Difference between revisions of "Quaternion"

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m (Quaternions moved to Quaternion: Singular is more general.)
(Added definition (cribbed from Wikipedia).)
 
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  +
== Definition ==
  +
  +
In mathematics, the '''quaternion''' extends the complex numbers. Quaternions were first described by Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quaternion as the quotient of two ''directed lines'' in a three-dimensional space, or, equivalently, as the quotient of two vectors. Multiplication of quaternions is noncommutative ('''i''' x '''j''' <font color="Red"> ≠ </font> '''j''' x '''i''').
  +
  +
Quaternions are generally represented in the form
  +
''a'' + ''b'' '''i''' + ''c'' '''j''' + ''d'' '''k'''
  +
  +
where ''a'', ''b'', ''c'', and ''d'' are ''real numbers''; whilst '''i''', '''j''', and '''k''' are the ''basic quaternions''.
  +
  +
Quaternions are used in pure mathematics, but also have practical uses in applied mathematics, particularly for calculations involving three-dimensional rotations, such as in three-dimensional computer graphics, computer vision, and crystallographic texture analysis. They can be used alongside other methods of rotation, such as Euler angles and rotation matrices, or as an alternative to them, depending on the application.
  +
:::::::::::::::from [https://en.wikipedia.org/wiki/Quaternion Wikipedia]
  +
 
== Overview ==
  
== Overview ==
   
Line 4: Line 16:
 
A quaternion is a set of four values (W X Y Z) that are used in Oolite to 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.
 
A quaternion is a set of four values (W X Y Z) that are used in Oolite to 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:
+
For a given axis (x y z) and angle (α), the quaternion representing a rotation of a degrees around the axis from the origin (0,0,0) to (x,y,z) is:
   
   
−
W = cosine( 0.5 * a )
+
W = cos (0.5 × α)
   
−
X = x * sine( 0.5 * a )
+
X = x × sin (0.5 × α)
   
−
Y = y * sine( 0.5 * a )
+
Y = y × sin (0.5 × α)
   
−
Z = z * sine( 0.5 * a )
+
Z = z × sin (0.5 × α)
   
   
Line 19: Line 31:
   
   
−
W = cosine ( 45 degrees ) = 0.707..
+
W = cos 45 ° = 0.707…
   
−
X = 0 * sine( 45 degrees ) = 0
+
X = 0 × sin 45 ° = 0
   
−
Y = 0 * sine( 45 degrees ) = 0
+
Y = 0 × sin 45 ° = 0
   
−
Z = 1 * sine( 45 degrees ) = 0.707..
+
Z = 1 × sin 45 ° = 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).
 +
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).
   
   
 
== Examples ==
  
== Examples ==
   
−
=== 90º turns about the z-axis ===
 +
=== 90 ° turns about the z-axis ===
   
 
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 relevant sub-entity entry for the [[Coriolis station (Oolite)|Coriolis station]]:
  
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 relevant sub-entity entry for the [[Coriolis station (Oolite)|Coriolis station]]:
Line 39: Line 51:
   
 
Code:
  
Code:
−
<key>subentities</key>
+
subentities =
−
<array>
+
(
−
<string>arc-detail 0 0 0 1 0 0 0</string>
+
"arc-detail 0 0 0 1 0 0 0",
−
<string>arc-detail 0 0 0 1 0 0 1</string>
+
"arc-detail 0 0 0 1 0 0 1",
−
<string>arc-detail 0 0 0 0 0 0 1</string>
+
"arc-detail 0 0 0 0 0 0 1"
−
<string>arc-detail 0 0 0 1 0 0 -1</string>
+
"arc-detail 0 0 0 1 0 0 -1"
−
<array>
 +
)
   
   
 
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 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).
   
−
=== 120º turns about the z-axis ===
 +
=== 120 ° turns about the z-axis ===
   
−
Similarly, the shipdata of the [[Weeviloid 2]] illustrates how to place sub-entities at 3 equilateral points:
 +
Similarly, the shipdata of the [[Weeviloid_Hunter|Weeviloid 2]] illustrates how to place sub-entities at 3 equilateral points:
   
   
 
Code:
  
Code:
−
<key>subentities</key>
 +
subentities =
−
<array>
 +
(
−
<string>weeviloid2-spine 0 0 0 1 0 0 0</string>
 +
"weeviloid2-spine 0 0 0 1 0 0 0",
−
<string>weeviloid2-spine 0 0 0 0.5 0 0 0.8660254</string>
 +
"weeviloid2-spine 0 0 0 0.5 0 0 0.8660254",
−
<string>weeviloid2-spine 0 0 0 0.5 0 0 -0.8660254</string>
 +
"weeviloid2-spine 0 0 0 0.5 0 0 -0.8660254"
−
<array>
 +
)
   
   
 
Given that the first entry (0 0 0 1 0 0 0) will place the sub-entity at the exact place that the model is situated (0 degrees), the next two lines reproduce it at 120 degrees and 240 degrees.
  
Given that the first entry (0 0 0 1 0 0 0) will place the sub-entity at the exact place that the model is situated (0 degrees), the next two lines reproduce it at 120 degrees and 240 degrees.
   
−  
  +
=== 22.5 ° turns about the z-axis ===
−
=== 22.5º turns about the z-axis ===
  
   
 
To display 16 sub-entities that join to make a ring, as done in the [[Ringpod]] and [[Torus station|Torus]] shipdata, these are the relevant lines:
 
To display 16 sub-entities that join to make a ring, as done in the [[Ringpod]] and [[Torus station|Torus]] shipdata, these are the relevant lines:
Line 73: Line 84:
   
 
Code:
  
Code:
−
<key>subentities</key>
 +
subentities =
−
<array>
 +
(
−
<string>torus_pod 0 0 0 1 0 0 0</string>
 +
"torus_pod 0 0 0 1 0 0 0", // 0 degree
−
<string>torus_pod 0 0 0 0.9808 0.0 0.0 0.1951</string><!-- 22.5 degrees -->
 +
"torus_pod 0 0 0 0.9808 0.0 0.0 0.1951", // 22.5 degrees
−
<string>torus_pod 0 0 0 0.9239 0.0 0.0 0.3827</string><!-- 45 degrees -->
 +
"torus_pod 0 0 0 0.9239 0.0 0.0 0.3827", // 45 degrees
−
<string>torus_pod 0 0 0 0.8315 0.0 0.0 0.5556</string><!-- 67.5 degrees -->
 +
"torus_pod 0 0 0 0.8315 0.0 0.0 0.5556", / /67.5 degrees
−
<string>torus_pod 0 0 0 0.7071 0.0 0.0 0.7071</string><!-- 90 degrees -->
 +
"torus_pod 0 0 0 0.7071 0.0 0.0 0.7071", // 90 degrees
−
<string>torus_pod 0 0 0 0.5556 0.0 0.0 0.83110</string><!-- 112.5 degrees -->
 +
"torus_pod 0 0 0 0.5556 0.0 0.0 0.83110", //112.5 degrees
−
<string>torus_pod 0 0 0 0.3827 0.0 0.0 0.9239</string><!-- 135 degrees -->
 +
"torus_pod 0 0 0 0.3827 0.0 0.0 0.9239", // 135 degrees
−
<string>torus_pod 0 0 0 0.1951 0.0 0.0 0.9808</string><!-- 157.5 degrees -->
 +
"torus_pod 0 0 0 0.1951 0.0 0.0 0.9808", // 157.5 degrees
−
<string>torus_pod 0 0 0 0 0 0 1</string><!-- 180 degrees -->
 +
"torus_pod 0 0 0 0 0 0 1", // 180 degrees
−
<string>torus_pod 0 0 0 -0.1951 0.0 0.0 0.9808</string><!-- 202.5 degrees -->
 +
"torus_pod 0 0 0 -0.1951 0.0 0.0 0.9808", // 202.5 degrees
−
<string>torus_pod 0 0 0 -0.3827 0.0 0.0 0.9239</string><!-- 225 degrees -->
 +
"torus_pod 0 0 0 -0.3827 0.0 0.0 0.9239", // 225 degrees
−
<string>torus_pod 0 0 0 -0.5556 0.0 0.0 0.83110</string><!-- 247.5 degrees -->
 +
"torus_pod 0 0 0 -0.5556 0.0 0.0 0.83110", // 247.5 degrees
−
<string>torus_pod 0 0 0 -0.7071 0.0 0.0 0.7071</string><!-- 270 degrees -->
 +
"torus_pod 0 0 0 -0.7071 0.0 0.0 0.7071", // 270 degrees
−
<string>torus_pod 0 0 0 -0.8315 0.0 0.0 0.5556</string><!-- 292.5 degrees -->
 +
"torus_pod 0 0 0 -0.8315 0.0 0.0 0.5556", // 292.5 degrees
−
<string>torus_pod 0 0 0 -0.9239 0.0 0.0 0.3827</string><!-- 315 degrees -->
 +
"torus_pod 0 0 0 -0.9239 0.0 0.0 0.3827", // 315 degrees
−
<string>torus_pod 0 0 0 -0.9808 0.0 0.0 0.1951</string><!-- 337.5 degrees -->
 +
"torus_pod 0 0 0 -0.9808 0.0 0.0 0.1951" // 337.5 degrees
−
<array>
 +
)
   
   
−
Again the 0º sub-entity will appear with the 0 0 0 1 0 0 0, and the next 15 lines place it at 22.5º, 45º, 67.5º, 90º, 112.5º, 135º, 157.5º, 180º (with 0 0 0 0 0 0 1), 202.5º, 225º, 247.5º, 270º, 292.5º, 315º and 337.5º.
 +
Again the 0 ° sub-entity will appear with the 0 0 0 1 0 0 0, and the next 15 lines place it at 22.5 °, 45 °, 67.5 °, 90 °, 112.5 °, 135 °, 157.5 °, 180 ° (with 0 0 0 0 0 0 1), 202.5 °, 225 °, 247.5 °, 270 °, 292.5 °, 315 ° and 337.5 °.
   
  +
=== using Java Script to calculate quaternions ===
  +
  +
Oolite contains a whole set of vector and quaternion methods than can be used for easy manipulating quaternions. To generate a quaternion for a certain rotation around the z-axis we can rotate the identity quaternion around the z-axis with the command:
  +
new Quaternion(1,0,0,0).rotate(new Vector3D(0,0,1), angle)
  +
With this command we can calculate our own quaternion for any rotation around the z-axis, or any other axis when desired. Angle is not in degree but in Math.PI
   
 
=== Quaternion calculus ===
  
=== Quaternion calculus ===
Line 101: Line 117:
 
To perform one rotation, and then another, one needs to multiply two quaternions together.
 
To perform one rotation, and then another, one needs to multiply two quaternions together.
   
−
Multiplying quaternions isn't commutative: Qa x Qb does not equal Qb x Qa
 +
Multiplying quaternions isn't commutative: Qa × Qb does not equal Qb × Qa
  +
  +
To multiply Q₁ (w₁ , x₁ , y₁ , z₁ ) by Q₂ (w₂, x₂, y₂, z₂):
  +
  +
W = w₁ × w₂ - x₁ × x₂ - y₁ × y₂ - z₁ × z₂
  +
  +
X = w₁ × x₂ + x₁ × w₂ + y₁ × z₂ - z₁ × y₂
  +
  +
Y = w₁ × y₂ + y₁ × w₂ + z₁ × x₂ - x₁ × z₂
  +
  +
Z = w₁ × z₂ + z₁ × w₂ + x₁ × y₂ - y₁ × x₂
  +
  +
  +
To determine the quaternion for a rotation of α degrees/radians around an axis defined by a vector (x, y, z):
  +
  +
W = cos (0.5 × α)
  +
  +
X = x × sin (0.5 × α)
   
−
To multiply Q1 (w1, x1, y1, z1) by Q2 (w2, x2, y2, z2):-
  
  +
Y = y × sin (0.5 × α)
   
−
W = w1 x w2 - x1 x x2 - y1 x y2 - z1 x z2
  
  +
Z = z × sin (0.5 × α)
   
−
X = w1 x x2 + x1 x w2 + y1 x z2 - z1 x y2
  
  +
''or'':
   
−
Y = w1 x y2 + y1 x w2 + z1 x x2 - x1 x z2
  
  +
W = 1 / tan (0.5 × α)
   
−
Z = w1 x z2 + z1 x w2 + x1 x y2 - y1 x x2
  
  +
X = x
   
  +
Y = y
   
−
To determine the quaternion for a rotation of A degrees/radians around an axis defined by a vector (x, y, z)
  
  +
Z = z
   
−
W = cosine ( 0.5 x A)
  
  +
These are equivalent because of the trigonometric identity sin x/cos x = tan x, and the fact that quaternion normalization means that multiplying each component by the same value (in this case, 1 / sin (0.5 × α)) does not affect the resulting orientation.
   
−
X = x * sine (0.5 x A)
  
  +
=== Oolite Specifics ===
   
−
Y = y * sine (0.5 x A)
  
  +
Although Quaternions are coordinate system indifferent, Oolite, because it makes use of OpenGL, uses a [http://en.wikipedia.org/wiki/Left-handed_coordinate_system#Orientation_and_handedness left-handed coordinate system]. This means that all positive quaternion rotations about an axis, as discussed in the [[#Overview|Overview]] section above, are anti-clockwise (counter-clockwise) when the axis of rotation is pointing towards you (clockwise if pointing away from you).
   
−
Z = z * sine (0.5 x A)
  
  +
Also all Oolite ships and other entities have to be designed with forwards pointing along the z-axis, up along the y-axis and right along the x-axis. These entities have an orientation in space is given by a quaternion (the [[Oolite_JavaScript_Reference:_Entity#orientation|.orientation]] JavaScript property) that describes the transformation from this initial design orientation to the required final orientation.
   
−
== The gory mathematical details ==
  
  +
== See Also ==
   
  +
* [[Oolite JavaScript Reference: Quaternion]]
  +
* [https://app.box.com/s/6usw7ozbvobze3hkouzc Quaternion calculator] - a very useful spreadsheet
 
* [http://en.wikipedia.org/wiki/Quaternion Wikipedia page on quaternions]
  
* [http://en.wikipedia.org/wiki/Quaternion Wikipedia page on quaternions]
  +
* [http://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation Wikipedia page on quaternions and spatial rotation]
 
* [http://skal.planet-d.net/demo/matrixfaq.htm The Matrix and Quaternion FAQ]
  
* [http://skal.planet-d.net/demo/matrixfaq.htm The Matrix and Quaternion FAQ]
  +
* [http://www.euclideanspace.com/maths/geometry/rotations/conversions/angleToQuaternion/program/index.htm Maths - Axis-Angle to Quaternion - Program ]
   
 
--------------------------------------------------------------------------------
  
--------------------------------------------------------------------------------
−
[[Category:Oolite]]
 +
[[Category:Oolite scripting]]

Latest revision as of 18:51, 18 November 2021

Definition

In mathematics, the quaternion  extends the complex numbers. Quaternions were first described by Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quaternion as the quotient of two directed lines in a three-dimensional space, or, equivalently, as the quotient of two vectors. Multiplication of quaternions is noncommutative (i x j  ≠  j x i).

Quaternions are generally represented in the form
  a + b i + c j + d k

  where a, b, c, and d  are real numbers; whilst i, j, and k are the basic quaternions.

Quaternions are used in pure mathematics, but also have practical uses in applied mathematics, particularly for calculations involving three-dimensional rotations, such as in three-dimensional computer graphics, computer vision, and crystallographic texture analysis. They can be used alongside other methods of rotation, such as Euler angles and rotation matrices, or as an alternative to them, depending on the application.
from Wikipedia

Overview

A quaternion is a set of four values (W X Y Z) that are used in Oolite to 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 (α), the quaternion representing a rotation of a degrees around the axis from the origin (0,0,0) to (x,y,z) is:


W = cos (0.5 × α)

X = x × sin (0.5 × α)

Y = y × sin (0.5 × α)

Z = z × sin (0.5 × α)


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


W = cos 45 ° = 0.707…

X = 0 × sin 45 ° = 0

Y = 0 × sin 45 ° = 0

Z = 1 × sin 45 ° = 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).


Examples

90 ° turns about the z-axis

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 relevant sub-entity entry for the Coriolis station:


Code: subentities = 

( 
   "arc-detail 0 0 0 1 0 0 0", 
   "arc-detail 0 0 0 1 0 0 1", 
   "arc-detail 0 0 0 0 0 0 1" 
   "arc-detail 0 0 0 1 0 0 -1" 
)


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).

120 ° turns about the z-axis

Similarly, the shipdata of the Weeviloid 2 illustrates how to place sub-entities at 3 equilateral points:


Code: 

subentities =
(
   "weeviloid2-spine	0 0 0 1 0 0 0",
   "weeviloid2-spine	0 0 0 0.5 0 0 0.8660254",
   "weeviloid2-spine	0 0 0 0.5 0 0 -0.8660254"
)


Given that the first entry (0 0 0 1 0 0 0) will place the sub-entity at the exact place that the model is situated (0 degrees), the next two lines reproduce it at 120 degrees and 240 degrees.

22.5 ° turns about the z-axis

To display 16 sub-entities that join to make a ring, as done in the Ringpod and Torus shipdata, these are the relevant lines:


Code: 

subentities =
(
   "torus_pod  0 0 0 1 0 0 0",                 // 0 degree
   "torus_pod  0 0 0 0.9808 0.0 0.0 0.1951", // 22.5 degrees
   "torus_pod  0 0 0 0.9239 0.0 0.0 0.3827", // 45 degrees 
   "torus_pod  0 0 0 0.8315 0.0 0.0 0.5556", / /67.5 degrees
   "torus_pod  0 0 0 0.7071 0.0 0.0 0.7071", // 90 degrees
   "torus_pod  0 0 0 0.5556 0.0 0.0 0.83110", //112.5 degrees
   "torus_pod  0 0 0 0.3827 0.0 0.0 0.9239", // 135 degrees
   "torus_pod  0 0 0 0.1951 0.0 0.0 0.9808", // 157.5 degrees
   "torus_pod  0 0 0 0 0 0 1",                 // 180 degrees
   "torus_pod  0 0 0 -0.1951 0.0 0.0 0.9808", // 202.5 degrees
   "torus_pod  0 0 0 -0.3827 0.0 0.0 0.9239", // 225 degrees
   "torus_pod  0 0 0 -0.5556 0.0 0.0 0.83110", // 247.5 degrees
   "torus_pod  0 0 0 -0.7071 0.0 0.0 0.7071", // 270 degrees
   "torus_pod  0 0 0 -0.8315 0.0 0.0 0.5556", // 292.5 degrees
   "torus_pod  0 0 0 -0.9239 0.0 0.0 0.3827", // 315 degrees
   "torus_pod  0 0 0 -0.9808 0.0 0.0 0.1951" // 337.5 degrees
)


Again the 0 ° sub-entity will appear with the 0 0 0 1 0 0 0, and the next 15 lines place it at 22.5 °, 45 °, 67.5 °, 90 °, 112.5 °, 135 °, 157.5 °, 180 ° (with 0 0 0 0 0 0 1), 202.5 °, 225 °, 247.5 °, 270 °, 292.5 °, 315 ° and 337.5 °.

using Java Script to calculate quaternions

Oolite contains a whole set of vector and quaternion methods than can be used for easy manipulating quaternions. To generate a quaternion for a certain rotation around the z-axis we can rotate the identity quaternion around the z-axis with the command: 

new Quaternion(1,0,0,0).rotate(new Vector3D(0,0,1), angle)

With this command we can calculate our own quaternion for any rotation around the z-axis, or any other axis when desired. Angle is not in degree but in Math.PI

Quaternion calculus

To perform one rotation, and then another, one needs to multiply two quaternions together.

Multiplying quaternions isn't commutative: Qa × Qb does not equal Qb × Qa

To multiply Q₁ (w₁ , x₁ , y₁ , z₁ ) by Q₂ (w₂, x₂, y₂, z₂):

W = w₁ × w₂ - x₁ × x₂ - y₁ × y₂ - z₁ × z₂

X = w₁ × x₂ + x₁ × w₂ + y₁ × z₂ - z₁ × y₂

Y = w₁ × y₂ + y₁ × w₂ + z₁ × x₂ - x₁ × z₂

Z = w₁ × z₂ + z₁ × w₂ + x₁ × y₂ - y₁ × x₂


To determine the quaternion for a rotation of α degrees/radians around an axis defined by a vector (x, y, z):

W = cos (0.5 × α)

X = x × sin (0.5 × α)

Y = y × sin (0.5 × α)

Z = z × sin (0.5 × α)

or:

W = 1 / tan (0.5 × α)

X = x

Y = y

Z = z

These are equivalent because of the trigonometric identity sin x/cos x = tan x, and the fact that quaternion normalization means that multiplying each component by the same value (in this case, 1 / sin (0.5 × α)) does not affect the resulting orientation.

Oolite Specifics

Although Quaternions are coordinate system indifferent, Oolite, because it makes use of OpenGL, uses a left-handed coordinate system. This means that all positive quaternion rotations about an axis, as discussed in the Overview section above, are anti-clockwise (counter-clockwise) when the axis of rotation is pointing towards you (clockwise if pointing away from you).

Also all Oolite ships and other entities have to be designed with forwards pointing along the z-axis, up along the y-axis and right along the x-axis. These entities have an orientation in space is given by a quaternion (the .orientation JavaScript property) that describes the transformation from this initial design orientation to the required final orientation.

See Also

Retrieved from "http://backup.witchspacewiki.org/index.php?title=Quaternion&oldid=70799"