# Slope and altitude based materials in Cycles

If you've used any graphics program that specializes in landscape generation, such as Terragen, you have probably seen that a core feature is the ability to automatically calculate the borders between snow, barren stone and grass from the geometry. Snow is only possible above a certain altitude where the temperature is below the freezing point, and even there snow will not stick on surfaces that slope too steeply. Similarly, grass will only be found below a certain altitude where it's warm enough, and will also not grow on too steeply sloping ground.

Previous proposed solutions use light groups, but I find these a bit unintuitive and unwieldy, so I use a more direct method by using the normal vectors to choose the material.

A suitable mountain terrain can be generated using the ANT Landscape add-on. There are numerous tutorials on using this add-on, so I will just provide a .blend file with a pregenerated terrain, as well as simple premade rock and snow materials.

## Using slope to choose texture

We'll start by showing the basic principle, which only requires a
few nodes. First, select the `Landscape` mesh object and
create a new material by pressing the `New` button in the
`Material` tab. Rename it to `Terrain`. A
`Diffuse BSDF` with a white color will be added by
default. Create a duplicate of the `Diffuse BSDF` by
selecting it, hitting `shift + D`, and dragging and dropping
it below the original copy. Change the color of the duplicate to any
color that contrasts well with white, such as deep blue. To mix both
colors, press `shift + A` and add a
`Shader`→`Mix Shader`. Connect the
`BSDF` outputs from the `Diffuse BSDF` nodes to
the `Shader` inputs of the `Mix Shader`, and
connect its `Shader` output to the `Surface`
port of the `Material Output` node.

We now want one of these two colors to be used at each point of the
mesh depending on the inclination at that point. To get access to
information on the geometry of the object, press `shift + A`
to add a new node and choose
`Input`→`Geometry`. Mathematically, we want
to look at the direction of the normal vector of the mesh. A normal
vector $\mathbf{\hat{n}}$ is a vector pointing straight out from a
surface to the direction that the surface is facing, as illustrated in
Figure 1. We are interested in the angle between this normal vector
and a fixed vector pointing straight upwards, which we call
$\mathbf{\hat{z}}$, as shown in Figure 2.

From linear algebra, we know that
$$
\mathbf{\hat{n}} \cdot \mathbf{\hat{z}} = \cos \theta
$$
where $\theta$ is the desired angle, and the dot denotes the dot
product. Solving for the angle gives
$$
\theta = \arccos (\mathbf{\hat{n}} \cdot \mathbf{\hat{z}})
$$
So, add a `Converter`→`Vector Math` node
and change the vector operation from the default `Add` to
`Dot Product` in the drop-down list inside the node. To
plug in the normal vector $\mathbf{\hat{n}}$, connect the `True
Normal` output from the `Geometry` node to the first
`Vector` input. The difference between the
`Normal` and `True Normal` ports will be
explained later in this tutorial when the distinction becomes
important. For the other `Vector` input, click on the
drop-down list for this port and manually enter the three numbers $(0,
0, 1)$. These are just the x, y and z components of the fixed
$\mathbf{\hat{z}}$ unit vector. Next, add a
`Converter`→`Math` node and change its
operation to `Arccosine`. Connect the gray
`Value` output of the `Dot Product` to the first
`Value` input of the `Arccosine` node.

We have now produced the angle $\theta$. To choose color based on
this angle, add another `Math` node and change its
operation to `Greater than`. Connect the output from the
`Arccosine` node to the first input of the `Greater
than` node, and connect its output to the `Fac` input
of the `Mix Shader`. The desired threshold angle could now
be typed in directly at the second input of the `Greater
than` node, but we would have to specify it in radians. To
be able to use degrees instead, add another `Math` node and
make it a `Multiply` node. Connect its output to the
`Greater than` node, then click on its second input port
and type `pi/180`

, the conversion factor from degrees to
radians. Note how the value of $\pi / 180$ is evaluated automatically
by Blender. Now the first input of this node can be fed with the
threshold angle in degrees. Enter `45`

for a first test,
and change `Viewport Shading` of the 3D View window to
`Rendered` to see the result. Figure 3 shows the full node
setup at this point, and Figure 4 shows the result. All parts of the
mountain with an inclination of 45° or less is white, and the rest is
blue.

Now that we have seen the working principle, let's switch to the
premade rock and snow materials in order to get something that
actually looks like a snow-covered mountain. Delete the two
`Diffuse BSDFs` and add a
`Group`→`Snow` to replace the white one and
a `Group`→`Rock` to replace the blue one
(connecting their outputs to the `Mix Shader`). Next, add a
`Input`→`TextureCoordinate` and connect its
`Generated` output to the `Vector` input of the
`Rock` node.

The mountain may now have a rock texture, but looks quite plastic
as you can see. This is because there is no displacement map in place,
so that the surface is completely smooth. The `Rock`
material group provides a displacement output, but if you connect it
directly to the `Displacement` port of the `Material
Output` node (try it!), it will affect all of the mountain, even
where the snow is. We already have a `Greater Than` node in
place that “selects” the parts that are not covered by
snow. What we need to do is to multiply the displacement value with
the output of the `Greater Than` node by adding
a `Math` node. Figure 5 shows what the node setup should
now look like, and Figure 6 shows a render at this point. At this
point it becomes important that we used the `True Normal`
output from the `Geometry`. The `True Normal`
port gives the normal of the raw mesh, while the `Normal`
port includes displacement maps and mesh smoothing. In this case, we
are using the normal vector port to calculate the displacement map, so
using the `True Normal` would create an infinite loop:
The `True Normal` would determine the displacement, but the
displacement would be included in the `True Normal`. I
don't know how Blender breaks free from such an infinite loop, I can
tell that it doesn't hang or crash, but the result wasn't good.

## Smoothing the snow lines

The border between rock and snow now looks very sharp and pixelated, so let's make it a bit smoother. The border
is sharp because the `Greater Than` node only outputs two
values, 0 (giving all snow) and 1 (giving no snow). The most common
way got get a softer transition is to use a `ColorRamp`
node with two color stops close to each other, as shown in Figure
7. But even though such a color ramp only takes a minute or two to
setup, I find that I want this effect so often that I have instead
created a reusable node group, composed of math nodes, for this
purpose. This also has the advantage that the threshold value and the
smoothness can be controlled by a variable from another node if
desired (though we won't do that in this tutorial). If you hate math
and love playing around with color ramps, you may skip this section
and do as shown in Figure 7 instead.

We need a function that stays at zero for some time, then gently
climbs up to one and again stays there. There are a number of such S-shaped mathematical functions, but I have chosen the
logistic function for its simplicity. It is given by
$$
f(x) = { 1 \over {1 + e^{s(m-x)}} }
$$
and is plotted in Figure 8. The constant $s$ controls sharpness of the
transition (lower values make it more smooth) and $m$ contains the
threshold value. So how do we translate this formula to Blender math
nodes? The key is to think about what operations are involved in the
formula and in what order they need to be performed. The part within
parentheses in a mathematical formula are always evaluated before the
part outside those parentheses, so we have to start with the $m-x$
part by using a `Subtract` node (all math nodes are created
by adding a `Converter`→`Math` node and
*then* choosing an operation in the drop-down list). The result
needs to be multiplied by $s$ using a `Multiply` node. I
haven't told you where to get the constant values from yet, leave them
unconnected and we will return to them in a minute. For the “$e$
to the power of” part, use a `Power` node, click on
its first input, and just type `e`

(Blender will again
substitute the value of Euler's number). Next use an `Add` node and
finally a `Divide` node. For the details of all
connections, refer to Figure 9.

Now select all those math nodes and click on
`Node`→`Make Group` or just press `ctrl
+ G`. Now that we have created a group, the function argument $x$
and the $s$ and $m$ constants can be made inputs of that group by
connecting each of them to the automatically created `Group
Input` node. Similarly, the final output from the
`Divide` node should be connected to the `Group
Output` node. Create new connections to these special
input/output nodes by dragging from the empty circle to another node
port. The inputs and outputs now also appears in the list called
`Interface` in the GUI. Click on each one of them and give
them a name, a default value, a minimum value and a maximum value. I'd
like to call the function argument `x` and give it a default
value of 0 (leave the min and max values as is). The $m$ constant can
be named `Midpoint` and have a default value of 0.5, a
minimum value of 0 and a maximum value of 1. The $s$ constant can be
named `Scale` and have a default value of 50, a minimum value
of 1, and a maximum value of 100. The output can be named `y`
and its other values left as is (I don't think they are used for
outputs). The node setup should now look exactly as in Figure
9. Finally, press `shift + tab` to exit the group, and with
the group still selected change its name under `Properties`
to `Logistic function.`

The node group that we just created can be reused in other .blend
files by using the `File`→`Link` or
`File`→`Append` commands. For now, let's
replace the `Greater Than` node with our new `Logistic
function` node. The output from the `Arccosine` node
should be connected to the `x` input port, the output from
the `Multiply` node should be connected to the
`Midpoint` input port, and the `Scale` input can
be set to a constant value of `30`

. The `y`
output should replace the connections from the `Greater
Than` node to both the `Mix Shader` and the
`Multiply` nodes. The snow border should now look smoother
and more natural than before. You can play with the `Scale`
value to very easily change the smoothness.

## Using altitude as well

Let's move on to limit the snow to only cover the mountain above
the snow line. The principle is the same, but now we use the
`Position` output from the `Geometry` node. Here
too we are only interested in the z component of the position vector
since this represents the altitude, so create a copy of the `Dot
Product` node by selecting it and hitting `shift + D`,
and connect the `Position` vector to its first input. Also
create a copy of the `Logistic function` and connect the
`Value` (gray) output of the `Dot Product` to
the `x` input of the `Logistic function`. Now
connect the output of the `Logistic function` to the
`Mix Shader` node, replacing the previous connection. We
got the opposite of what we wanted - only the bottom part of the
mountain is covered by snow, so we must invert the signal. Insert a
`Subtract` math node, set its first input to a constant 1,
and connect the new `Logistic function` to its second
input. Connect its output to both the `Mix Shader` and the
`Multiply` node for the displacement.

The top part of the mountain should now be completely covered by
snow. We still don't want the steep slopes to be covered, so we now
need to combine the two conditions. If snow was always represented by
1 and bare rock was always represented by 0 it would just be a matter
of multiplying the two signals. However, most of the signals work the
other way around. Delete the `Subtract` node we just added
and add a new one to instead invert the output from the old
`Logistic function`. Combine the two signals using a new
`Multiply` node, invert its output with another
`Subtract` node, and connect the final output to the
`Mix Shader` and the displacement `Multiply`
node. The nodes should now look as in Figure 10.

## A finishing touch

The spots of snow are now very big, which looks unrealistic. This
is due to the limited resolution of the mesh. We can easily cheat and
simulate a more detailed mesh by adding a random component to the snow
line. Figure 11 shows the new nodes needed colored in red. The
`Noise Texture` node simulates the detailed variations in
slope of the mountain. Since it will provide only positive values
between 0 and 1, it would only increase the slope, so we subtract 0.5
to have it both increase and decrease it. We then multiply it by 2.5
to fine tune the impact of this new random component, and finally add
it to the slope value that we had before. Figure 12 shows the rendered
scene after this last modification.

We are not aiming for photo realism here, but I hope to have provided some insights in getting the node editor to select materials based on your desired conditions, and that I managed to explain some of the mathematics inherent in the node editor. It is often possible to accomplish the wanted results without understanding the math, but getting this understanding can be helpful and satisfying nevertheless.

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