// radius range
maxNoise += maxNoiseIncrement;
delta = ofNoise(maxNoise) * maxDelta;
// each layer corresponds to a radius
for(int i=0; i<numlayers; (int="" i++){="" i++){="" i<numlayers;="" radnoise[i]="" radnoise[i] +=="" raddoiseincrement;="" layerradi[i]="minRad" laverradi[i])="" minrad="ofNoise(radNoise[i])" raddoise[i])="" {="" }="">
 
Genera Esfera: Interacting with a Trackball Mapped onto a Sphere to Explore
 
Generative Visual Worlds
 
In this case we use 1-dimensional Perlin noise, a function of one single parameter that is increased by a small constant, sometimes called "noise detail", at every call. This gives a random but organic appearance to the movement. The arguments of these Perlin noise calls, together with the number of points and the maximum distance to which two points can connect, define the visual appearance of the sphere. Neither the distribution of points in the layers nor their distribution on the corresponding sphere change. (See figure 3.)
 
Marbles. The second version of Genera Esfera gets its name from its resemblance to the colored glass swirls of a marble. It is built by means of a particle system, the movement of the particles being ruled by a Perlin noise vector field. The vector field is a field defined on a rectangle and wrapped onto the sphere via an equirectangular map. Changing the vector field parameters has a great impact on the particles' trajectories, as shown in Figure 4.
 
This version has been technically challenging, because the aesthetic result we wanted needs a lot more computational performance. We work with a set of 1024 leading particles, evolving over the sphere. At every frame, for each leading particle we draw a tail of 512 particles. Therefore, in real-time, more than half a million particles are drawn. This is accomplished by making use of the OpenGL data structure VBO, or Vertex Buffer Object [3].
 
![img-3.jpeg](images/img003.jpeg)
Figure 4: Same code, different settings, give really distinct appearances. In the settings, the colors and the constant parameters that affect the vector field behavior are set up.(Screenshots.)
 
![img-4.jpeg](images/img004.jpeg)
 
Next we describe how the noise field is computed and how it determines the behavior of the particles. The field is not a simulation of any real vector field, and it is not defined by its physics. Instead, it is algorithmically defined making extensive use of the Perlin noise function. The following steps are performed at every frame for every particle.
 
- The elapsed time is represented by  $t$ . At every frame it is increased by the constant timeStep.
Each particle has what we call "field coordinates"  $(x,y)\in [0,w]\times [0,h]$ , which evolve over time.
Each coordinate of the vector field  $(f_x, f_y) \in [0, 1]^2$  is returned by a Perlin noise function call:
 
$$
f _ {x} = N \left(t + p h a s e, c o m p l e x i t y \cdot \frac {x}{w} + p h a s e, c o m p l e x i t y \cdot \frac {y}{h} + p h a s e\right)
$$
 
$$
f _ {y} = N \left(t - p h a s e, c o m p l e x i t y \cdot \frac {x}{w} - p h a s e, c o m p l e x i t y \cdot \frac {y}{h} - p h a s e\right)
$$
 
The parameter complexity affects the general evolution of subsequent Perlin noise calls, while the parameter phase serves to separate the arguments that define the coordinates  $f_{x}$  and  $f_{y}$ , making them more independent. Notice that we denote by  $N$  the calls to the 3-dimensional Perlin noise function.
 
Barrière and Carreras
 
- The velocity of particle $i$ comes from $f_{x}, f_{y}$ and the time $t$, through a new call to Perlin noise that adds some more randomness. It is also affected by a parameter called pollenMass, a measure of dispersion:
 
$$
\left\{
\begin{array}{l}
v_{x} = speed \cdot (2f_{x} - 1) \\
v_{y} = speed \cdot (2f_{y} - 1)
\end{array}
\right\}, \text{ where } speed = \frac{1 + N(t + i, f_{x}, f_{y})}{pollenMass}
$$
 
- After these computations, the field coordinates of the particle are set to $(x + v_{x}, y + v_{y})$.
- The angles $\theta$ and $\phi$ that give the spherical coordinates of the particle position are obtained from the field coordinates by the equirectangular map $[0, w] \times [0, h] \to [0, 2\pi] \times [-\frac{\pi}{2}, \frac{\pi}{2}]$.
 
Adjusting the parameters timeStep, $w$, $h$, complexity, phase, and pollenMass of the vector field allows us to define a wide range of aesthetically different behaviors (see Figure 4).
 
### Random Distributions and Perlin noise
 
In this section we explain the two types of randomness used in the project. The first type is the uniformly distributed random numbers which are used in many ways over the application, especially when distributing points on the sphere's surface. And the second type is the Perlin noise function, a kind of "controlled randomness", used in the radius variation in Genera Esfera Estrident and in the vector field computation, in Genera Esfera Caniques.
 
![img-5.jpeg](images/img005.jpeg)
Figure 5: Two different textures to see the difference between random (left) and Perlin noise (right).
 
![img-6.jpeg](images/img006.jpeg)
 
**Random numbers.** A random number generator gives pseudo-random numbers uniformly distributed in $[0,1]$. As in most of the generative artworks, this function is intensively used.
 
It is worth mentioning how this function is used to distribute points on the sphere's surface.
 
Notice that choosing the spherical coordinates of a point, $\theta$ and $\phi$, uniformly distributed in $[0, 2\pi]$ and $[0, \pi]$, respectively, does not give a uniform distribution on the sphere. Indeed, with this distribution, we get a higher concentration of points near the poles. Despite this lack of uniformity, we use this distribution in some of our scenes, for aesthetical reasons. For instance, in both screenshots in Figure 4.
 
A uniform distribution is obtained by taking $u$ and $v$ uniformly distributed in $[0,1]$, and then by defining the spherical coordinates as:
 
$$
\theta = 2\pi u
$$
 
$$
\phi = \arccos(2v - 1)
$$
 
This gives us the coordinates $(x, y, z) = (\cos \theta \sin \phi, \sin \theta \sin \phi, \cos \phi,)$ of a random point on the unit sphere.
 
**Observation.** The uniform distribution on the unit sphere has density function $f(\Omega) = \frac{1}{4\pi}$, where $\Omega$ denotes the solid angle [1] (p. 69). Using spherical coordinates, the differential solid angle is $d\Omega = \sin \phi \, d\theta \, d\phi$,
 
252
 
Genera Esfera: Interacting with a Trackball Mapped onto a Sphere to Explore Generative Visual Worlds
 
which gives us the distribution $f(\theta, \phi) = \frac{\sin\phi}{4\pi}$ for the 2-dimensional variable $(\Theta, \Phi)$. Then, the marginal densities are:
 
$$
f_{\Phi}(\phi) = \int_{0}^{2\pi} f(\theta, \phi) \, d\theta = \frac{\sin\phi}{2}, \quad \text{and} \quad f_{\Theta}(\theta) = \int_{0}^{\pi} f(\theta, \phi) \, d\phi = \frac{1}{2\pi}
$$
 
Therefore, the cumulative density functions are:
 
$$
v = F_{\Phi}(\phi) = \int_{0}^{\phi} f_{\Phi}(\phi) \, d\phi = \frac{1 - \cos\phi}{2}, \quad \text{and} \quad u = F_{\Theta}(\theta) = \int_{0}^{\theta} f_{\Theta}(\theta) \, d\theta = \frac{\theta}{2\pi}
$$
 
Our uniform distribution is obtained by choosing $u$ and $v$ uniformly distributed in $[0,1]$, and then define $\theta$ and $\phi$ by $v = F_{\Phi}(\phi)$ and $u = F_{\Theta}(\theta)$.
 
As one of the reviewers of this paper pointed out, there is another elegant method of uniformly distributing points on a sphere, a particular case of Monte Carlo sampling called rejection sampling, consisting of choosing points on a cube centered at the origin, rejecting those that are not inside the sphere with the same center and diameter of the cube, and then normalizing the accepted ones [14]. As this method does not make any use of trigonometric functions, it is likely to be more efficient than the one we use. However, this is not an issue, because the uniform distribution of points on the sphere is performed only at initialization time.
 
**Perlin noise.** The second kind of randomness is called Perlin noise. Ken Perlin is a mathematician who worked in the simulation of textures for computer graphics in films, starting back in 1981 for the known science fiction film *Tron*. In 1985 he devised the so-called Perlin noise algorithm, which has been and still is widely used [11]. In 1997, he won an Academy Award for Technical Achievement from the Academy of Motion Picture Arts and Sciences, an Oscar, for his noise and turbulence procedural texturing techniques in the mentioned film *Tron*.
 
Perlin noise is widely used to define textures and give the objects more natural appearance, to generate organic movement and, in general, to make randomness smooth. It works with one, two or three arguments, and always returns a number between 0 and 1. By means of increasing its arguments, a series of pseudo-random values is produced. Small increases give smoother results, textures and movements.
 
To see how Perlin noise works, we focus on its 3-dimensional version. Perlin noise $N(a,b,c)$ is determined by computing a pseudo-random gradient at each of the eight nearest vertices of $(a,b,c)$ on the integer cubic lattice and then doing spline interpolation.
 
More precisely, for each of the eight points in the set $\{(i,j,k) \mid i \in \{\lfloor a \rfloor, \lfloor a \rfloor + 1\}, j \in \{\lfloor b \rfloor, \lfloor b \rfloor + 1\}, k \in \{\lfloor k \rfloor, \lfloor k \rfloor + 1\}\}$, a pseudo-random gradient $\vec{g}_{i,j,k}$ is generated. Then, the eight values obtained by the dot products $\vec{g}_{i,j,k} \cdot (a - i, b - j, c - k)$ are trilinearly interpolated by $s(a - \lfloor a \rfloor)$, $s(b - \lfloor b \rfloor)$, and $s(c - \lfloor c \rfloor)$, where $s(t) = 6t^5 - 15t^4 + 10t^3$.
 
For more details on Perlin noise, see [11, 12]. Figure 5 illustrates the difference between random and Perlin noise, with a textures drawing example.
 
### Further work
 
Genera Esfera is a work in progress. Still in its infancy, it has enormous potential for future growth. We have four lines of future work:
 
- We have already started a new version of Genera Esfera to explore new animations and visuals.
- Although the vector field we use results in interesting aesthetic variations, other vector fields over the sphere may give interesting results as well.
 
- We plan to recruit musicians to prepare a playlist specially for Genera Esfera. It may be a selection of existing songs, or a list of songs created for the project.
- Finally, we want Genera Esfera to be open to the collaboration of other visual artists. So we plan to involve other creative coders in the design of other versions of the sphere animation.
 
### Conclusion
 
We have presented Genera Esfera, an interactive installation to explore generative worlds, its design process and the steps to develop it. A trackball with two buttons is used to perform the interaction, making the action possibilities readily perceivable by any potential user. Our work shows that clearly mapped interaction and evolving visuals are interesting enough for a specialized visual arts festival and can attract its visitors.
 
We have detailed how random numbers and Perlin noise have been used. By taking two different mathematical approaches, we have been able to create two distinct visual worlds: one, consisting of a set of points randomly distributed over a sphere’s surface and in several layers, wire-connected when at some distance range; the other, consisting of a particle system over the sphere, driven by a noise field. This was one of our initial goals: the generation of different and interesting aesthetics.
 
The installation is the starting point to explore new visual graphic results, to find new uses of random numbers and Perlin noise, to add new mathematical tools to our set, and to exploit their potentiality to create generative worlds.
 
Acknowledgements. The authors are grateful to the two anonymous reviewers, whose comments helped to greatly improve this paper.
 
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