
azimuth measured--that is, where is azimuth zero? The
rotation of the heavens (and the fact most humanity
lives north of the equator) suggests (for surveyor-type
measurements) the northward direction, and this is indeed
the usual zero point. The azimuth angle (viewed from the north)
is measured Mathematicians however prefer their own notation and replace
"latitude" (or elevation) l with - [
And in case you have to know: In referring (r,
q, f)
to cartesian (x,y,z) with the same origin, q
is measured from the z-axis and f
is measured in the (x,y) plane, counterclockwise from the x-axis.] |

**Next Stop: #6 The Calendar**

Author and curator: David P. Stern

Last updated 3 April 1999