Temp Directory?: /var/www/html/mathimages/imgUpload/tmp Harmonic Warping - Math Images

# Harmonic Warping

(Difference between revisions)
 Revision as of 10:04, 24 June 2009 (edit)← Previous diff Revision as of 10:29, 24 June 2009 (edit) (undo)Next diff → Line 2: Line 2: |ImageName=Harmonic Warping of Blue Wash |ImageName=Harmonic Warping of Blue Wash |Image=Harmonic warp.jpg |Image=Harmonic warp.jpg - |ImageIntro=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. + |ImageIntro=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 within a finite space. |ImageDescElem= |ImageDescElem= - 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. + 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. The border of the image is infinite so that the tiling continues unendingly and the tiles become eternally smaller. Line 12: Line 12: |ImageDesc= |ImageDesc= - [[Image:HarmonicWarp.png|thumb|300px]] + To create this image, a harmonic warping operation was used to map the infinite tiling of the source image onto a finite plane. This operation essentially took the entire infinite Euclidean plane and squashed it into a rectangular. This type of operation can be called a ''distance compressing warp''. - [[Image:UnionFlag.gif|300px]] + - [[Image:UnionFlag_Rectangular.jpg|300px]] + - Essentially, an equation was used to map the points of values + + [[Image:HarmonicWarp.png|thumb|right|300px|Harmonic Warping Equation]] + The equation used to perform the harmonic warp is show in a graph to the right and is as follows: + + :$d(x) = 1 - \frac{1}{1+x}$ + + :$d(y) = 1 - \frac{1}{1+y}$ + + You can see that for both of these equations, as x and y go to infinity, d(x) and d(y) both approach a limit of 1. + {{HideThis|1=Limit|2= + $\lim_{x \rightarrow \infty} 1 - \frac{1}{1+x}$ + $\lim_{x \rightarrow \infty} 1 - \frac{1}{1+\infty}$ + $\lim_{x \rightarrow \infty} 1 - \frac{1}{\infty}$ + $\lim_{x \rightarrow \infty} 1 - 0}$ + $\lim_{x \rightarrow \infty} 1$ + }} + + + + [[Image:UnionFlag.gif|300px]] + [[Image:UnionFlag_Rectangular.jpg|300px]] - *equation $d(x) = 1 - \frac{1}{1+x}$, limit is 1 - $d(y) = 1 - \frac{1}{1+y}$, 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 *mapping (x,y) from Euclidean plane unto (d(x),d(y)) in rectangle Line 59: Line 72:
Four Infinite PolesFour Infinite Poles Tiling [[Image:StAndrews_4Polar.jpg|200px]] [[Image:StAndrews_4Polar.jpg|200px]] [[Image:StGeorges_4Polar.jpg|200px]] [[Image:StGeorges_4Polar.jpg|200px]]
- +

## Revision as of 10:29, 24 June 2009

Harmonic Warping of Blue Wash
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 within a finite space.

# 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. The border of the image is infinite so that the tiling continues unendingly and the tiles become eternally 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

To create this image, a harmonic warping operation was used to map the infinite tiling of the source [...]

To create this image, a harmonic warping operation was used to map the infinite tiling of the source image onto a finite plane. This operation essentially took the entire infinite Euclidean plane and squashed it into a rectangular. This type of operation can be called a distance compressing warp.

Harmonic Warping Equation

The equation used to perform the harmonic warp is show in a graph to the right and is as follows:

$d(x) = 1 - \frac{1}{1+x}$
$d(y) = 1 - \frac{1}{1+y}$

You can see that for both of these equations, as x and y go to infinity, d(x) and d(y) both approach a limit of 1.

$\lim_{x \rightarrow \infty} 1 - \frac{1}{1+x}$ $\lim_{x \rightarrow \infty} 1 - \frac{1}{1+\infty}$ $\lim_{x \rightarrow \infty} 1 - \frac{1}{\infty}$ Failed to parse (syntax error): \lim_{x \rightarrow \infty} 1 - 0}

$\lim_{x \rightarrow \infty} 1$

• mapping (x,y) from Euclidean plane unto (d(x),d(y)) in rectangle

## Four Infinite Poles

Big table of rectangular, polar, cardinal 4 poles for both flag!

 Saint Andrew's Flag Saint George's Flag Original Flag Rectangular Tiling Polar Tiling Four Infinite Poles Tiling

# 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.