Egg (3D Mapping) Tutorial by Robert (Red) Williams --

It is possible, with some attention to detail, to remain in complete
control of the 3D mapping transformation even when using it in
conjunction with the kaleidoscope or tiling transformations.
The secret lies in creating a correctly scaled and oriented image
using your usual techniques and then carefully completing a specific
sequence of steps to map the image onto the surface of the egg.

To demonstrate this, I took Faye Williams' "We Three Kings" and mapped
it onto an egg.  I chose it because it's a fairly complex, 3-layer
image, two of which are tiled while the other is untransformed.  The
enclosed parameters are for a six-layer image.  The bottom three are
Faye's original image which was retained for comparison purposes.

The first step is to "erect" all three layers of the original image so
they will be aligned with the axis of the egg when they are mapped.  This
requires changing < Rotation angle: > to 30 for each of the three
layers.  Also, reduce the magnification to a value of  2 to avoid
cropping the image - it can be adjusted later.

Next, load the 3D mapping transformation (from the fs folder) into each
layer.  As you do so, click and drag it to the top, deactivate it and
select the "egg" shape.  Enter "2" in the "Z Translation" box and make
sure the rest of the translation and rotation boxes are set to "0"
(zero).

Now, for each layer in turn, Copy the Center coordinates from the
Location Tab and Paste them into the "Fractal Center" boxes on the
Mapping Tab.

Similarly, transfer the Magnification and Rotation Angle values from the
Location Tab to the "Fractal Magnification" and "Fractal Rotation" boxes
on the Mapping Tab.

Now, for each layer in turn, on the Location Tab, set the Center
coordinates to (0,0), the Magnification to "1" and the Rotation angle to
"0" (zero).

Finally, activate the 3D Mapping transformation on all three layers and
if you've done everything exactly right, you will have a vertical egg,
centered in your screen and with Faye's image mapped perfectly onto it.

Now that you have done this, you can:

Rotate the egg around its own axes using the X, Y, and Z rotation
settings.

Relocate the egg using the X and Y translation settings.

Adjust the overall scale of the image using the Z translation setting
(larger number - smaller image).

Adjust the size of the fractal image relative to the size of the egg
using the Fractal Magnification setting.

Rotate the fractal image relative to the axis of the egg using the
Fractal Rotation setting.

I used these settings to approximate the angles of Faye's original
image.  If you reduce the Fractal Magnification settings to "1" on
each layer, I think it makes a "nicer" egg although it isn't as faithful
to Faye's original creation.

As far as I can tell, this procedure works for any image once you have
created it using your normal techniques.  When using the kaleidoscope
transformation, you must be careful to disable the "Use Screen Center"
prior to selecting an off-center portion of the image, but once you have
the desired image centered and vertically aligned, the rest of the
process is just following the steps outlined above.

If you want to preview the scale, simply load the 3D mapping
transformation into any convenient image (the default works nicely).  
On the Location tab, set the center to (0,0),  Rotation Angle to "0"and 
Magnification to "1".  On the Mapping Tab, set the Z Translation to "2" 
and the remainder of the Translation and Rotation settings to "0" (zero).  
Activate the transformation to see the egg.

Hope this helps - have fun.

EggTutorial {
; Image entitled "We Three Kings" created by
; Faye Williams is used here by robert williams
;(no relation) to demonstrate the use of  3D
;Mapping transformations
fractal:
  title="EggTutorial" width=480 height=540 author="robert williams"
  created="May, 1999" numlayers=6
layer:
  method=linear caption="Egg3" opacity=86 visible=yes alpha=no
  mergemode=subtraction
mapping:
  center=0/0 magn=1 angle=0 numtransforms=2
transform:
  filename="fs.uxf" entry="fs-3d-map" p_shape="Egg" p_rotx=15 p_roty=-10
  p_rotz=10 p_transx=0 p_transy=0 p_transz=2
  p_fraccenter=-1.03702647929377398/-0.865990894114283686 p_fracmagn=2
  p_fracangle=30
transform:
  filename="sg.uxf" entry="Hexagonal_tiling" p_mode="hexagonal tiling"
  p_tw=1 p_center=-0.995699697066326529/-0.918447624362244897
formula:
  filename="bwp.ufm" entry="bwp8" maxiter=100 percheck=normal
  p_InitValue=0/0 p_Exponent=1.2345/-1.2345 p_Bailout=4
inside:
  transfer=none repeat=yes
outside:
  filename="lkm.ucl" entry="range-lite" transfer=linear repeat=yes
  p_rangecenter=1 p_rangewidth=0.1 p_colorby="last angle"
gradient:
  smooth=yes numnodes=3 index=68 color=0 index=330 color=0 index=386
  color=3579367
layer:
  method=linear caption="Egg2" opacity=86 visible=yes alpha=no
  mergemode=luminance
mapping:
  center=0/0 magn=1 angle=0 numtransforms=2
transform:
  filename="fs.uxf" entry="fs-3d-map" p_shape="Egg" p_rotx=15 p_roty=-10
  p_rotz=10 p_transx=0 p_transy=0 p_transz=2
  p_fraccenter=-1.03702647929377398/-0.865990894114283686 p_fracmagn=2
  p_fracangle=30
transform:
  filename="sg.uxf" entry="Hexagonal_tiling" p_mode="hexagonal tiling"
  p_tw=1 p_center=-0.995699697066326529/-0.918447624362244897
formula:
  filename="bwp.ufm" entry="bwp8" maxiter=100 percheck=normal
  p_InitValue=0/0 p_Exponent=1.2345/-1.2345 p_Bailout=4
inside:
  transfer=none repeat=yes
outside:
  filename="lkm.ucl" entry="range-lite" transfer=linear repeat=yes
  p_rangecenter=1 p_rangewidth=0.1 p_colorby="last angle"
gradient:
  smooth=yes numnodes=6 index=3 color=7429937 index=51 color=12973542
  index=95 color=3555693 index=210 color=12632294 index=289 color=0
  index=377 color=0
layer:
  method=linear caption="Egg1" opacity=86 visible=yes alpha=no
  mergemode=multiply
mapping:
  center=0/0 magn=1 angle=0 numtransforms=2
transform:
  filename="fs.uxf" entry="fs-3d-map" p_shape="Egg" p_rotx=15 p_roty=-10
  p_rotz=10 p_transx=0 p_transy=0 p_transz=2
  p_fraccenter=-1.03702647929377398/-0.865990894114283686 p_fracmagn=2
  p_fracangle=30
transform:
  filename="sg.uxf" entry="Hexagonal_tiling" enabled=no
  p_mode="hexagonal tiling" p_tw=1
  p_center=-0.995699697066326529/-0.918447624362244897
formula:
  filename="bwp.ufm" entry="bwp8" maxiter=100 percheck=normal
  p_InitValue=0/0 p_Exponent=1.2345/-1.2345 p_Bailout=4
inside:
  transfer=none repeat=yes
outside:
  filename="lkm.ucl" entry="range-lite" transfer=linear repeat=yes
  p_rangecenter=1 p_rangewidth=0.1 p_colorby="last angle"
gradient:
  smooth=yes numnodes=6 index=3 color=7429937 index=51 color=12973542
  index=95 color=3555693 index=210 color=12632294 index=289 color=0
  index=377 color=0
layer:
  method=linear caption="New Layer 1" opacity=86 visible=no alpha=no
  mergemode=subtraction
mapping:
  center=-1.03702647929377398/-0.865990894114283686
  magn=3.08627623205628321 angle=319.140515240496038 numtransforms=1
transform:
  filename="sg.uxf" entry="Hexagonal_tiling" p_mode="hexagonal tiling"
  p_tw=1 p_center=-0.995699697066326529/-0.918447624362244897
formula:
  filename="bwp.ufm" entry="bwp8" maxiter=100 percheck=normal
  p_InitValue=0/0 p_Exponent=1.2345/-1.2345 p_Bailout=4
inside:
  transfer=none repeat=yes
outside:
  filename="lkm.ucl" entry="range-lite" transfer=linear repeat=yes
  p_rangecenter=1 p_rangewidth=0.1 p_colorby="last angle"
gradient:
  smooth=yes numnodes=3 index=68 color=0 index=330 color=0 index=386
  color=3579367
layer:
  method=linear caption="Layer 2" opacity=86 visible=no alpha=no
  mergemode=luminance
mapping:
  center=-1.03702647929377398/-0.865990894114283686
  magn=3.08627623205628321 angle=319.140515240496038 numtransforms=1
transform:
  filename="sg.uxf" entry="Hexagonal_tiling" p_mode="hexagonal tiling"
  p_tw=1 p_center=-0.995699697066326529/-0.918447624362244897
formula:
  filename="bwp.ufm" entry="bwp8" maxiter=100 percheck=normal
  p_InitValue=0/0 p_Exponent=1.2345/-1.2345 p_Bailout=4
inside:
  transfer=none repeat=yes
outside:
  filename="lkm.ucl" entry="range-lite" transfer=linear repeat=yes
  p_rangecenter=1 p_rangewidth=0.1 p_colorby="last angle"
gradient:
  smooth=yes numnodes=6 index=3 color=7429937 index=51 color=12973542
  index=95 color=3555693 index=210 color=12632294 index=289 color=0
  index=377 color=0
layer:
  method=linear caption="Layer 1" opacity=86 visible=no alpha=no
  mergemode=multiply
mapping:
  center=-1.03702647929377398/-0.865990894114283686
  magn=3.08627623205628321 angle=319.140515240496038 numtransforms=1
transform:
  filename="sg.uxf" entry="Hexagonal_tiling" enabled=no
  p_mode="hexagonal tiling" p_tw=1
  p_center=-0.995699697066326529/-0.918447624362244897
formula:
  filename="bwp.ufm" entry="bwp8" maxiter=100 percheck=normal
  p_InitValue=0/0 p_Exponent=1.2345/-1.2345 p_Bailout=4
inside:
  transfer=none repeat=yes
outside:
  filename="lkm.ucl" entry="range-lite" transfer=linear repeat=yes
  p_rangecenter=1 p_rangewidth=0.1 p_colorby="last angle"
gradient:
  smooth=yes numnodes=6 index=3 color=7429937 index=51 color=12973542
  index=95 color=3555693 index=210 color=12632294 index=289 color=0
  index=377 color=0
}