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A periodic shift in Polar Roses for Valentines Day
|| 2/14/2007 || 6:28 am || Comments Off on A periodic shift in Polar Roses for Valentines Day || ||

Original Process:

Shifted to the right (or left)
Zoom of Grant Park Quilt

Last month I thoroughly explored the applications of geometric tessellations using the polar coordinate filter. After doing a bit more research I discovered how similar these “amazing circles” appear to be very similar to the sinusoidal curves of a Polar Rose. They use the equation:

In the original process I used to construct last month’s Polar Roses each interation added one tile to the entire panorama. I believe this represents the a in the equation. And after each iteration, there would be a new pedal added.

Today’s discovery is the realization of how these panoramas are wrapped around a cylinder to produce the pedals when the polar coordinate system is used. I’ve been working in whole numbers only, yet if the tiles are wrapped in a cylindrically fashion using a half shift, there will be a change in the pedal’s layout.

So to create this shift, half of one tile is removed on each end (notice above), and when they connect after being wrapped around a cylinder they still form a perfect tessellation and thus new pedal configuration when the filter is applied.

The result of this polar shift is quite interesting- a 90 degree rotation for each iteration.

View the 90 degree difference:
Old:

Shifted 90 degrees:

I can now apply this periodic shift to each of the geometries to produce a completely new set of Polar Roses. This should be interesting.



Polar Coordinates of the Delmar Loop in University City, Missouri [icositetragon]
|| 1/23/2007 || 8:38 am || Comments Off on Polar Coordinates of the Delmar Loop in University City, Missouri [icositetragon] || ||


7 tiled map

8 polar coordinate maps using a zoom of Infinite Delmar Loop.

Read more about the process involved to make these here.

View other maps of the Delmar Loop in University City, Missouri.

+ MORE



Polar Coordinates of Harvard University [dodecagon]
|| 1/22/2007 || 9:31 am || Comments Off on Polar Coordinates of Harvard University [dodecagon] || ||


4 tiled map

8 polar coordinate maps using a zoom of Harvard Quilt – NW.

Read more about the process involved to make these here.

View other maps of Harvard University.

+ MORE



Polar Coordinates of the George Washington University [octagon]
|| 1/21/2007 || 9:24 am || Comments Off on Polar Coordinates of the George Washington University [octagon] || ||


5 tiled map

8 polar coordinate maps using a zoom of GWU Quilt #3.

Read more about the process involved to make these here.

View other maps of the George Washington University.

+ MORE



Polar Coordinates of the Super Dome in New Orleans, Louisiana [hexagon]
|| 1/20/2007 || 9:15 am || Comments Off on Polar Coordinates of the Super Dome in New Orleans, Louisiana [hexagon] || ||


6 tiled map

8 polar coordinate maps using a zoom of Superdome Quilt.

Read more about the process involved to make these here.

View other maps of New Orleans, Louisiana.

+ MORE



Polar Coordinates of Charleston, West Virginia [diamond]
|| 1/19/2007 || 10:03 am || Comments Off on Polar Coordinates of Charleston, West Virginia [diamond] || ||


3 tiled map

8 polar coordinate maps using a zoom of Charleston Quilt #2.

Read more about the process involved to make these here.

View other maps of Charleston, West Virginia.

+ MORE



Polar Coordinates of Austin, Texas [square]
|| 1/18/2007 || 4:27 pm || Comments Off on Polar Coordinates of Austin, Texas [square] || ||


8 tiled map

I recently read through the work of Dr. Andrew Hudson-Smith (located at the Center for Advanced Spatial Analysis in London). I was intrigued by his use of Polar Coordinates to create panoramas and synthetic planets. I had been shown how to make the panoramas before, but I hadn’t actually figured out how I could incorporate my current work with this design technique. After some experimenting I figured out some interesting new methods to display tessellated aerial photography using the Polar Coordinates filter. The results are quite amazing and show definite promise for further research…

For this experiment I used the “zoom” of Austin Quilt #2 (a square quilt projection) as my test tessellation to begin working with. I could have used any zoom from any previously made map, but since it was snowing in Austin yesterday, I thought it would be ironic to use the Polar Coordinates filter.

The process to create the panorama planets is relatively simple:
– Stitch together series of photographs
– Reduce the panorama (by default a rectangle) to a square
– Apply Polar Coordinates filter in Photoshop

What makes my experiments unique is that instead of stitching together a photograph, I am I am stitching together perfectly symmetrical square tiles (composed of tessellated aerial photography) to create a rectangle. Moreover, I am maintaining perfect ratios throughout: 1:1, 2:1, 3:1, 4:1, 5:1, 6:1, 7:1, and 8:1. For example, the source tile is 800×800 and for each polar coordinate created (below) I merely added one more tile to rectangle ([1600, 2400, 3200, 4000, 4800, 5600, 6400]x800).

For each ratio there are three images shown below:
– Rectangle with number of tiles
– Conversion of the Rectangle to Square
– Application of Polar Coordinates filter

Note: The first image of each ratio below appears to reduce in size after each tile has been been added. In order to maintain a constant width on my website, I’ve chosen to reduce each rectangle to 800 pixels in width (otherwise this entry would be 6400 pixels wide right now!). When originally manipulating the rectangles they were not reduced in size. Essentially, it was done to show the scale of the panorama at a fixed width of 800 pixels.

1 : 1

:Original:

:Converts To:

2 : 1

:Rectangle 2 Tiles Wide (reduced from: 1600×800):

:Rectangle reduced to a Square:

:Converts To:

3 : 1

:Rectangle 3 Tiles Wide (reduced from:2400×800):

:Rectangle reduced to a Square:

:Converts To:

4 : 1

:Rectangle 4 Tiles Wide (reduced from: 3200×800):

:Rectangle reduced to a Square:

:Converts To:

5 : 1

:Rectangle 5 Tiles Wide (reduced from: 4000×800):

:Rectangle reduced to a Square:

:Converts To:

6 : 1

:Rectangle 6 Tiles Wide (reduced from: 4800×800):

:Rectangle reduced to a Square:

:Converts To:

7 : 1

:Rectangle 7 Tiles Wide (reduced from: 5600×800):

:Rectangle reduced to a Square:

:Converts To:

8 : 1

:Rectangle 8 Tiles Wide (reduced from: 6400×800):

:Rectangle reduced to a Square:

:Converts To:

I am excited about what directions this technique might take my cartographic activities :) My next experiment will be to see how the application of the Polar Coordinates filter will effect each geometry of the quilt projection. This experiment was conducted using the square quilt projection, up next with be the diamond. The results should be interesting.

Related:





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Nikolas Schiller is a second-class American citizen living in America's last colony, Washington, DC. This blog is my on-line repository of what I have created or found on-line since May of 2004. If you have any questions or comments, please contact:

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