glChAoS.P - Rampe class of attractors

Rampe class of attractors

Ten types of attractors with similar characteristics

Rampe1 / Rampe2 / Rampe3 / Rampe3mod. / Rampe4 / Rampe5 / Rampe6 / Rampe7 / Rampe8 / Rampe9 / Rampe10

Rampe3 mod.
preset values

Rampe class

a group of attractors with similar characteristics

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) rappresent 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 ∈ R³ 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{x,y,z} = k[j](x,y,z)
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)

Resolution: X render in new window

touchScreen Full Rendering Mode

Rampe1

\begin{align} x_{i+1} & = z_i \centerdot sin(k_{0x} \centerdot x_i) + cos(k_{1x} \centerdot y_i) \\ y_{i+1} & = x_i \centerdot sin(k_{0y} \centerdot y_i) + cos(k_{1y} \centerdot z_i) \\ z_{i+1} & = y_i \centerdot sin(k_{0z} \centerdot z_i) + cos(k_{1z} \centerdot x_i) \\ \end{align}

explore

    pNew.x = p.z*sin(k[0].x*p.x)+cos(k[1].x*p.y);
    pNew.y = p.x*sin(k[0].y*p.y)+cos(k[1].y*p.z);
    pNew.z = p.y*sin(k[0].z*p.z)+cos(k[1].z*p.x);

Rampe2

\begin{align} x_{i+1} & = z_i \centerdot sin(k_{0x} \centerdot x_i) + arccos(k_{1x} \centerdot y_i) \\ y_{i+1} & = x_i \centerdot sin(k_{0y} \centerdot y_i) + arccos(k_{1y} \centerdot z_i) \\ z_{i+1} & = y_i \centerdot sin(k_{0z} \centerdot z_i) + arccos(k_{1z} \centerdot x_i) \\ \end{align}

explore

    pNew.x = p.z*sin(k[0].x*p.x)+acos(k[1].x*p.y);
    pNew.y = p.x*sin(k[0].y*p.y)+acos(k[1].y*p.z);
    pNew.z = p.y*sin(k[0].z*p.z)+acos(k[1].z*p.x);

Rampe3

\begin{align} x_{i+1} & = x_i z_i \centerdot sin(k_{0x} \centerdot x_i) - \arccos(k_{1x} \centerdot y_i) \\ y_{i+1} & = y_i x_i \centerdot sin(k_{0y} \centerdot y_i) - \arccos(k_{1y} \centerdot z_i) \\ z_{i+1} & = z_i y_i \centerdot sin(k_{0z} \centerdot z_i) - \arccos(k_{1z} \centerdot x_i) \\ \end{align}

explore

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

Rampe3 mod.

\begin{align} x_{i+1} & = z_i^2 \centerdot sin(k_{0x} \centerdot x_i) - \arccos(k_{1x} \centerdot y_i) \\ y_{i+1} & = x_i^2 \centerdot sin(k_{0y} \centerdot y_i) - \arccos(k_{1y} \centerdot z_i) \\ z_{i+1} & = y_i^2 \centerdot sin(k_{0z} \centerdot z_i) - \arccos(k_{1z} \centerdot x_i) \\ \end{align}

explore

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

Rampe4

\begin{align} x_{i+1} & = x_i \centerdot sin(k_{0x} \centerdot x_i) + cos(k_{1x} \centerdot y_i) \\ y_{i+1} & = y_i \centerdot sin(k_{0y} \centerdot y_i) + cos(k_{1y} \centerdot z_i) \\ z_{i+1} & = z_i \centerdot sin(k_{0z} \centerdot z_i) + cos(k_{1z} \centerdot x_i) \\ \end{align}

explore

    pNew.x = p.x*sin(k[0].x*p.x)+cos(k[1].x*p.y);
    pNew.y = p.y*sin(k[0].y*p.y)+cos(k[1].y*p.z);
    pNew.z = p.z*sin(k[0].z*p.z)+cos(k[1].z*p.x);

Rampe5

\begin{align} x_{i+1} & = z_i \centerdot sin(k_{0x} \centerdot x_i) + cos(k_{1x} \centerdot y_i) + sin(k_{2x} \centerdot z_i)\\ y_{i+1} & = x_i \centerdot sin(k_{0y} \centerdot x_i) + cos(k_{1y} \centerdot y_i) + sin(k_{2y} \centerdot z_i)\\ z_{i+1} & = y_i \centerdot sin(k_{0z} \centerdot x_i) + cos(k_{1z} \centerdot y_i) + sin(k_{2z} \centerdot z_i)\\ \end{align}

explore

    pNew.x = p.z*sin(k[0].x*p.x)+cos(k[1].x*p.y)+sin(k[2].x*p.z);
    pNew.y = p.x*sin(k[0].y*p.x)+cos(k[1].y*p.y)+sin(k[2].y*p.z);
    pNew.z = p.y*sin(k[0].z*p.x)+cos(k[1].z*p.y)+sin(k[2].z*p.z);

Rampe6

\begin{align} x_{i+1} & = z_i \centerdot sin(k_{0x} \centerdot x_i) - cos(k_{1x} \centerdot y_i) \\ y_{i+1} & = x_i \centerdot sin(k_{0y} \centerdot y_i) + cos(k_{1y} \centerdot z_i) \\ z_{i+1} & = y_i \centerdot sin(k_{0z} \centerdot z_i) - cos(k_{1z} \centerdot x_i) \\ \end{align}

explore

    pNew.x = p.z*sin(k[0].x*p.x)-cos(k[1].x*p.y);
    pNew.y = p.x*sin(k[0].y*p.y)+cos(k[1].y*p.z);
    pNew.z = p.y*sin(k[0].z*p.z)-cos(k[1].z*p.x)

Rampe7

\begin{align} x_{i+1} & = z_i \centerdot sin(k_{0x} \centerdot x_i) - cos(k_{1x} \centerdot y_i) \\ y_{i+1} & = x_i \centerdot cos(k_{0y} \centerdot y_i) + sin(k_{1y} \centerdot z_i) \\ z_{i+1} & = y_i \centerdot sin(k_{0z} \centerdot z_i) - cos(k_{1z} \centerdot x_i) \\ \end{align}

explore

    pNew.x = p.z*sin(k[0].x*p.x)-cos(k[1].x*p.y);
    pNew.y = p.x*cos(k[0].y*p.y)+sin(k[1].y*p.z);
    pNew.z = p.y*sin(k[0].z*p.z)-cos(k[1].z*p.x);

Rampe8

\begin{align} x_{i+1} & = z_i \centerdot sin(k_{0x} \centerdot x_i) - cos(y_i) \\ y_{i+1} & = x_i \centerdot cos(k_{0y} \centerdot y_i) + sin(z_i) \\ z_{i+1} & = y_i \centerdot sin(k_{0z} \centerdot z_i) - cos(x_i) \\ \end{align}

explore

    pNew.x = p.z*sin(k[0].x*p.x)-cos(p.y);
    pNew.y = p.x*cos(k[0].y*p.y)+sin(p.z);
    pNew.z = p.y*sin(k[0].z*p.z)-cos(p.x);

Rampe9

\begin{align} x_{i+1} & = z_i \centerdot sin(k_{0x} \centerdot x_i) - \arccos(k_{1x} \centerdot y_i) + sin(k_{2x} \centerdot z_i)\\ y_{i+1} & = x_i \centerdot sin(k_{0y} \centerdot x_i) - \arccos(k_{1y} \centerdot y_i) + sin(k_{2y} \centerdot z_i)\\ z_{i+1} & = y_i \centerdot sin(k_{0z} \centerdot x_i) - \arccos(k_{1z} \centerdot y_i) + sin(k_{2z} \centerdot z_i)\\ \end{align}

explore

    pNew.x = p.z*sin(k[0].x*p.x)-acos(k[1].x*p.y)+sin(k[2].x*p.z);
    pNew.y = p.x*sin(k[0].y*p.x)-acos(k[1].y*p.y)+sin(k[2].y*p.z);
    pNew.z = p.y*sin(k[0].z*p.x)-acos(k[1].z*p.y)+sin(k[2].z*p.z);

Rampe10

\begin{align} x_{i+1} & = z_i \centerdot sin(k_{0x} \centerdot x_i) - cos(k_{1x} \centerdot y_i) + \arcsin(k_{2x} \centerdot z_i)\\ y_{i+1} & = x_i \centerdot sin(k_{0y} \centerdot x_i) - cos(k_{1y} \centerdot y_i) + sin(k_{2y} \centerdot z_i)\\ z_{i+1} & = y_i \centerdot sin(k_{0z} \centerdot x_i) - cos(k_{1z} \centerdot y_i) + sin(k_{2z} \centerdot z_i)\\ \end{align}

explore

        pNew.x = p.z*p.y*sin(k[0].x*p.x)-cos(k[1].x*p.y)+asin(k[2].x*p.z);
        pNew.y = p.x*p.z*sin(k[0].y*p.x)-cos(k[1].y*p.y)+ sin(k[2].y*p.z);
        pNew.z = p.y*p.x*sin(k[0].z*p.x)-cos(k[1].z*p.y)+ sin(k[2].z*p.z);