function subdivideArc(cx, cy, r, start, end, n, invertDirection) {
var diff = (invertDirection) ? start-end : end-start;
var theta = diff / (n - 1);
cx = split(cx);
cy = split(cy);
r = split(r);
var tangent = split(tan(theta));
var radial = split(cos(theta));
var x = mul(r, split(cos(start)));
var y = mul(r, split(sin(start)));
var ret = Array(n);
for (var i=0; i<n; i++) {
ret[i] = [
num(add(x, cx)),
num(add(y, cy))
];
var tx = scale(y, -1);
var ty = x;
x = mul(add(x, mul(tx, tangent)), radial);
y = mul(add(y, mul(ty, tangent)), radial);
}
return ret;
}
function selfIntersections(poly, filterFn) {
var seen = {};
var l = poly.length;
var isects = [];
for (var o=0; o<l; o++) {
var s0 = poly[o];
var e0 = poly[(o+1) % l];
arrayOrObject(s0, oc);
arrayOrObject(e0, on);
for (var p=0; p<l; p++) {
if (o === p) { continue; }
var s1 = poly[p]
var e1 = poly[(p+1) % l];
arrayOrObject(s1, pc);
arrayOrObject(e1, pn);
if (cmp(pc, oc) || cmp(pc, on) || cmp(pn, oc) || cmp(pn, on)) {
continue;
}
var r = isect(oc, on, pc, pn);
// since these are homogeneous vectors, if the last component `w` is 0
// then we've done something wrong
var wraw = r[2];
if (wraw.length === 1 && !wraw[0]) {
continue;
}
var w = float(r[2]);
r[0] = float(r[0]) / w;
r[1] = float(r[1]) / w;
r.pop();
if (cmp(r, oc) || cmp(r, on) || cmp(r, pc) || cmp(r, pn)) {
continue;
}
var key = r+'';
var unique = !seen[key];
if (unique) {
seen[key] = true;
}
var collect = unique;
if (filterFn) {
collect = filterFn(r, o, s0, e0, p, s1, e1, unique);
}
if (collect) {
isects.push(r);
}
}
}
return isects;
}