2-4 sheets of 30x40 cardboard or whatever you can find in the dumpster :)
Jig saw or band saw (band saw is better)
Step 1: Figure Out Your Shape
Anyway, you can rough test the curve sitting on the bed, leaning against it. That will help you to refine it to the proper height without wasting any time. After you have a good guess of your ideal shape, trace it to another scrap, and wrap it with a 9" strip, something you can lean against. Refine until you have the perfect shape. It is important to work it into a smooth, flowing curve free of sharp transitions.
Step 2: Cutting Ribs
Cut out cardboard rectangles 1/2" wider and longer than the final template. Glue them together in a stack with (2) small blobs between each piece. I Built ribs 3" on center, but you may want to alter this to 4 or 5" O.C. if cutting them is laborious.
Stack laid out in front of you, tape down the template carefully and trace. Remove the template but keep it in case you need more ribs. Cut it in one shot and pull the shapes apart.
Step 3: Assembling the Skeleton Frame
Drag lines across the width at 3, 4 or 5" intervals. This is a guide for gluing down the ribs.
Lines drawn, divide this piece into (4) 3"Wx30"L strips.
These strips will be glued along top and bottom of the ribs. Align them and tape down to a square table. Glue down the outside ribs first. It is important that these are square to the strips. Apply a 3" long bead of hot glue, then place down on the ends of your upper and lower strip. The rest of the ribs can be placed by eye on the construction lines that you drew.
Make a second skeleton frame with your remaining parts.
Step 4: Stretching a Skin
The length of each skin should be 30", with the grain oriented along the length. Figuring out the width is tricky. Stretch a length of yarn around the curve and mark with a sharpie. Starting with a 1" wide scrap, cut 1/8" wide grooves spaced the length of the yarn + 1/8". Leave 3" tabs beyond the grooves. Roll the strip around the edge of your table to soften it for a curve. Test-fit by wrapping around the skeleton. Once you get a test strip that fits tight around the curve, cut out the full-sized skins.
"break" a curve into the skins every inch.
I clamped a board across the table, square to the edge. The board sufficed as a guide to keep the folds/breaks along the grain. Place the skin against your guide board, hanging over the edge of the table. Press a 30" board on the edge, and roll it over to snap a fold. Once you get comfortable doing this, you can quickly slide the cardboard out another 1" and snap, slide and snap.
Dry-fit the skins to skeletons. If they fit taught, you are ready to glue.
Apply a heavy glue bead to the curve edges,with yellow wood glue. Glue will soak into the end grain, so you want to apply a second pass of glue beads 3 min later.
Place a skin concave side up. Apply 6 evenly spaced hot glue blobs to the back side of one of the alignment rails. Squeeze a 3" tab over these glue blobs until the glue cools. Apply 6 evenly spaced glue blobs to the other alignment rail...roll the frame over the skin and wrap the 3" tab over these blobs, press and hold.
I will add a video of this. At the time it seemed self explanatory, now it doesn't see so clear.
Tape the skin to the end ribs. A quality seam at the end will greatly increase the longevity of your headboard.
Step 5: Joining the Two Sides
I haven't attached the headboard to anything, but if you wish, it can be hung on two screws. Glue a small piece of thin plywood inside of the alignment rail and drill a 3/8" hole. The little white line is a plywood edge.
Its worthwhile to learn how to build with this method. You can apply it to any shape - chase lounges, giant sculpture, anything large extruded shape - have fun! As with anything cardboard, you can finish with with primer and paint or wrap it with foam and cloth - spray mount or staple to the rails.