Transformations

EXTRA STEP 9 : Transform the background image

Select menu option Transform / Background image for transforming  a background image to a circular pattern or a spiral pattern. (You do not need to select or open a tessellation.) Optionally change the background image in advance with menu option Tessellation / Load Background, as in STEP 5. Note that the whole background image is used for the transformation, so its size may differ from the view size. The example image at the right has size 1000 x 640 pixels.
A dialog asks for certain parameters, refer to the Documentation page. For unregistered users the dialog is skipped, and the output image contains a watermark.
The results of 4 circle transformations are shown below. The size of the output image is 1200 x 900 pixels. The circle center is moved 200 pixels to the right. The input background image can be rotated programmatically in multiples of 90 degrees via the dialog parameter Image rotations.
Tissellator maintains the aspect ratio of the background image, as you can see clearly from the 4 transformations below. Click on an image for enlargement.

boat background image

Image rotations : 0 Image rotations : 1 Image rotations : 2 Image rotations : 3
boat circle transformation, rotations=0 boat circle transformation, rotations=1 boat circle transformation, rotations=2 boat circle transformation, rotations=3

The single spiral, the double spiral, and the poly spiral transformations shown below have 5 and 7 spiral arms (just count them); the multi spiral has 7 and 5 arms, whereas the sphere spiral has 11 and 3 arms. The Double Spiral transformation has Image rotations : 1. The Poly Spiral transformation has Frequency : 3. Click on an image for enlargement.

Spiral Double Spiral Multi Spiral Poly Spiral Sphere Spiral
boat Spiral transformation boat Double Spiral transformation boat Multi Spiral transformation boat Poly Spiral transformation boat Sphere Spiral transformation

Note: For unregistered users some transformations are limited.

EXTRA STEP 10 : Transform the tessellated image

Any (multi) tessellated image from the 36 polygon (triangle, quadrilateral, pentagon, hexagon) tessellations can be transformed to a circular pattern or a spiral pattern. The alternative is to create a circle or spiral tessellation directly as a new project; in that case the tiles are rendered just as designed, but this is only possible for a subset of configurations. On the other hand, a transformation of a (multi) tessellated image deforms the tiles. The kind of deformation depends on the type of tessellation, and in particular on the form of the repeating part of a tessellated image being a rectangle or a parallelogram. For circle transformations the deformation is always limited; for spiral transformations the deformation increases with sharper parallelogram angles (due to a different algorithm).
A dialog asks for certain parameters, refer to the Documentation page. For unregistered users the dialog is skipped, and the output image contains a watermark.
The 6 types of circle and spiral transformations are shown below for the bird's multi tessellated image at step 8. The circle transformation has 7 radials. The spiral transformation has 13 and 8 arms. The double spiral transformation has 5 and 8 arms. The multi spiral transformation has 5 and 3 arms. The poly spiral transformation has 13 and 8 arms, and frequency 2.5 . The sphere spiral transformation has 11 and 3 arms. Click on an image for enlargement.

You can reproduce these images using the project file from the above Express Route and the 4 bird images at step 7. Now do the actions:

Circle Spiral Double Spiral
bird circle transformation, rotations=0 bird spiral transformation bird double spiral transformation
Multi Spiral Poly Spiral Sphere Spiral
bird multi spiral transformation, rotations=0 bird poly spiral transformation bird sphere spiral transformation

Note: For unregistered users some transformations are limited.

The repeating pattern of the bird's multi tessellation is a square. A repeating pattern of a parallelogram is demonstrated by a dragon pentagon tessellation of Heesch type 21. This tessellation has always a corner of 60 degrees, so that the parallelogram has corners of 60 degrees and 120 degrees.

The image at the right shows the multi tessellated image including the parallelogram at the top left, that is used by Tissellator for the circle and spiral transformations.

The 6 types of circle and spiral transformations are shown below. The circle transformation has 7 radials. The spiral transformation has 13 and 8 arms. The double spiral transformation has 11 and 7 arms. The multi spiral transformation has 9 and 3 arms. The poly spiral transformation has 11 and 8 arms, and frequency 3. The sphere spiral transformation has 11 and 2 arms. Click on an image for enlargement.

dragon multi tessellated image

Circle Spiral Double Spiral
dragon circle transformation dragon spiral transformation dragon double spiral transformation
Multi Spiral Poly Spiral Sphere Spiral
dragon multi spiral transformation dragon poly spiral transformation dragon sphere spiral transformation


EXTRA STEP 11 : Hyperbolic transformation of a tessellated image

Hyperbolic transformations to a Poincaré disk can be created for tessellations having rotation symmetry. The table below contains their Heesch types. Depending on that Heesch type the tiles can be painted with 1 image, a subset of images, or all possible images.
In case of tessellations with 1 image the background image is used; so firstly execute Tessellation/Create Tessellation , then Transform/Tessellated image/Hyperbolic Tessellation. A dialog asks for certain parameters, refer to the Documentation page. For unregistered users the dialog is skipped, and the output image contains a watermark.
When more than 1 image is used, those images must be selected in menu Tessellation/Load Multi Background . In case of a subset of images the order in the table below (with a capital representing an image) must be applied for having a correct result. Now, firstly execute Tessellation/Create Multi Tessellation, then execute Transform/Tessellated image/Hyperbolic Tessellation. In case of all images several tiles are combined to a 'super-tile' for rendering the hyperbolic picture; in case of IH79 or IH79MCE, 4 tiles form a super-tile. Hence, the tiles in the resulting Poincaré disk are smaller compared to the tiles in the case of 1 image.

Heesch type
1 image subset of images all images
IH7 (hexagon)
-
IH21 (pentagon)
-
IH31 (quadrilateral)
-
IH33 (quadrilateral)
pattern: A B B -
IH39 (triangle)
-
IH55 (quadrilateral)
-
IH61 (quadrilateral)
-
IH79 (triangle),
IH79MCE (triangle)
pattern: A A B B B B A A, or
pattern:  A B C D D C B A
-
IH88 (triangle)
-

N.B. For multi color images of IH33 the rendering only succeeds when the deformation of the outline is limited.

Hyperbolic transformations may take a very long time. A lot of small triangles must be created near the border of the Poincaré disk. Therefore, the user can control the number of iterations: at each iteration the next level of neighbor triangles is generated. Just start with a small value, e.g. 4, and inspect how the disk is filled. Increase then the number of iterations. If a value of 0 iterations is entered in the dialog, then Tissellator chooses a suitable number of iterations. The chosen value is shown in the Status Bar, and can still be incremented for creating a Poincaré disk with more details at the border.

The hyperbolic transformations are shown below for the bird's (multi) tessellated image at steps 6 and 8. The number of triangles are 8, 6, 6, computed with 9 iterations. For the multi tessellated image the translate vector is ( 100, 50 ). Click on an image for enlargement.

You can reproduce these images using the project file from the above Express Route and the 4 bird images at step 7. Now do the actions:


1 image all images
Poincaré disk for 1 image Poincaré disk for all images