Geometric algebra - Revision history

A1210435: /* See also */

2013-11-04T05:00:40Z

A1210435 at 05:00, 28 October 2013

2013-10-28T05:00:54Z

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	'''Rationale'''		'''Rationale'''

	As the main approach inherent in this project involved the use of rotations in Geometric Algebra, we were required to learn the basics of GA with the help of Dr James ~~Chapel~~.		As the main approach inherent in this project involved the use of rotations in Geometric Algebra, we were required to learn the basics of GA with the help of Dr James Chappell.
	The following description/brief introduction to the fundamental elements of GA is a combination of James' two documents below and numerous other sources. James graciously provided us with his personal powerpoint slides that proved to be an invaluable resource in learning the new materiala as well as a brief worksheet to test your knowledge on GA. Both are provided below in the [https://www.eleceng.adelaide.edu.au/personal/dabbott/wiki/index.php/Geometric_algebra#Useful_Resources Useful Resources].		The following description/brief introduction to the fundamental elements of GA is a combination of James' two documents below and numerous other sources. James graciously provided us with his personal powerpoint slides that proved to be an invaluable resource in learning the new materiala as well as a brief worksheet to test your knowledge on GA. Both are provided below in the [https://www.eleceng.adelaide.edu.au/personal/dabbott/wiki/index.php/Geometric_algebra#Useful_Resources Useful Resources].

A1210435 at 01:28, 28 October 2013

2013-10-28T01:28:05Z

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	As the main approach inherent in this project involved the use of rotations in Geometric Algebra, we were required to learn the basics of GA with the help of Dr James Chapel.		As the main approach inherent in this project involved the use of rotations in Geometric Algebra, we were required to learn the basics of GA with the help of Dr James Chapel.
	The following description/brief introduction to the fundamental elements of GA is a combination of James' two documents below and numerous other sources. James graciously provided us with his personal powerpoint slides that proved to be an invaluable resource in learning the new materiala as well as a brief worksheet to test your knowledge on GA. Both are provided below in the [Useful Resources].		The following description/brief introduction to the fundamental elements of GA is a combination of James' two documents below and numerous other sources. James graciously provided us with his personal powerpoint slides that proved to be an invaluable resource in learning the new materiala as well as a brief worksheet to test your knowledge on GA. Both are provided below in the [https://www.eleceng.adelaide.edu.au/personal/dabbott/wiki/index.php/Geometric_algebra#Useful_Resources Useful Resources].

A1210435 at 01:27, 28 October 2013

2013-10-28T01:27:52Z

← Older revision		Revision as of 10:57, 28 October 2013
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	As the main approach inherent in this project involved the use of rotations in Geometric Algebra, we were required to learn the basics of GA with the help of Dr James Chapel.		As the main approach inherent in this project involved the use of rotations in Geometric Algebra, we were required to learn the basics of GA with the help of Dr James Chapel.
	The following description/brief introduction to the fundamental elements of GA is a combination of James' two documents below and numerous other sources. James graciously provided us with his personal powerpoint slides that proved to be an invaluable resource in learning the new materiala as well as a brief worksheet to test your knowledge on GA. Both are provided below.		The following description/brief introduction to the fundamental elements of GA is a combination of James' two documents below and numerous other sources. James graciously provided us with his personal powerpoint slides that proved to be an invaluable resource in learning the new materiala as well as a brief worksheet to test your knowledge on GA. Both are provided below in the [Useful Resources].

A1210435 at 01:26, 28 October 2013

2013-10-28T01:26:03Z

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	For a much more detailed explanation of the proof for 3D and the difficulties that arose in 4D, please read the [https://www.eleceng.adelaide.edu.au/personal/dabbott/wiki/index.php/Semester_B_Final_Report_2013_-_Secure_communications_without_key_exchange%3F_2013#2D_.26_3D_Rotations Final Report].		For a much more detailed explanation of the proof for 3D and the difficulties that arose in 4D, please read the [https://www.eleceng.adelaide.edu.au/personal/dabbott/wiki/index.php/Semester_B_Final_Report_2013_-_Secure_communications_without_key_exchange%3F_2013#2D_.26_3D_Rotations Final Report].

	==Resources==		==Useful Resources==
			* [[Media:GASlides.pdf\|Intro to Clifford's Geometric Algebra]]
			* [[Media:GAWorksheet.pdf\|Worksheet #1 in Geometric Algebra]]

			Another key document was [[Media:threedspace.pdf\|''Geometric Algebra: A natural representation of three-space'']],by James M. Chappell, Derek Abbott and Azhar Iqbal.

	==See also==		==See also==

A1210435: /* Our Work */

2013-10-28T01:05:07Z

Our Work

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	==Resources==		==Resources==


	~~==Our Work==~~

A1210435: /* How we used it? */

2013-10-28T01:04:39Z

How we used it?

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	[[Image:AliceBob3.png\|center\|710px]]		[[Image:AliceBob3.png\|center\|710px]]


			For a much more detailed explanation of the proof for 3D and the difficulties that arose in 4D, please read the [https://www.eleceng.adelaide.edu.au/personal/dabbott/wiki/index.php/Semester_B_Final_Report_2013_-_Secure_communications_without_key_exchange%3F_2013#2D_.26_3D_Rotations Final Report].

	==Resources==		==Resources==

A1210435: /* How we used it? */

2013-10-28T01:01:18Z

How we used it?

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	[[Image:AliceBob2.png\|center\|~~560px~~]]		[[Image:AliceBob2.png\|center\|590px]]

A1210435: /* How we used it? */

2013-10-28T01:00:52Z

How we used it?

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	[[Image:AliceBob2.png\|center\|560px]]		[[Image:AliceBob2.png\|center\|560px]]

	However as we can see through the application of Demoivres theorem, this breaks down to the form shown ~~on the slide~~, which implies that the cross product of the 2 rotation axes must be 0, and hence the 2 rotation axes of Alice and Bob must be parallel. However in order for Alice and Bob to use parallel Rotation Axes, a preferred direction would need to be communicated beforehand, however this would make the process vulnerable and goes against the premise of the double padlock protocol.
			However as we can see through the application of Demoivres theorem, this breaks down to the form shown below, which implies that the cross product of the 2 rotation axes must be 0, and hence the 2 rotation axes of Alice and Bob must be parallel. However in order for Alice and Bob to use parallel Rotation Axes, a preferred direction would need to be communicated beforehand, however this would make the process vulnerable and goes against the premise of the double padlock protocol.


			[[Image:AliceBob3.png\|center\|710px]]

	==Resources==		==Resources==

A1210435: /* How we used it? */

2013-10-28T00:57:18Z

How we used it?

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	[[Image:RotationPicture.png\|center\|700px]]		[[Image:RotationPicture.png\|center\|700px]]

	So for an initial message vector '''m''' in the form shown below, we apply a '''Rotation Operator''' or '''Rotor''' R to the message in the shown form, wit a corresponding '''Reversion Operator''' applied to opposite side of the message vector as shown.		So in 3D Rotations for an initial message vector '''m''' in the form shown below, we apply a '''Rotation Operator''' or '''Rotor''' R to the message in the shown form, wit a corresponding '''Reversion Operator''' applied to opposite side of the message vector as shown.

	[[Image:proof1a.png\|center\|450px]]		[[Image:proof1a.png\|center\|450px]]
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	[[Image:AliceBob2.png\|center\|~~525px~~]]		[[Image:AliceBob2.png\|center\|560px]]

			However as we can see through the application of Demoivres theorem, this breaks down to the form shown on the slide, which implies that the cross product of the 2 rotation axes must be 0, and hence the 2 rotation axes of Alice and Bob must be parallel. However in order for Alice and Bob to use parallel Rotation Axes, a preferred direction would need to be communicated beforehand, however this would make the process vulnerable and goes against the premise of the double padlock protocol.

	==Resources==		==Resources==

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	* [[Geometric_algebra\|Geometric Algebra]]		* [[Geometric_algebra\|Geometric Algebra]]
	* [[CLUViz Geometric Algebra Animation Software]]		* [[CLUViz Geometric Algebra Animation Software]]


	* [[Secure communications without key exchange]] (main page)		* [[Secure communications without key exchange]] (main page)