I’ve been fooling around with this for years. I have a better solution.
Rather than a rectangular pyramid tent, I have a half pyramid with a triangular vestibule door.


I used to lay out the door with the ridge on the grain. Cut out of the roll of fabric like:

The side panel I sewed the ridge to was along the grain, so I made the door panel also along the grain to match. Also, it just barely fit onto my 58 inch wide fabric.
When I set up the tent, it looked like:

That red line is the dotted “line along grain” in the schematic above. The zipper is the front right ridge – goes all the way from the ground to the peak.
Fabric stretches a lot less along the grain, so it pulled the side the side panel down about 3 inches. So it’s in my face when I’m sleeping.
I slept many nights just fine, but it bugged me.
So I cut another panel out differently:

The door panel no longer fit onto my 58 inch wide fabric, so I had to patch on an extra piece. No big deal.
Note the two dotted lines “lines along grain”.
When I set up the tent, it no longer pulled the side panel into my face, mission accomplished:

Those two red lines correspond to the dotted “lines along grain” in the schematic. That is a little funky, because rather than taut panels, you get some wrinkles in weird areas. Not to worry.
I did a 1.5 inch deflection catenary curve (using Roger’s spreadsheet) along the ridge on the door panel. There was no catenary curve on the side panel. I just sewed on the door panel starting at the peak. As I got down to the ground, the door panel was stretched a little relative to the side panel. No big deal, I had anticipated this. I just left the door panel unfinished along the ground. After I sewed the door panel to the side panel, I just marked a straight line on the door panel to match the bottom of the side panel, and hemmed it, so it was then even.
I didn’t do any catenary curve on the other two edges of the door panel – zipper edge or ground edge.
If you do a rectangular pyramid tent, you don’t have to worry about any of this. The half pyramid design weighs a little less and uses a little less ground area – a little easier site placement.
Maybe this would be applicable for a vestibule door on other tent designs.

