Projecting an object onto a surface

Hello, tell me, please, who has experience. How can you project an object on a plane without distortion. In my case, weaving in the form of a vine on the plane of the chair. I try to apply Shrinkwrap, the shape of the vine is not preserved. Or, maybe someone will tell another way of weaving (maps are not suitable, geometry is needed). Thank.

you might want to try out tissue addon for that