There are lots of ways to make these, old bike wheels seeming to be the most popular. This is an alternative method that has a lot of better features - lighter, can be hand manipulated, and it has two individual LED sections. Read on and light paint! I added a video of the dome being used - you'll notice that on each "pass" the dome gets brighter and brighter - that's why it's important to time the photo right - you only want one pass.
Step 1: Assembling the Disc
You can purchase a disc like this that's used for crafting wreathes - or cut it yourself. This was about $5 at Michael's. Sand the outer edge as smooth as possible. If you have sanding sealer, use it to seal the edge - this helps with adhesion, but is not necessary.
I assume you have some experience with soldering and using LED strips, so won't go into those details.
Measure the circumference of the disc and cut an RGB LED strip as close to this as possible (remember to cut on "cut" lines on the strip!) A small overlap is OK, but you don't want a gap greater than 1/4" or there will be a dead spot in the light.
Solder on the four wires and make a note of which wire is which - the ends on the LED won't be visible. Test the strip.
Peel off the adhesive and press the LED strip into place. If you have any "problem" areas where the LED won't stick, you can use a clamp to apply more pressure. Fold over the wires and tape them down. I used Gorilla tape cut to narrow strips. Test the LED's again (I do this before every major step - I've had solder break off.)
Cover the entire thing in rubber, leaving the LED strip alone. You can use spray-on or paint on (liquid electrical tape.) If spraying, cover the LED's in tape. Test the tape first - some will BOND to the waterproof LED covering! I do cover the edges of the LED's - this helps bond the LED's to the disc.
Step 2: Assembling the Handle
I used 1/2" PVC pipe. It's easy to cut and ridged enough. This central part is glued together - the central "handle" that goes into this is not - you want it to be removable. The ends are BOLTED on with short 1/8" thick bolts, and the ends loc-tighted.
Step 3: Wire Up the Outer Disc
You need 12-18 volts to power the 12V LED's. 18V will overdrive them, but not burn them out. You'll probably shorten the LED lifespan to 10,000 hours. So you can use two 9V in series (18V - my favorite), a 9V and two AA in series (12V.) I used whatever was lying around for this - the old film container actually houses the 9V battery. You can run wires down the inside of the PVC. Use what you want. I have the "handle" attached to make working on this step easier.
Step 4: Lighting the Cross Piece
You can use RGB or single color LED's. I went with blue. I painted the PVC black, and stuck on the LED's (yes, measure, cut, solder, test first. Since the PVC is round, the LED's have little surface to stick to. I used strips of Gorilla tape between each LED. You can also use small tie-straps - BUT - these WILL dig into the soft silicone covering on the LED's and may lead to future problems. I've had this happen - the clear covering wound up being cut into chunks by the straps - took about two months.
I used an A23 12V battery to power the center strip - You'll get over 30 hours of use. These lack the power to fire up the RGB strips, but for a shorter one color, they are great.
So: one 23A battery powers the center strip, and a 12V or 18V battery combination powers the outer circle. The RGB controller also controls the power for outer ring. I use a small rocker switch epoxied to the end of the 23A battery holder to control the power for the center row.
In the photos you can see that Gorilla tape and liquid electrical tape were my go-to adhesives for this. BTW, two years later and it's still doing great.
Step 5: The Results
So what will this do? Using the handle, you can spin, wave and move it around to create light patterns, shapes, etc. Any long exposure photo will yield nice results.
Left: Outer ring set to green, blue crosspiece lit.
Next: Some random wobbles.
Domes - 1st one, Both set to blue, 2nd one, only the crosspiece lit. Third dome - Outer LED on green, crosspiece on. Used a 10" handle, set on ground and pushed. Changing the length of the handle changes the dome height and shape.
Next (big oval in air): Outer set to color pulse, crosspiece on, used handle to spin while I moved it. Think Spirograph!
Red photos: Just playing around.
Last: Spun back and forth, blue crosspiece on, outer ring set to red.