This chapter is about mapping textures to model faces.
i will explain the basics of UV mapping in theory.
If you first want to see UV unwrapping in action, then
you may want to jump right away to the next chapter.
And get back here later if necessary.

So, the goal is to add color and texture to this model’s surface.
This is where UV Maps and UV unwrapping come into play.

We already know that this model is made out of a set of connected faces,
typically Quads and Triangles.

As a matter of fact most graphic engines work only with triangles.
fortunately the triangulation of quads and more complex polygons
is always possible.
In fact the mesh  importer
always triangulates the models before they get imported.
Thus at the end of the day we always end up with a bunch of triangles.

Now, triangles have one very important property:
they always have an absolute plain surface.
And plain surfaces are essentially 2 dimensional.

In practice this means that
we can use simple images for our texturing,
so all we have to do is to specify
which part of an image shall be used
to texturize which triangle on the model.

Let me pick up this single triangle from our model.
In the following i will name this: a model face.

And here i have prepared a simple 2 dimensional testgrid image.
I have placed another triangle on top of the image.
This is: a UV face.
And finally i also have specified which corner of
the UV face corresponds to which corner of the model face.

that’s all we need. Now my computer is able to
cut the UV face out of the image,
then transform, scale and rotate it
until it fits exactly into the model face.

This transformation works independent from the shape of the UV Face.
In fact we can move the vertices of the UV Face wherever we
want. The area marked by the UV Face will always be transformed
such that it exactly matches the model face.

The relation between the corners of the UV Face
and the model face is also named: a mapping.

Note that the UV Face is always flat,
so we only need a 2 dimensional coordinate system
to specify its vertex locations.

The 2 dimensions of the UV Face are called: “u” and “v”,
while the 3 dimensions of the model face are called: “x”, “y”, and “z”.

mathematically we have defined a mapping from a 2 dimensional image
to our 3 dimensional model. This mapping is named: UV mapping.
And now you also know why UV faces are named UV faces.

So far we have only specified the mapping
between one UV face and a corresponding Model face.
If we want to texturize our entire model,
then we have to add a similar corelation
from each individual model face to a corresponding UV face.
The entire set of these corelations is named: UV Map.
And the process of defining the corelations is named: UV unwrapping.

In practice it does not make much sense to define the UV mapping for
each face by hand. This can be done much better by a computer.
And Blender provides some tools which make UV unwrapping easy.

Actually i already have used blender’s UV tools to create the single
UV face for this demonstration.
Let me show you how i did that as a warm-up exercise.
Then i will finish the complete unwrap of our model in the next chapter.

So, You find the UV tools when you are in edit mode.
Enable face select mode and then select one single face.
Now press “u”.
this opens the UV mapping option menu.
From all available options select the first one: unwrap.
By now the UV Map has already been created.
And we can make it visible as follows:

Switch to the UV Editing screen layout,
Now you see the UV editor on the left side.
And therein is the UV map section,
where the “U” axis goes from left to right,
and the “V” axis goes from bottom to top.

The UV Map contains a single highlighted UV face.
And this UV Face is already mapped to the model face.

So by now we have the mapping,
and now we have to create a test texture for the model.
In the image editor select: image, new image.
name the image: “test grid”, and check the “UV test grid” option.
this generates the test grid image.

Now we have the texture, but we do not see it on the object.
This is just a display setting, we only need to switch the viewport shading to: texture.

And finally we can see it.
The single UV face now gets correctly mapped
from the texture to the Object surface.

But hold on. The object is not nicely illuminated.
This is because we have not yet set up our light environment.
I will get back to this in a later chapter.
For now i show you a quick step for checking your unwrap.
This method is independent from the light set up.

Go back to viewport-shading solid.
Now the object light setup does no longer matter,
but the texture is gone.
We can fix that as follows:
open the left properties sidebar.
Search the display section.
and there enable textured solid.

By now you know everything that is needed to understand the
principles of UV mapping. And you can define the UV Map for
individual faces, and you can make your texture visible on
the model face.

In the next chapter, i will proceed by defining a simple workflow
for creating a complete UV-Map for your model.