3D attractors are point clouds generate from sequences of numbers p_n{x_n,y_n,z_n} ⇒ p_n ∈ R³, n ∈ N, where n_0→∞ denotes the step of the iteration process starting from a initial p₀{x₀,y₀,z₀} point.
In the cloud each next point is function of the previous one:

\[ \eqalign { x_{i+1} = \xi(x_i, y_i, z_i) & \\ y_{i+1} = \phi(x_i, y_i, z_i) & \\ z_{i+1} = \psi(x_i, y_i, z_i) & } \qquad \Bigg\{ \eqalign { & x, y, z \in R \\ & [0, i, n_{\rightarrow\infty}[ \text{   } \Rightarrow i,n \in N \\ } \]

In the computational code:

• p(x,y,z) represent the i-th point p_i{x_i,y_i,z_i}

• k_j→m are constant values characteristic of any single attractor, where k ∈ R and [0,j,m] ⇒ j,m ∈ N

• pNew ∈ R³ is the new point: p_i+1{x_i+1,y_i+1,z_i+1} that will calculated

In the ATTRACTORS window of glChAoS.P:

• left side panel contains starting point coordinates p₀{x₀,y₀,z₀}

• right side panel contains constant values used in the expression of the current attractor, where: k_j = k[j]

• k_j values can also be generated randomly between [min, k_j, max] interval.

Colors are indicative of point speed: distance between p_i and p_i+1
 
You can to start wglChAoS.P with a specific attractor directly from  explore  button. Select lowResources for low resources devices (e.g. mobile devices)
 

Pickover

\begin{align} x_{i+1} & = sin(k_0 \centerdot y_i) + z_i \centerdot cos(k_1 \centerdot x_i) \\ y_{i+1} & = z_i \centerdot sin(k_2 \centerdot x_i) - cos(k_3 \centerdot y_i) \\ z_{i+1} & = sin( x_i)\\ \end{align}

 

explore

    pNew.x =     sin(kVal[0]*p.y) - p.z*cos(kVal[1]*p.x);
    pNew.y = p.z*sin(kVal[2]*p.x) -     cos(kVal[3]*p.y);
    pNew.z =     sin(p.x)                               ;

SinCos

\begin{align} x_{i+1} & = & cos(k_0 \centerdot x_i) + sin(k_1 \centerdot y_i) - sin(k_2 \centerdot z) \\ y_{i+1} & = & sin(k_3 \centerdot x_i) - cos(k_4 \centerdot y_i) + sin(k_5 \centerdot z) \\ z_{i+1} & = & -cos(k_6 \centerdot x_i) + cos(k_7 \centerdot y_i) + cos(k_8 \centerdot z) \\ \end{align}

 

explore

    pNew.x =  cos(kVal[0]*p.x) + sin(kVal[1]*p.y) - sin(kVal[2]*p.z);
    pNew.y =  sin(kVal[3]*p.x) - cos(kVal[4]*p.y) + sin(kVal[5]*p.z);
    pNew.z = -cos(kVal[6]*p.x) + cos(kVal[7]*p.y) + cos(kVal[8]*p.z);

KingsDream

\begin{align} x_{i+1} & = sin(z_i \centerdot k_0) + k_3 \centerdot sin(x_i \centerdot k_0) \\ y_{i+1} & = sin(x_i \centerdot k_1) + k_4 \centerdot sin(y_i \centerdot k_1) \\ z_{i+1} & = sin(y_i \centerdot k_2) + k_5 \centerdot sin(z_i \centerdot k_2) \\ \end{align}

 

explore

    pNew.x = sin(p.z * kVal[0]) + kVal[3] * sin(p.x * kVal[0]);
    pNew.y = sin(p.x * kVal[1]) + kVal[4] * sin(p.y * kVal[1]);
    pNew.z = sin(p.y * kVal[2]) + kVal[5] * sin(p.z * kVal[2]);