Separating circle created from Octahedron by removing 6 of its 12 edges

octahedron.separating_circle.scad

look_inside=false;
diff_last=false;
$vpr = [355, 55, 40];
$fn = 25;
$vpt = [0,0,0];
function map_3D(c) = [cos(c[0])*sin(c[1]), sin(c[0])*sin(c[1]), cos(c[1])];
sc = 7.745966692414834 ;
coords =[
[0,90]
, [90,90]
, [180,90]
, [180,180]
, [270,90]
, [0,0]
];
module vertex(_v, c, half=false) {
    p = coords[_v];
    v = map_3D(p) * sc;
    difference(){
        color(c) translate(v) sphere(0.5);
        if (half) {
            la1 = p[0];
            ph1 = 90 - p[1];
            translate([0, 0, 0]) rotate([0, 0, la1]) rotate([0, -ph1, 0])
                translate([sc+0.5, 0]) rotate([90,0,90]) color([0,0,0])
            translate([-0.5,-0.5,-1]) cube([1,1,0.4]);
        }
    }
}
module vtxt(_p1, num) {
    p1 = coords[_p1];
    la1 = p1[0];
    ph1 = 90 - p1[1];
    translate([0, 0, 0]) rotate([0, 0, la1]) rotate([0, -ph1, 0])
        translate([sc+0.5, 0]) rotate([90,0,90]) color([0,0,0])
            linear_extrude(0.01)
    text(str(num), size=0.5, halign="center", valign="center");
}
module edge2(_p1,_p2,_e) {
    p1 = coords[_p1];
    p2 = coords[_p2];
    // al/la/ph: alpha/lambda/phi | lxy/sxy: delta lambda_xy/sigma_xy
    // https://en.wikipedia.org/wiki/Great-circle_navigation#Course
    la1 = p1[0];
    la2 = p2[0];
    l12 = la2 - la1;
    ph1 = 90 - p1[1];
    ph2 = 90 - p2[1];
    al1 = atan2(cos(ph2)*sin(l12), cos(ph1)*sin(ph2)-sin(ph1)*cos(ph2)*cos(l12));
    // delta sigma_12
    // https://en.wikipedia.org/wiki/Great-circle_distance#Formulae
    s12 = acos(sin(ph1)*sin(ph2)+cos(ph1)*cos(ph2)*cos(l12));
    translate([0, 0, 0]) rotate([0, 0, la1]) rotate([0, -ph1, 0])
      rotate([90 - al1, 0, 0])
        rotate_extrude(angle=s12, convexity=10, $fn=100)
            translate([sc, 0]) circle(0.1, $fn=25);
}
difference(){
rotate([0,-$t*360,0]) union(){
color([0,0,1])
edge2( 0 , 1 , 0 );
color([0,0,1])
edge2( 1 , 2 , 1 );
color([0,0,1])
edge2( 2 , 3 , 2 );
color([0,0,1])
edge2( 3 , 4 , 3 );
color([0,0,1])
edge2( 4 , 5 , 4 );
color([0,0,1])
edge2( 5 , 0 , 5 );
vertex( 0 , [0,1,0] , true );
vertex( 1 , [0,1,0] , true );
vertex( 2 , [0,1,0] , true );
vertex( 3 , [0,1,0] , true );
vertex( 4 , [0,1,0] , true );
vertex( 5 , [0,1,0] , true );
difference(){
color([1,1,1, 0.7 ]) translate([0,0,0]) sphere(sc, $fn=180);
if (!diff_last) translate([0,0,0]) sphere(sc-0.1, $fn=180);
}
vtxt( 0 , 0 );
vtxt( 1 , 1 );
vtxt( 2 , 2 );
vtxt( 3 , 3 );
vtxt( 4 , 4 );
vtxt( 5 , 5 );
}
if (diff_last) translate([0,0,0]) sphere(sc-0.1, $fn=180);
if (look_inside) translate([0,0,0]) cube([ 7.745966692414834 , 7.745966692414834 , 7.745966692414834 ]);
}

Left side of 0-1-2-3-4-5-0 spans 0°..270° range.
After adding 360° to coordinates <=180°, right side of 0-1-2-3-4-5-0 spans 270°..540° range.
Solving system of linear equations "mod 2π" is outlined here (Z/pZ with prime p=62831849):
https://math.stackexchange.com/a/4507047/1084297
Fixating only the 6 pole vertices (to span [-1..1] range for x, y and z axis), computing all other sphere polar coordinate vertex positions can be done with solving single system of linear equations "mod 2π".