You have to explode the image so it reads as a texture.
Then you need to fuss with its position.
I drew a line from the end of the curves to position it perpendicular to the spring point of the arc.
I then positioned the texture so it was parallel with this spring line.
Then paint bucket, eyedropper to select the texture and paint the rectangle.
You’ll note that the image doesn’t quite fit with the shape you drew on the curved surface.
*You don’t have to do the above step, but if you have a really skewed face you’ll get weird results: