Ver. 1 - Default UV
Plane (mesh) + BazierCurve (path) + Hair ParticleSystem (textures) PLANE (mesh)
note
Location of plane and curves (for curves any transform only in edit mode) should be x0/y0/z0
1st vertices must be in the zero position on the Y-axis
Edit plane only via edit mode in a specific axis (Y) -> G-Y-move mouse
add vert only via Ctrl+R

mod Plane UV like

Modifiers (deform - Y)

BAZIER CURVE (PATH)
Hair ParticleSystem (textures)