Trillions of triangles

I have been stitching my way through strips and strips of triangles for my big project.

My ideas are fairly fixed now, and the next big step is for me to cut a large curve through the strips and just stitch it to the checkerboard.  I've been researching ways to cut and stitch curves, but most of them seem to be on a small scale, or freehand and shallow, so I'm now referring to my dressmaking sources for suggestions on how to stitch precise, longer curves.  I may go with stay stitching the edges before I stitch the curved edges together.