The Uzebox does not have a frame buffer so you can't arbitrarily put pixels everywhere you'd like. That said, this can be done to some extent with the use of ramtiles (check the wiki for more on the subject). Currently this can in theory be done in video mode 3 but due to the very low number of ramtiles available, it's not useful. Now only mode 6 have a putpixel function in the demo project...check /demos/VectorDemo/VectorDemo.c for this (note that mode 6 is monochrome 1 bpp but supports 255 ramtiles).
The Uzebox is not well suited for "vector" graphics (i.e: dynamic lines, circles, points,etc). You would be better off creating the spiral/logo in a drawing program like Gimp and convert it to tiles with the gconvert