How to progressively map a plane to a semisphere?

[PhotoComix]

this is somehow related to this question
http://www.gimptalk.com/forum/%22ben...re-t39911.html

Anyway my qustion may be a bit different:
suppose a plane, as a rectangular and a bit elastic grid,as a net

Now from below the grid ,in its very center a sphere, no larger then the plane, start to pop out,springing upward , pushing up the net

Beyond a certain point of push the elasticity of the net is no sufficient, starting where tension is higher )the net anchors to the plane broke

More the net is free more it wrap around the sphere

But here the difference with the original question, i would like this applied to the Semisphere not to the sphere

meaning that when half of the sphere pop out, the net should be totally wrapped around the surface of the semisphere:

all anchor points previously on the periphery of the rectangle transferred on the Equator, on the maximum circumference of the Sphere, where intersect the plane.

Now you may replace the word "grid" with "original image"
to understand better the point

well i would really like have a similar mapping script and what i found difficult is not map the plane to the semi-sphere but all the "in betweens", the progression

I believe a script for Mathmap will be the easier and more handy solution (as name said is a gimp plugin created for mapping, and ,bigger point support animation and customized script )

But anyway if that may solved by a gimp script or plugin would be the same till the output may be a animation.

I just think that MathMap, and soon GMIC offer a easier and more flexible solution,and what relevant a good preview of the results, but even as a gimp script will be handy

But i'm not good to write scripts, even less in this case because i can't figure out a formula for a similar mapping

I just hope cartographers may ,if not actually code (will be the best ) found the formula to be applied

[PhotoComix]

PS about why i prefer map to a semisphere and not to the sphere

A flat projection of the semisphere will contain all the original image distorted how i wish while in case of a sphere only the half on the upper half of the sphere will be usable (and distortions too extreme )

UPDATE i may have part of the solution how map a square to a semi sphere and a rectangle to the correspondent semi ellipsoid

I failed to find the discussion on the formula to be applied but if may help i have as Mathmap code

filter rel2ellv3(image in)
xprime=x/sqrt(1-4*(y/H)*(y/H));
yprime=y/sqrt(1-4*(x/W)*(x/W));
in(xy:[xprime,yprime])
end

The remaing problem is the progression a smooth transition from the Original Image to the resulting semi sphere or ellipsoid

( And that if the rectangle is not a square result will be a semi ellipsoid and not a semisphere, but that do not worry me so much
...when and if needed may be corrected simply scaling down only the larger side ,to make image's H and W equal)

[RobA]

When doing any such transformations, the issue is (I believe) whether you are striving for an equal area transformation, or something else, such as equal lengths or maintaining angles.

I think you want something like this (made manually using warp):
warp.jpg

The mapping you asked for only defines what happens at the outer edges, not in the center It could be stretched, shrunk or bent...

-Rob A>

[ravells]

Wouldn't it be easier to use the image as a texture on a sphere or hemisphere in blender or some other 3d app and just take a render of it?

I know it's not what you want, as you seem to be streets ahead of me on the maths, but perhaps the obvious solution was overlooked...

[Redrobes]

Had to admit that I was thinking much the same thing. Get a grid, texture it with UV coords, create a morph target of a hemisphere and then set the anim off.

[RobA]

Would a 3d program allow you to map the perimeter of the square to the perimeter of the hemisphere?

-Rob A>

[ravells]

Hmm best I could do was to warp the bounding square into a circle and then map that onto the hemisphere.

[PhotoComix]

Thank for the quick replies

RobA, Ravell
yes i want something as in your examples, i was thinking from a POV
from the top as in RobA example

About how to treat the center both RobA and Ravell guessed correctly what i wish:

Only a mild distortion on the center, much more severe to the periphery ,and ,if possible confined very closed to the perimeter

this seems fine

EDIT

Seems fine at thumb size, at original size much less but i supposed was intended to be seen at thumb size.
anyway a very little bit more distortion on the center, a bit more smooth transition with more severe distortion to the periphery



BUT the real problem is how to make a smooth progression from the image to the left to the image to the right ?


About the suggestion to use blender or other 3D app:
May be very good, but i have no any previous experience of 3D app,
more i do not use so muck hot key, while the constant use of hot key seems strictly needed in blender.

Do you think blender may solve the problem of the transition?
(NOT by fading! i need a "progressive mapping" )

PS OOOOPSS

I do not know why RobA example come out so big, at this size seems quite wrong but looked fine before at little size

[PhotoComix]

Image are more clear then word

here the original



here mapped



But how to slowly move from the first to the second ?

[guyanonymous]

Ah...you mean in small changes....

You're talking about an animation, perhaps with 10 frames where it morphs (changes) into the other image 10% at a time?