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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?

John
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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
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/
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Associated Topics:
High School Coordinate Plane Geometry
High School Definitions
High School Geometry
High School Higher-Dimensional Geometry
High School Practical Geometry

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