Quaternion - Revision history

Cholmondely: Added picture!

2023-03-05T10:14:38Z

Added picture!

Cholmondely: Added note

2023-03-04T19:23:44Z

Added note

Cholmondely: Added definition (cribbed from Wikipedia).

2021-11-18T17:51:14Z

Added definition (cribbed from Wikipedia).

Cholmondely: /* See Also */ Added another (Griff's: http://www.aegidian.org/bb/viewtopic.php?p=31017#p31017)

2021-10-11T00:51:52Z

‎See Also: Added another (Griff's: http://www.aegidian.org/bb/viewtopic.php?p=31017#p31017)

Norby: link to Quaternion calculator

2014-06-20T10:00:51Z

link to Quaternion calculator

PG1: Undo revision 35431 by PG1 (talk)

2013-01-10T22:29:58Z

Undo revision 35431 by PG1 (talk)

PG1: /* Oolite Specifics */

2013-01-10T21:32:50Z

‎Oolite Specifics

PG1: /* Oolite Specifics */

2013-01-10T19:12:12Z

‎Oolite Specifics

PG1: Added Oolite Specifics section

2013-01-10T19:08:03Z

Added Oolite Specifics section

Eric Walch: Converted examples from XML to open-step. Added JS rotation example.

2011-10-01T10:03:00Z

Converted examples from XML to open-step. Added JS rotation example.

← Older revision		Revision as of 10:14, 5 March 2023
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		+	[[File:Quaternion Eagle.jpg\|thumb\|right\|400px\|A "Quaternion Eagle" from the Holy Roman Empire (each quaternion being represented by four coats of arms on the imperial eagle's remiges) Hans Burgkmair, c. 1510. Twelve quaternions are shown, as follows (eight dukes being divided into two quaternions called "pillars" and "vicars", respectively): Seill ("pillars"), Vicari ("vicars"), Marggrauen (margraves), Lantgrauen (landgraves), Burggrauen (burggraves), Grauen (counts), Semper freie (nobles), Ritter (knights), Stett (cities), Dörfer (villages), Bauern (peasants), Birg (castles).]]
	== Definition ==		== Definition ==

@@ Line 10: / Line 10: @@
   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 ==

← Older revision		Revision as of 17:51, 18 November 2021
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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 153: / Line 153: @@
 * [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]
 --------------------------------------------------------------------------------
 [[Category:Oolite scripting]]

@@ Line 149: / Line 149: @@
 * [[Oolite JavaScript Reference: Quaternion]]
 * [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]

@@ Line 142: / Line 142: @@
 === Oolite Specifics ===
-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 clockwise when the axis of rotation is pointing towards you (anti-clockwise if pointing away from you).
+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).
 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.

@@ Line 144: / Line 144: @@
 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).
-Also all Oolite ships and other entities have to be designed with the forward points 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.
+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.
 == See Also ==

← Older revision		Revision as of 19:08, 10 January 2013
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	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.		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 [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).
		+
		+	Also all Oolite ships and other entities have to be designed with the forward points 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.

	== See Also ==		== See Also ==

@@ Line 39: / Line 39: @@
 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>
@@ Line 56: / Line 56: @@
 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>
@@ Line 72: / Line 72: @@
 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 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 °.
 === Quaternion calculus ===
@@ Line 134: / Line 139: @@
 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.
 == See Also ==