Getting a flat image onto a curved surface

UV unwrap lab

A texture is a flat rectangle; a model is not. UV mapping is the dictionary between them — every vertex carries a coordinate saying which point of the image lands here. Drag the handles in the layout panel and watch the texture slide over the model. Then switch to a projection and see what you get when nobody unwraps anything by hand.

The model
Drag to rotate
seam — an edge where the UVs jump the triangle under your pointer
The UV layout
Drag a handle to move a vertex

This is the texture image with the model's surface flattened onto it. Anywhere two triangles that touch on the model sit apart here, the model has a seam.

Triangles
Seam edges
Texture area used
Worst stretch
Display overlays
What this mapping does

Why a seam is unavoidable. You cannot flatten a closed surface onto a rectangle without cutting it — the same reason every world map lies about something. So the question is never whether to cut, only where: down the back of a character, along a hemline, inside a wheel arch. Every cut duplicates the vertices along it, which is why a "24-vertex" cube is really 24 and not 8 the moment it carries a texture: the eight corners each need several different UV coordinates, and a vertex can only hold one.