Harmonic Warping
From Math Images
(Difference between revisions)
Line 23: | Line 23: | ||
*infinite tiling of Euclidean plane mapped onto a rectangle (or ellipse) | *infinite tiling of Euclidean plane mapped onto a rectangle (or ellipse) | ||
*mapping (x,y) from Euclidean plane unto (d(x),d(y)) in rectangle | *mapping (x,y) from Euclidean plane unto (d(x),d(y)) in rectangle | ||
+ | |||
==Polar Harmonic Warping== | ==Polar Harmonic Warping== | ||
- | + | Link to [[Polar Coordinates]] | |
- | Link to [[ | + | |
- | |||
- | |||
+ | ==Four Infinite Poles== | ||
+ | Link to [[Hyperbolic Geometry]] | ||
- | |||
+ | Big table of rectangular, polar, cardinal 4 poles for both flag! | ||
+ | <table border="1" cellpadding="12"> | ||
+ | <tr> | ||
+ | <th>Name</th> | ||
+ | <th>Flag</th> | ||
+ | <th>Rectangular Tiling</th> | ||
+ | <th>Polar Tiling</th> | ||
+ | <th>Four Infinite Poles</th> | ||
+ | </tr> | ||
+ | <tr> | ||
+ | <td> Saint Andrew's Flag </td> | ||
+ | <td> [[Image:StAndrews_Flag.png]]</td> | ||
+ | <td> [[Image:StAndrews_Rectangular.jpg]]</td> | ||
+ | <td> [[Image:StAndrews_Polar.jpg]]</td> | ||
+ | <td> [[Image:StAndrews_4Polar.jpg]]</td> | ||
+ | </tr> | ||
+ | <tr> | ||
+ | <td> Saint George's Flag </td> | ||
+ | <td> [[Image:StGeorges_Flag.png]]</td> | ||
+ | <td> [[Image:StGeorges_Rectangular.jpg]]</td> | ||
+ | <td> [[Image:StGeorges_Polar.jpg]]</td> | ||
+ | <td> [[Image:StGeorges_4Polar.jpg]]</td> | ||
+ | </tr> | ||
+ | </table> | ||
|other=Single Variable Calculus | |other=Single Variable Calculus |
Revision as of 15:03, 23 June 2009
- This image is a tiling based on harmonic warping operations. These operations take a source image and compress it to show the infinite tiling of the source image into a finite space.
Harmonic Warping of Blue Wash |
---|
Contents |
Basic Description
This image is an infinite tiling. If you look closely at the edges of the image, you can see that the tiles become smaller and smaller and seem to fade into the edges. This is true. The border of the image is infinite so that the tiling is infinite and the tiles become infinitely smaller.
The source image used for this tiling is another image that is mathematically interesting and is also featured on this website. See Blue Wash for more information about how the source image was created.
A More Mathematical Explanation
- Note: understanding of this explanation requires: *Single Variable Calculus
Essentially, an equation was used to map the points of
- equation , limit is 1
, limit is 1
- distance compressing warp
- infinite tiling of Euclidean plane mapped onto a rectangle (or ellipse)
- mapping (x,y) from Euclidean plane unto (d(x),d(y)) in rectangle
Polar Harmonic Warping
Link to Polar Coordinates
Four Infinite Poles
Link to Hyperbolic Geometry
Big table of rectangular, polar, cardinal 4 poles for both flag!
Name | Flag | Rectangular Tiling | Polar Tiling | Four Infinite Poles |
---|---|---|---|---|
Saint Andrew's Flag | ||||
Saint George's Flag |
Teaching Materials
- There are currently no teaching materials for this page. Add teaching materials.
About the Creator of this Image
Paul Cockshott is a computer scientist and a reader at the University of Glasgow. The various math images featured on this page were originally produced for his research.
Leave a message on the discussion page by clicking the 'discussion' tab at the top of this image page.