Coordinate Systems, Longitude, Latitude

Date: 04/19/97 at 08:13:36
From: John Harris
Subject: Math terms

I know that it takes six numbers to locate and orient an object in 
space at a given time. Could you tell me the names of these six 
numbers?  The terms "height," "width," and "depth" don't seem 
appropriate, being more applicable to finding the area of a box.  I've 
heard the names "pitch," "roll," and "yaw," but don't know how these 
are defined. Could you help?  

Thanks in advance,
John

Date: 04/19/97 at 18:21:14
From: Doctor Toby
Subject: Re: Math terms

Dear John, 

There are many different ways to use 6 numbers for this, depending on 
what coordinate system you use.

First, I'll describe different ways to locate an object.

The most common method, invented by Rene Descartes in the 17th 
century, is now called "Cartesian coordinates" after him. Here you 
have 3 axes perpendicular to each other, the same axes on which you 
measure length, width, and height. These axes are often labeled by the 
variables x, y, and z, so most people just refer to these variables as 
the "x-coordinate", "y-coordinate", and so on. The fancy names which 
Descartes introduced, however, are "abscissa" (for x), "ordinate" 
(for y), and "altitude" (for z). If you want to locate objects in a 
rectangular room, put the origin at the southwest corner of the floor. 
Then the location of an object is given by the distance east of this 
corner (the abscissa), the distance north (the ordinate), and the 
height above the floor (the altitude).

If you just want to locate an object on a plane, where there are only 
two coordinates, you only use the abscissa and the ordinate. You can 
also use polar coordinates way to locate an object on a plane. The 
distance of a point from the origin is called "radius", which is 
usually labeled by the variable r or the Greek letter rho. If you draw 
a line from the origin to the object, the angle this line makes with 
the x-axis is called "bearing" or "heading", because it indicates a 
direction, and is labeled by the Greek letters theta or phi. The 
radius and the bearing are the 2 coordinates that locate the object.

There are two different, common ways to generalize polar coordinates 
to 3-dimensional space: "cylindrical" and "spherical" coordinates.
Cylindrical coordinates are easy; you just combine a polar plane with 
the z axis, so the coordinates are the radius, the bearing, and the 
altitude. The radius here is called "axial radius", because it's not 
the total distance from the origin to the object (which is the square 
root of r^2 + z^2), only the distance from the object to the z-axis.

Another coordinate you can use is called "latitude". To find this, 
draw a line from the origin to the object. The angle between this line 
and the ground is the latitude. Your coordinates could be the axial 
radius, the bearing, and the latitude. But it's more common, when 
using the latitude, to use the total radius, the distance of the 
object to the origin. The radius, the bearing, and the latitude are 
spherical coordinates.

Spherical coordinates are ideal for describing locations on Earth,
because Earth is (to a very good approximation) a sphere. In this 
context, the bearing is usually called "longitude" and the radius is 
called "altitude". But on Earth, the distance from the center of Earth 
(which is the origin) is not used as the altitude; rather, the 
distance from sea level is used as the altitude. An object below sea 
level has a negative altitude. An object in the northern hemisphere 
has a positive latitude, and an object in the southern hemisphere has 
a negative latitude. An object in the eastern hemisphere has a 
positive longitude, while an object in the western hemisphere has a 
negative longitude.

You can make the longitude in the western hemisphere positive by 
adding 360 degrees to it, treating it as an east latitude which 
happens to be more than 180 degrees east of the Greenwich meridian.
To make the latitude positive, use the colatitude, which is 90 degrees 
minus the latitude. The north pole has a latitude of 90 degrees and a 
colatitude of 0; the equator has a latitude of 0 and a colatitude of 
90 degrees; the south pole's latitude is -90 degrees, and its 
colatitude is 180 degrees.

The variables used to label spherical coordinates are not consistent.
If the axial radius is labeled by r, the radius is labeled by rho; if 
the axial radius is labeled by rho, the radius is labeled by r. If the 
bearing is labeled by theta, the colatitude is labeled by phi; if the 
bearing is labeled by phi, the colatitude is labeled by theta. 
Physicists usually use r for the radius and theta for the colatitude,
while mathematicians use r for the axial radius and theta for the 
bearing. I don't know what real people (engineers) use, since I'm not 
one of them, but they probably follow the physicists. Personally, I 
prefer the physicists' labeling, even though I usually side with 
mathematicians on notational matters.

To summarize, suppose you have a plane flying on Earth. If the area 
the plane might cover is limited, you can set up an origin at home 
base, and describe the plane's location with Cartesian coordinates
(the abcsissa, the ordinate, and the altitude) or with cylindrical 
coordinates (the axial radius, the bearing, and the altitude). If the 
plane might go anywhere on Earth, however, you'll want to put the 
origin at the centre of Earth and use spherical coordinates (the 
latitude, the longitude, and the altitude).

Now suppose you want to describe the orientation of the plane. You can 
do this in part by describing the direction the plane is headed. Set 
up a system of spherical coordinates with the plane at the origin. The 
direction the plane is headed gives a value of bearing and latitude.
This bearing, when used to describe the orientation, is called "yaw".
This latitude, when used to describe the orientation, is called 
"pitch".

If the plane is flying due east, its yaw is 0.
If the plane is flying due north, its yaw is 90 degrees.
If the plane is flying due south, its yaw is -90 degrees.
If the plane is flying due west, its yaw is 180 degrees (which is the 
   same as a yaw of -180 degrees).
If the plane is flying level, its pitch is 0.
If the plane is flying upwards, its pitch is positive.
If the plane is flying downwards, its pitch is negative.

Now consider a plane flying with a steady pitch and yaw, but spinning 
around its line of flight like a dart. Clearly, you need one more 
number to describe the plane's orientation, the number which changes 
when the plane spins. This extra number is called "roll".

A plane with a roll of 0 isn't leaning left or right.
A plane leaning to the right has a positive roll.
A plane leaning to the left has a negative roll.
A plane flying upside down has a roll of 180 degrees, or -180 degrees.

The terms "pitch", "yaw", and "roll" are usually used in real planes
to describe a *change* in the plane's orientation. But if you want to 
use them to describe orientation absolutely, you have to decide on an 
orientation for which these values are 0. (I chose when the plane is 
level, not leaning, and facing east.) Then rotate the plane by the 
right values of yaw, then pitch, and then roll. You must do this in 
the right order! For example, if you turn the plane first by pitch, to 
90 degress (so it's flying straight up), and then turn it through a 
yaw of 90 degrees, it will be headed due north and leaning on its left 
side. In other words, its pitch is 0, its yaw is 90 degrees, and its 
roll is -90 degrees. So, to calculate the absolute orientation of a 
plane, you need to do these changes in orienation in the right order.

Another way to describe orientation is called "Euler angles". Leonhard 
Euler invented these angles (named after him at a later date, of 
course) in the 18th century. Start with a plane in the standard 
orientation (level and facing due east). Turn the plane through a 
certain amount of yaw to get the Euler angle labeled phi. Then turn it 
through a certain amount of roll to get the Euler angle labeled theta. 
(Yes, this contradicts the labels used to describe location in 
spherical coordinates.) Finally, turn it through a certain amount of 
yaw again to get the Euler angle labeled by the Greek letter psi. Even 
though you never turned the plane through a pitch, you can get a 
pitch, in effect, by doing the right yaw, roll, and yaw. For example, 
a plane headed straight up and not leaning (with a yaw of 0, a pitch 
of 90 degrees, and a roll of 0) has a phi value of 90 degrees and 
theta and psi values of -90 degrees. Try this yourself with a model 
plane (such as a pencil) and see!

Euler angles are used a lot by physicists studying angular momentum.
Physicists doing quantum mechanics usually make the standard 
orientation facing north (along the y-axis) rather than facing east 
(along the x-axis), because this fits in better with some other 
quantum mechanical notations. Otherwise, they do the angles the way 
Euler originally defined them, which has the standard orientation 
facing east. Personally, I think physicists should just use yaw, 
pitch, and roll, like ordinary airplane pilots do. But nobody ever 
listens to me :-).

-Doctor Toby,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/