If you can't find someone to borrow a projector from, there are a few things you can, depending on how much effort you're willing to expend, and how precise you need it. (for kids, getting it recognizable is often enough).
For a one-off project, it's likely not worth making a camera lucida.
If you draw a grid over your design, you can press some faint lines into the cake, and use that as a reference for how to draw the artwork on the cake. Wait until the icing has crusted over, and press in lightly with a straight edge ... enough so you can see it if you shine a light at a severe angle.
If you can make a copy of the design of the size that's needed for the cake, you can either cut it out so that you can trace around it (just scoring lightly so that you can then go back and fill in after you remove the pattern). You can then either cut out portions of different color and align those and trace again, or you can sometimes press lightly through the paper and transfer marks well enough that you can see it to trace in icing.
Another method of transferring a design of the same size is to use a pin and poke through the paper repeatedly along the line you're interested in transferring. I don't know how visible that would be on a cake, though -- you might have to use something larger (eg, a push-pin) and I'd be afraid of having to push down so hard that you get the paper stuck on the icing.
If you do want to try this, spin the pin as you're pushing through the paper. (this reduces the friction when pinning into poster boards ... I would hope it would do the same for the paper)
Yet another method is a cross between the full-sized copy, and the grid technique. Make two copies of the design, and draw series of parallel lines on one (spacing depends on the precision you need). Use a knife to cut each of the lines, but do not go all the way to the edge of the paper. Use your knife to then cut out every other strip. Place the cut up paper on the other page, and transfer the lines for reference. Place the cut paper on the cake, and the full picture to mark what you can in the area from the slots cut in the paper. Remove the paper, and mark the area that's now uncovered.