Quaternion

Overview

A quaternion is a set of four values (W X Y Z) that specify a rotation in 3D space. To specify a particular rotation you need to think about the axis about which the rotation is made and the angle or amount by which the model is to be rotated.

For a given axis (x y z) and angle (a), the quaternion representing a rotation of a degrees around the axis from the origin (0,0,0) to (x,y,x) is:

W = cosine( 0.5 * a )

X = x * sine( 0.5 * a )

Y = y * sine( 0.5 * a )

Z = z * sine( 0.5 * a )

So a rotation of 90 degrees about the z axis (0 0 1) would be:

W = cosine ( 45 degrees ) = 0.707..

X = 0 * sine( 45 degrees ) = 0

Y = 0 * sine( 45 degrees ) = 0

Z = 1 * sine( 45 degrees ) = 0.707..

Now because quaternions are normalised (adjusted so that W * W + X * X + Y * Y + Z * Z = 1) when Oolite reads them in, you can multiply each part of a quaternion by the same value and still have it represent the same angle. So this rotation can also be represented as W = 1, X = 0, Y = 0, Z= 1 (let's use [1 0 0 1] as shorthand).

Example

Oolite uses quaternions to specify rotations in some parts of shipdata.plist, most notably in the subentities part of an entry, like here in the entry for coriolis-station:

Code:

<string>new_coriolis.dat</string> 
<key>name</key> 
<string>Coriolis Station</string> 
<key>port_radius</key> 
<real>500</real> 
<key>roles</key> 
<string>coriolis station</string> 
<key>subentities</key> 
<array> 
   <string>dock 0 0 500 1 0 0 0</string> 
   <string>arc-detail 0 0 0 1 0 0 0</string> 
   <string>arc-detail 0 0 0 1 0 0 1</string> 
   <string>arc-detail 0 0 0 0 0 0 1</string> 
   <string>arc-detail 0 0 0 1 0 0 -1</string> 

The last four numbers after the four 'arc-detail' lines are the W X Y and Z of quaternions representing rotations about the z-axis of 0 degrees, 90 degrees, 180 degrees, and 270 degrees (the first three numbers are the subentity's position relative to the station, in this case all are at the same place at the station's origin).

The gory mathematical details

• Wikipedia page on quaternions
• The Matrix and Quaternion FAQ

---