glChAoS.P - Magnetic class of attractors

Magnetic class of attractors

with point components rotation/permutation

Magnetic base algorithm

MagneticRight
Flower_0009nd.sca

Magnetic Left

magnetic class with point components rotation/permutation

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}
m is the number of magnets, with m ∈ N
v_j→m are constant values characteristic of any single attractor, where v ∈ R³ and [0,j,m] ⇒ j ∈ N
k_j→m are constant values characteristic of any single attractor, where k ∈ R³ and [0,j,m] ⇒ j ∈ 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:

starting point coordinates are p₀{x₀,y₀,z₀} = {0,0,0}
left side panel contains constant values used in the expression of the current attractor, where: v_j{x,y,z} = v[j](x,y,z)
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)
v_j and k_j values can also be generated randomly between [min, v_j, max] and [min, k_j, max] interval.

Colors are indicative of point speed: distance between p_i and p_i+1

Magnetic base algorithm

\[ \eqalign { p_{i+1} & = \sum_{j=0}^{m-1} \Bigg( \eqalign { \vec a & = \vec v_j - \vec p_i, \text{ } & \vec b & = \vec k_j \centerdot \eqalign { \vec a \above 1pt (\vec a \odot \vec a) }, \text{ } & f(\vec b) & } } \Bigg) \quad \Rightarrow \quad \Biggm\{ \eqalign { & [0, i, n_{\rightarrow\infty}[ \text{ } \Rightarrow i,n \in N \\ & [0, j, m[ \text{ } \Rightarrow j,m \in N \rightarrow m = \text{ number of magnets } \\ & \vec a,\vec b,\vec p,\vec v,\vec k \in R^3 \\ & f(\vec w) \rightarrow \text{ transformation function } } \]

where f(w) is, in base to selection, a functuon that shift/rotate/permutate the vector/point components, in base to iteration index i:

Magnetic rotation right

switch(i%3) {
  case 0 : pNew = p;
  case 1 : pNew = vec3(p.z,p.x,p.y);
  case 2 : pNew = vec3(p.y,p.z,p.x);
}

Magnetic rotation left

switch(i%3) {
  case 0 : pNew = p;
  case 1 : pNew = vec3(p.y,p.z,p.x);
  case 2 : pNew = vec3(p.z,p.x,p.y);
}

Magnetic full permutated

switch(i%6) {
  case 0 : pNew = p;
  case 1 : pNew = vec3(p.y,p.z,p.x);
  case 2 : pNew = vec3(p.z,p.x,p.y);
  case 3 : pNew = vec3(p.x,p.z,p.y);
  case 4 : pNew = vec3(p.z,p.y,p.x);
  case 5 : pNew = vec3(p.y,p.x,p.z);}

Magnetic straight

// ever... no transformation !
  pNew = p;

You can to start wglChAoS.P with a specific attractor directly from explore button. Select lowResources for low resources devices (e.g. mobile devices)

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Other examples of attractors belonging to this class:

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