As
I described in my previous post, I set out to procedurally generate a sky that
reflects the current weather conditions over Stockholm. At first, it was hard
to grasp the project to the whole extent. For that reason and to become
accustomed to OpenGL in general, I started off with basic sphere as can be seen
in Figure 1. Afterwards, the surface was adapted to obtain a realistic-looking
shape with irregularities. Finally, the shape is colored and blended.
The cloud starts as a sphere
The first step towards a complex cloud shape is a
geometrical primitive – a sphere. To construct a sphere from a given number of
rings and sectors, sphere coordinates were used. Additionally, each four
adjacent vertices are connected to two triangles which are then rendered to the
screen by OpenGL. The more vertices the sphere includes the smoother is the
outline. The sphere shown in Figure 1 has 100 rings and 100 sectors.
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| Figure 1 A sphere decomposed into triangles |
Adding noise
In order to distort the primitive shape, some variation has
to be applied to the surface. The basic idea is to apply a displacement factor D thus:
Pnew = P + PC * D
Where P is the original vertex position on the outline of
the sphere, PC is the vector from the center of the sphere to P and Pnew
is the resulting displaced vector. Different approaches that determine the
displacement will have different impacts on the shape. To get a feeling for the
implications of the displacement, a random value was assigned to the
displacement factor D. Figure 2 shows a sphere with randomly displaced
vertices. Obviously, this technique does not yield satisfying results. The
surface is very edgy and the impact of single extreme displacements diminishes
with the number of total vertices. Roughly speaking, the high number of
vertices that is needed for a smoothly curved sphere counterbalances the
randomness.
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| Figure 2 Distorted surface with random noise |
A noise function ought to achieve better results because it
regulates the displacement factor in a more controlled manner. Moreover,
another function is incorporated that creates a turbulence-like approximation
as described by G. Taxén in Cloud
Modeling for Computer Graphics:
float Turbulence (vector p) {t = 0;scale = 1;while (scale > pixelsize) {t += abs (Noise (p / scale) * scale);scale /= 2;}return t;}
Together the
noise and the turbulence function create a more sophisticated displacement
factor D such D = turbulence + noise. The turbulence function takes a 4-dimensional
input consisting of the vertex normal and a time variable to achieve animated
fluctuations over time. The noise function comes from the implementation of
Perlin Noise 4D available at GitHub (https://github.com/ashima/webgl-noise).
The improved result can be seen in Figure 4 (whereas Figure 3 shows a test run
with Perlin Noise 3D which generated slightly worse displacements).
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| Figure 3 Distorted surface with 3D Perlin Noise and turbulence |
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| Figure 4 Distorted surface with 4D Perlin Noise and turbulence |
Finally, the parameters of both functions were tweaked in
many weary tests to achieve a cloud-like shape (Figure 7 displays a snapshot
from the final cloud including the adapted parameters).
Fading the edges
So far, the surface of the cloud has begun to look realistic
with the irregular turbulence. Problematic is that authentic clouds are to some
extent translucent which the current model does not simulate. It is not
sufficient to simply apply a transparency value to the whole shape but instead
the outer edges can be faded out towards the visible background. To create the
illusion of fading edges two imaginary layers are created orthogonal to the
view direction of the camera. The normal of the created planes are parallel to
the view direction and point towards the camera. Figure 5 shows a schematic
depiction of how to select vertices for fading. The distance to the plane of
each point on the sphere can be calculated from the plane equations using the
normals. Each vertex on the sphere that is between the camera and the dashed
plane is visible, each vertex between the planes will be faded and the rest is
invisible.
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| Figure 5 Fading schema with two planes orthogonal to the camera that determine the visibility of vertices |
A percentage value L that describes the location of a vertex
between the planes can be derived from both of the previously calculated
distances. These percentages are mapped to alpha values on a scale from zero to
one for blending. Therefore, a simple linear blending function b(x) = x was
used but did not result in a natural appearance of the cloud due to the curved
shape. Instead, the fading was enhanced with a composed cubical blending
function:
b(x) = 1 - 2 * p^2 if x < 0.5
or b(x) = 2 * p^2 - 4 * p + 2 otherwise
(Mind that this function is continuous and differentiable in
the considered range.)
Figure 6 shows the obtained results on a greyscale where 1
is white and 0 is black. The final version in Figure 7 shows the fading effect
in colored version.
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| Figure 6 Cubical fading for the outer vertices of the cloud (the higher the determined transparency the darker the area) |
Adding color
To colorize the cloud, two different approaches where
chosen. The top of the cloud ought to be a little brighter than the bottom so
depending on the vertical position of a fragment the color is interpolated
between the two defaults which were picked from a picture of a real cloud.
Additionally, the color is based on the noise that transformed a specific realm
of the shape. Therefore, the noise value between zero and one was incorporated
into the process. Figure 7 and Figure 8 show the resulting cloud at different
times.
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| Figure 7 Final cloud including turbulence, fading and color |
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| Figure 8 Final cloud including turbulence, fading and color (this image was captured slightly after Figure 7 showing how the cloud transforms over time) |
The cloud evolved from a geometric primitive to a somewhat
realistic model. In the next step, several basic cloud models are combined to
obtain more complex structures.
References
- Cloud Modeling for Computer Graphics, G. Taxén (1999)
- Realistic and Fast Cloud Rendering, Niniane Wang (2003)
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