Make a Mini Planet

Goal

Turn a panoramic or regular landscape photo into a mini planet, like this one:

by nate bolt, on Flickr

Walkthrough

1. I chose a picture of some sheep in a field. This picture has a fairly flat horizon, featureless sky, and a bottom section without structures, which is good because the bottom will get distorted a lot in order to make it the center of my planet. I'm pretty sure that a picture without an uninterrupted band of sky at the top won't look good, but I can't definitively say so. Open this picture in GIMP.

2. Duplicate the layer so you have the option of starting over at any time.
3. To keep a version of the picture you can edit again later, save it in xcf format, GIMP's native format that preserves all the image's layers. Go to File > Save and in the popup window, choose a folder and set the name to planet.xcf (only the .xcf part matters). Click File > Save often during every project, in case GIMP or a plugin crashes.
4. Click Filters > Distorts > Polar Coordinates... in the menu bar. Expect this plugin to take longer than expected to load. It brings you to a window with a preview and five configuration options. Set the circle depth at 100% to give the planet a circular border, offset angle at 278.98 to put the flagpole at the top, Map backwards unchecked, Map from top unchecked, To polar checked. Then click OK to apply the filter.
   
5. Now I have an ugly line that used to be the edges of my picture that I need to take care of. I've decided to worry about the sky later, and focus on the interrupted treeline and field. In this instance, I'm going to duplicate the planet, flip it, line up both planets' seams, and make the bottom copy partially visible. This will make more sense when I break it down:
6. Duplicate the planet layer and drag the new copy to below the planet layer in the layer list.
7. Click the eye icon next to the top layer to make it temporarily invisible.
8. Select the lower planet layer so the little thumbnail has a highlighted box around it. Go to the toolbox on the left and select the Flip Tool, or go to Tools > Transform Tools > Flip in the menu bar. Go to the Tool Options tab and set Affect to the first icon (Layer). The Flip Type doesn't matter in this case. Then click on the image and watch it flip.
9. Select the top layer again and drag in the Opacity bar at the top of the layer list to set the opacity to a level where you can see both planet layers at once.
10. Select the lower planet layer, and then switch to the Rotate Tool in the toolbox or at Tools > Transform Tools > Rotate. Click your main image. In Tool Options, drag the Image opacity bar to around 50% so you can see both layers at the same time and accurately line up the top one. Then drag in the image or drag the slider in the popup window to rotate the layer so that the two seams are close but not touching. If they lined up perfectly, the planet would look like someone had placed a mirror along it. It would look more unnatural than it already does.
11. Select the top layer and bring its opacity back up to 100%. Now, add a layer mask by going to Layer > Mask > Add Layer Mask... in the menu bar. In the popup window, choose White (full opacity). The image doesn't look any different... yet. A layer mask is extremely useful; it allows you to create transparent areas in your layers without erasing parts of the layer itself, which means you can adjust the transparent areas as much as you want. You perform adjustments with the Paintbrush Tool, and black or white "paint."
    
12. Under the toolbox there are two rectangles, one on top of the other, black and white by default. To the upper right there is a two headed arrow, which is used for switching the foreground color and background color. To the lower left there is a tiny black square on top of a tiny white square, which returns the colors to the defaults. Click this icon with the tiny squares. Now click the tool in the toolbox that looks like a paintbrush. Hover the mouse pointer over your image. The pointer will have a circle around it. If the circle is too big or small to paint effectively, go to the Tool Options and drag in the Size slider. It's pretty versatile: you can make the size go all the way from 1 pixel to 10,000 pixels.
    
13. The layer mask, a white rectangle next to the top layer in the layer list, should be selected. Paint on the picture now to reveal the lower layer. Hit the X key or the double-headed arrow above the active colors to switch to painting opacity instead of transparency. If the edge of the paintbrush is too hard, click the white square at the left in the Tool Options tab, and click a fuzzier or harder brush. The confetti-type brushes aren't useful for this project.
14. Keep painting with black and white until the horizon seems to blend together. It helps the illusion to reveal full structures like trees, buildings or sheep when masking, and make their outlines as hard as possible. Don't worry about the center or the sky yet. I'll think of some way to fix those.
15. Merge the top two layers by selecting the top one and clicking Layer > Merge Down.
16. Activate the Healing Tool, which looks like a band-aid cross in the toolbox. Hold down the Command key adjacent to the space bar and click a place on the picture that you want to start cloning from. I picked a spot just above the center. Because this is relatively featureless, this is easy. I set the brush size and dragged from the center of the picture out along the seam to the horizon. The tool shows two circles while it works; one is around your pointer, and the other is moving out from where you Command-clicked and copying that part of the image. Repeat this process to heal any unnatural edges or areas.
17. Duplicate the planet layer, put it under the old one, flip it horizontally and rotate it so the seams line up.
18. Add a layer mask to the top layer, with full opacity.. Then select the layer itself, not the layer mask. Use the fuzzy select tool, which looks like a wand in the toolbox, to select the sky but not the planet. Adjust the Threshhold bar in the Tool Options if it selects too much or not enough with one click on the sky.
19. Select the layer mask. Use a hard-edged black brush to draw along one side of the seam in the sky, with one edge of the brush on the seam itself. This should leave just a tiny band of lighter or darker blue between two equally blue bands. Switch to a soft brush and keep painting in the sky (because you selected it, the planet won't get painted over). Only paint the sky until the color of the sky you're revealing with the brush matches the sky color of the top layer. The sky should seamlessly blend at this point. Switch to a white brush and fix any parts of the planet that got erased.
20. Merge the top layer down again. Use the heal tool or the clone tool, which is similar, to heal the small band in the sky just like how I healed the grass.
21. Click Rectangle Select Tool in the toolbox and drag inside your circle of sky until the rectangle is fully within the circle of sky. This will make it look better than having a white background. To crop it, click Image > Crop to Selection.
22. The picture should look finished at this point. Delete the backup layer you created at the very start, so just the planet layer is left, and export the image.

Result

A very small planet.

Learn More

GIMP has an official page for this plugin, which explains its intricacies.
The first tutorial to pop up in my search results is this one from Photojojo. It includes a few unnecessary steps, like rotating and resizing, but if you're stuck here it'll help.