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To generate 3D-structures I have written a short algorihm. This I would like to discuss. The algorihm uses vectors, which are combined by the dot product (scalar multiplication). Since the scalar product is defined in any dimension, for example, the program also generates structures in 4D - space. The results are attractors of a sort of self-organization.

The sorce file „GraphTest.txt“ generates pixels (x, y). The plot of this points shows the image  „Graph3D.jpg“. The Java-program „Graph4D.jar“ (Graph.zip) creates 4D-structures, which are projected to a 2D picture.



public class Main {

    public static void main(String[] args) {

        /* Example Program Graph
         * Three - Node - Graph  1, 2, 3
         * Two Edges 1 -> 2, 1 -> 3
         * Output: Position (x,y) of Nodes 1,2,3 */


        //********************************************************//
        //  These parameters can be changed:                      //
                     double e = 5.0, s = 1.0;                     //
        //********************************************************//


       
        double radi, dr, x;

        //Initial conditions (x, y, z)
        //Node1
        double x1 = 0.01,  y1 = -0.02, z1 = 0.04;
        //Node2
        double x2 = -0.04, y2 = -0.05, z2 = 0.01;
        //Node3
        double x3 = 0.05,  y3 = -0.04, z3 = -0.01;
        //Edge12
        double rq12, r12, scalar12, dx12 = 1.0, dy12 = -1.0, dz12 = 1.0;
        //Edge13
        double rq13, r13, scalar13, dx13 = -1.0, dy13 = 1.0, dz13 = -1.0;

        double eist = 1.0;
       
        int  N = 150000;  //steps

     //-------------------------------------------------------------------------



     for(int i = 0; i < N; i++) {

     //Edge12-------------------------------------------------------------------
             rq12 = (x2-x1)*(x2-x1)+(y2-y1)*(y2-y1)+(z2-z1)*(z2-z1);

                r12 = Math.sqrt(rq12);
                scalar12 = dx12*(x2-x1)+dy12*(y2-y1)+dz12*(z2-z1);

                if(scalar12 < 0 && eist > e && r12 > s) {
                    radi = (rq12/(s*s)-1)*(1.0-(scalar12*scalar12)/rq12);
                    if(radi < 0.0)  radi= 0.0;

                    x = -scalar12/r12+Math.sqrt(radi);

               dx12 = dx12-x*(x1-x2)/r12;
                    dy12 = dy12-x*(y1-y2)/r12;
                    dz12 = dz12-x*(z1-z2)/r12;

                    dr = Math.sqrt(dx12*dx12+dy12*dy12+dz12*dz12);

                    dx12 = dx12/dr;
                    dy12 = dy12/dr;
                    dz12 = dz12/dr;
          }

    //Edge13------------------------------------------------------------------
                rq13 = (x3-x1)*(x3-x1)+(y3-y1)*(y3-y1)+(z3-z1)*(z3-z1);

                r13 = Math.sqrt(rq13);
                scalar13 = dx13*(x3-x1)+dy13*(y3-y1)+dz13*(z3-z1);

                if(scalar13 < 0 && eist > e && r13 > s) {
                    radi = (rq13/(s*s)-1)*(1.0-(scalar13*scalar13)/rq13);
                    if(radi < 0.0)  radi = 0.0;

                    x = -scalar13/r13+Math.sqrt(radi);

               dx13 = dx13-x*(x1-x3)/r13;
                    dy13 = dy13-x*(y1-y3)/r13;
                    dz13 = dz13-x*(z1-z3)/r13;

                    dr = Math.sqrt(dx13*dx13+dy13*dy13+dz13*dz13);

                    dx13 = dx13/dr;
                    dy13 = dy13/dr;
                    dz13 = dz13/dr;
          }
       //-----------------------------------------------------------------------

         //Boundary value
          eist = r12+r13;

        //Superposition
          x1 = x1+dx12+dx13;
          y1 = y1+dy12+dy13;
          z1 = z1+dz12+dz13;

          x2 = x2-dx12;
          y2 = y2-dy12;
          z2 = z2-dz12;

          x3 = x3-dx13;
          y3 = y3-dy13;
          z3 = z3-dz13;

        //Position of node1, node2 and node3
          System.out.format("%10f %10f %10f %10f %10f %10f %n", x1, y1, x2, y2, x3, y3);
         
          /* Plot (x1, y1)
           *      (x2, y2)
           *      (x3, y3)
           */
       
        }
    }

}

A test render would be appreciated