Log polar mapping - Revision history

Apm: Category:Surprising facts

2026-03-29T20:38:45Z

← Older revision		Revision as of 22:38, 29 March 2026
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	* https://en.wikipedia.org/wiki/Mercator_projection <br>{{wikitodo\|show that in the limit it gets equivalent to log polar - does it?}}		* https://en.wikipedia.org/wiki/Mercator_projection <br>{{wikitodo\|show that in the limit it gets equivalent to log polar - does it?}}

			[[Category:Surprising facts]]

Apm: /* Related / Intuitively understanding the size of an atom

2025-11-20T16:47:36Z

← Older revision		Revision as of 18:47, 20 November 2025
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	* [[Visualization methods for gemstone metamaterial factories]]		* [[Visualization methods for gemstone metamaterial factories]]
	* [[Challenges in the visualization of gem-gum factories]]		* [[Challenges in the visualization of gem-gum factories]]
			----
			* [[Intuitively understanding the size of an atom]]

	== External links ==		== External links ==

Apm: /* Inverse transformation */ added 2D backtransform

2024-02-07T12:47:47Z

Inverse transformation: added 2D backtransform

← Older revision		Revision as of 14:47, 7 February 2024
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	Using just the positive half-space though is plenty for illustrating convergent assembly where there is a bottom.		Using just the positive half-space though is plenty for illustrating convergent assembly where there is a bottom.

	=== Inverse ~~transformation~~ ===		=== Inverse transformations ===

			For the 2D case this is rather trivial:
			* x = e^z' * cos(x')
			* z = e^z' * sin(x')

			Foe the 3D case not so much. <br>
	{{todo\|inverse of log polar mapping generalized to 3D}}		{{todo\|inverse of log polar mapping generalized to 3D}}

Apm: /* Displaying many scales and their relation simultaneously */

2024-02-07T12:42:39Z

Displaying many scales and their relation simultaneously

← Older revision		Revision as of 14:42, 7 February 2024
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	Logistics and things like [[level crossing lag]] to the first initial output.		Logistics and things like [[level crossing lag]] to the first initial output.

	== Displaying many scales and their relation simultaneously ==		'''Displaying many scales and their relation simultaneously.''' <br>
			This can be done by '''generalizing log polar mapping to 3D''' like in the following:

			=== Log polar mapping generalized to 3D ===

	~~This can be done by '''generalizing log polar mapping to 3D''' like so:~~
	* x'(x,y,z) = pi/2 - atan2( z, sqrt(pow(x,2) + pow(y,2)) ) * cos(atan2(y,x)))		* x'(x,y,z) = pi/2 - atan2( z, sqrt(pow(x,2) + pow(y,2)) ) * cos(atan2(y,x)))
	* y'(x,y,z) = pi/2 - atan2( z, sqrt(pow(x,2) + pow(y,2)) ) * sin(atan2(y,x)))		* y'(x,y,z) = pi/2 - atan2( z, sqrt(pow(x,2) + pow(y,2)) ) * sin(atan2(y,x)))
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	This is a natural generalization of the much more commonly used 2D case.		This is a natural generalization of the much more commonly used 2D case.

	=== ~~Relation to more common 2D log~~ polar mapping ===		=== Log polar mapping – conventional 2D case ===

	Using z instead of y to better match the analogy.		Using z instead of y to better match the analogy.
	* x' = atan2(y,x)		* x' = atan2(y,x)
	* z' = log( sqrt(pow(x,2) + pow(z,2)) ) / log(base)		* z' = log( sqrt(pow(x,2) + pow(z,2)) ) / log(base)

			== Relation between conventional 2D and generalized 3D case ==

	'''In both 2D and 3D case:''' z' corresponds to the radius of a circle or sphere (respectively) just logarithmized. <br>		'''In both 2D and 3D case:''' z' corresponds to the radius of a circle or sphere (respectively) just logarithmized. <br>
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	(to the point that the 1D linear negative z-axis is blown up to a 2D cylinder wall). <br>		(to the point that the 1D linear negative z-axis is blown up to a 2D cylinder wall). <br>
	Using just the positive half-space though is plenty for illustrating convergent assembly where there is a bottom.		Using just the positive half-space though is plenty for illustrating convergent assembly where there is a bottom.

			=== Inverse transformation ===

			{{todo\|inverse of log polar mapping generalized to 3D}}

	== Related ==		== Related ==

Apm: /* Motivation */

2024-02-07T12:28:42Z

Motivation

← Older revision		Revision as of 14:28, 7 February 2024
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	'''Log polar mapping should be well suitable for a static poster showing all relevant internals of a [[nanofactory]] simultaneously.''' <br>		'''Log polar mapping should be well suitable for a static poster showing all relevant internals of a [[nanofactory]] simultaneously.''' <br>
	Being able to display/visualize all the convergent assembly ~~levels~~ (and routing ~~levels~~) of an entire nanofactory all at once (in 3D) without needing to resort to a animation/video format doing some zooming in/out. Displaying more at once also makes more apparent that smaller machinery operates at higher frequency (not higher speed though - careful!).		Being able to display/visualize all the convergent [[assembly level]]s (and [[routing level]]s) of an entire nanofactory all at once (in 3D) without needing to resort to a animation/video format doing some zooming in/out. Displaying more at once also makes more apparent that smaller machinery operates at higher frequency (not higher speed though - careful!).

	Animation can still be added showing the activity at all scales at once and making the continuity of matter transport more obvious.		Animation can still be added showing the activity at all scales at once and making the continuity of matter transport more obvious.

Apm at 12:28, 7 February 2024

2024-02-07T12:28:00Z

← Older revision		Revision as of 14:28, 7 February 2024
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	== Motivation ==		== Motivation ==

	'''Log polar mapping should be well suitable for a static poster showing all relevant internals of a nanofactory simultaneously.''' <br>		'''Log polar mapping should be well suitable for a static poster showing all relevant internals of a [[nanofactory]] simultaneously.''' <br>
	Being able to display/visualize all the convergent assembly levels (and routing levels) of an entire nanofactory all at once (in 3D) without needing to resort to a animation/video format doing some zooming in/out. Displaying more at once also makes more apparent that smaller machinery operates at higher frequency (not higher speed though - careful!).		Being able to display/visualize all the convergent assembly levels (and routing levels) of an entire nanofactory all at once (in 3D) without needing to resort to a animation/video format doing some zooming in/out. Displaying more at once also makes more apparent that smaller machinery operates at higher frequency (not higher speed though - careful!).

Apm: /* Motivation */

2024-02-07T12:27:43Z

Motivation

← Older revision		Revision as of 14:27, 7 February 2024
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	== Motivation ==		== Motivation ==

	'''Log polar mapping should be well suitable for a static poster showing all relevant internals of a nanofactory.''' <br>		'''Log polar mapping should be well suitable for a static poster showing all relevant internals of a nanofactory simultaneously.''' <br>
	Being able to display/visualize all the convergent assembly levels (and routing levels) of an entire nanofactory all at once (in 3D) without needing to resort to a animation/video format doing some zooming in/out. Displaying more at once also makes more apparent that smaller machinery operates at higher frequency (not higher speed though - careful!).		Being able to display/visualize all the convergent assembly levels (and routing levels) of an entire nanofactory all at once (in 3D) without needing to resort to a animation/video format doing some zooming in/out. Displaying more at once also makes more apparent that smaller machinery operates at higher frequency (not higher speed though - careful!).

Apm: /* External links */

2024-02-07T12:26:19Z

External links

← Older revision		Revision as of 14:26, 7 February 2024
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	* [https://www.researchgate.net/publication/255591825_An_Introduction_to_the_Log-Polar_Mapping 1996 – An Introduction to the Log-Polar Mapping – Helder J. AraujoJorge Miranda DiasJorge Miranda Dias – ResearchGate]		* [https://www.researchgate.net/publication/255591825_An_Introduction_to_the_Log-Polar_Mapping 1996 – An Introduction to the Log-Polar Mapping – Helder J. AraujoJorge Miranda DiasJorge Miranda Dias – ResearchGate]

	Not log polar mapping but ~~mercator~~ projection to the extreme. <br>		=== Relation to Mercator projection taken to the extreme ends ===

			Not log polar mapping but Mercator projection to the extreme. <br>
	This should be locally similar to log polar mapping in that math in the limit becomes identical. <br>		This should be locally similar to log polar mapping in that math in the limit becomes identical. <br>
	https://mrgris.com/projects/merc-extreme/		https://mrgris.com/projects/merc-extreme/

	* https://en.wikipedia.org/wiki/Mercator_projection <br>{{wikitodo\|show that in the limit it gets equivalent to log polar - does it?}}		* https://en.wikipedia.org/wiki/Mercator_projection <br>{{wikitodo\|show that in the limit it gets equivalent to log polar - does it?}}

Apm: /* External links */

2024-02-07T12:25:20Z

External links

← Older revision		Revision as of 14:25, 7 February 2024
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	https://mrgris.com/projects/merc-extreme/		https://mrgris.com/projects/merc-extreme/

	* https://en.wikipedia.org/wiki/Mercator_projection {{wikitodo\|show that in the limit it gets equivalent to log polar - does it?}}		* https://en.wikipedia.org/wiki/Mercator_projection <br>{{wikitodo\|show that in the limit it gets equivalent to log polar - does it?}}

Apm: /* External links */

2024-02-07T12:25:11Z

External links

← Older revision		Revision as of 14:25, 7 February 2024
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	This should be locally similar to log polar mapping in that math in the limit becomes identical. <br>		This should be locally similar to log polar mapping in that math in the limit becomes identical. <br>
	https://mrgris.com/projects/merc-extreme/		https://mrgris.com/projects/merc-extreme/

			* https://en.wikipedia.org/wiki/Mercator_projection {{wikitodo\|show that in the limit it gets equivalent to log polar - does it?}}

Log polar mapping - Revision history

Apm: Category:Surprising facts

Apm: /* Related */ * Intuitively understanding the size of an atom

Apm: /* Inverse transformation */ added 2D backtransform

Apm: /* Displaying many scales and their relation simultaneously */

Apm: /* Motivation */

Apm at 12:28, 7 February 2024

Apm: /* Motivation */

Apm: /* External links */

Apm: /* External links */

Apm: /* External links */

Apm: /* Related / Intuitively understanding the size of an atom