I have been building my cat tarp and started a bridge hammock (in other words I have been busy making cat cuts)
I have found a very easy and efficient way to cut great looking catenary curves....
First, it is important to not that I have a shop with a sheet rock wall.

Step 1:
To make the cat curve I started by measuring up 65" from the floor in the shop in two different locations on the same wall. I then use a chalkline to mark a level line between these points.

Step 2:
Next I use modelling pins (you could use thumb tacks) to hang the factory edge of the fabric along the chalkline mark.

Step 3:
At this point if my fabric is not cut to length I do this by measuring to the desired length and then use a plumb bob to ensure a straight line (gravity please don't ever fail me ) I make several dots along the line and then remove the plumb bob and use a ruler or straight edge to connect the dots.

Step 4:
I then find the middle of the fabric. I place a mark on the wall where the mid point is.
At the mid-point I use a measuring tape to determine the maximum amount of curve I want (6" or 10" etc, etc) I make a mark on the fabric at this point.

Step 5:
I then use a piece of string (usually the chalk line) and pin one end of the string to the end of the fabric. Then do the same to the other end of the fabric, but I make sure that the string touches the mark on the fabric I made at the mid-point.

Step 6:
With the string still in place I carefully (without moving the string) make several tiny dots on the fabric along the string with a sharpie or other marking device.

Step 7:
Remove the string.

Step 8:
Connect the dots using a ruler or straight edge.

VILOA perfect cat cut pattern.

To ease the cutting process I leave the fabric pinned to the wall while I use a soldering iron to cut and fuse the edges of the fabric.

Warning:
Be careful using a soldering iron for anything other than its intended purpose.
Also...Using a soldering iron on sheet rock (or plaster walls) will leave permanent marks.