Magnetic class of attractors


with point components rotation/permutation



MagneticRight
Flower_0009nd.sca

Magnetic Left

magnetic class with point components rotation/permutation

3D attractors are point clouds generate from sequences of numbers pn{xn,yn,zn} ⇒ pnR3, nN, where n0→∞ 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:


 

In the ATTRACTORS window of glChAoS.P:


Colors are indicative of point speed: distance between pi and pi+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)
 

Resolution: X        fixed canvas
 
  touchScreen         lowResources

 

 

 

 

 

 

 

Other examples of attractors belonging to this class: