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polygone.js
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polygone.js
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/*
[x,y] is a point (array of two reals)
[x1,y1,x2,y2] is a segment (array of four reals)
[x1,y1,x2,y2,...,xn,yn] is a polygon (array of 2n reals)
[m11,m21,m12,m22] is 2x2 matrix (array of four reals)
*/
// -------------------------------------------------
// add the new array a to this
Array.prototype.pushArray = function(a){
Array.prototype.push.apply(this,a);
}
// -------------------------------------------------
// return the determinant of a matrix
function determinant(m11,m21,m12,m22)
{
return m11*m22 - m12*m21;
}
// -------------------------------------------------
// return the dual point of a segment (line) s
// if the line pass throw zero returns [] (empty array)
function dualPoint(x1,y1,x2,y2) {
var d = determinant(x1,y1,x2,y2);
return d ? [(y2-y1)/d, (x1-x2)/d] : []
}
// -------------------------------------------------
// return the dual polygon of a polygon `poly`
function dualPolygon(poly) {
var n = poly ? poly.length : 0; // number of vertices = number of edges
var dualPoly = []; // the dual polygon will be stored inside
for (var i = n-2, j = 0; i >= 0; i-=2, j = i+2) {
dualPoly.pushArray(dualPoint(poly[i],poly[i+1],poly[j],poly[j+1]));
}
return dualPoly;
}
// -------------------------------------------------
// return the volume of a polygon `poly`
function volume(poly){
var n = poly.length; // number of points
var sumdet = 0; // the volume x2 will be here
for (var i = n-2, j = 0; i >= 0; i-=2, j = i+2) {
sumdet += Math.abs(determinant(poly[i],poly[i+1],poly[j],poly[j+1]));
}
return sumdet/2;
}
// -------------------------------------------------
// return the centroid of a polygon `poly`
function centroid(poly){
var n = poly.length; // number of points
var d; // temporary determinant
var a = 0; // the total area (if 0 is inside)
var x = 0, y = 0; // (x/3/a,y/3/a) will be the centroid
for (var i = n-2, j = 0; i >= 0; i-=2, j = i+2) {
d = Math.abs(determinant(poly[i],poly[i+1],poly[j],poly[j+1]));
a += d;
x += (poly[i]+poly[j])*d;
y += (poly[i+1]+poly[j+1])*d;
}
return a ? [x/3/a, y/3/a] : [];
}
// -------------------------------------------------
// apply linear transform m to polygon `poly`
function transform(poly,m){
var n = poly.length; // number of points
var x,y; // temp variables
for (var i = 0; i < n-1; i+=2) {
x = poly[i];
y = poly[i+1];
poly[i] = m[0]*x + m[2]*y;
poly[i+1] = m[1]*x + m[3]*y;
}
}
// -------------------------------------------------
// translate by v the polygon `poly`
function translate(poly,v){
var n = poly.length; // number of points
for (var i = 0; i < n-1; i+=2) {
poly[i] = poly[i]+v[0];
poly[i+1] = poly[i+1]+v[1];
}
}
// -------------------------------------------------
// put the point pt at 0 = translate by -pt the polygon `poly`
function atzero(poly,pt){
var n = poly.length; // number of points
for (var i = 0; i < n-1; i+=2) {
poly[i] = poly[i]-pt[0];
poly[i+1] = poly[i+1]-pt[1];
}
}
// =================================================
// Hessian calculations (Minkowski tensor)
// =================================================
// -------------------------------------------------
// functions used in xzx
function x2(px,qx) {
return px*px+px*qx+qx*qx;
}
function xy(px,py,qx,qy) {
return px*py+(px*qy+qx*py)/2+qx*qy;
}
// -------------------------------------------------
// return the Minkowski tensor Z.ZdZ
function zxz(poly){
var n = poly.length; // number of points
var d; // temporary determinant
var z2x2 = 0, z2y2 = 0, z2xy=0; // the zxz will be [x2 xy; xy y2]
for (var i = n-2, j = 0; i >= 0; i-=2, j = i+2) {
d = Math.abs(determinant(poly[i],poly[i+1],poly[j],poly[j+1]));
z2x2 += d*x2(poly[i],poly[j]);
z2y2 += d*x2(poly[i+1],poly[j+1]);
z2xy += d*xy(poly[i],poly[i+1],poly[j],poly[j+1]);
}
return [z2x2/12,z2xy/12,z2xy/12,z2y2/12];
}
// -------------------------------------------------
// return the Lutwak-Yang-Zhang tensor
function lyz(poly){
var n = poly.length; // number of points
var d; // temporary determinant
var a = 0; // the total area (if 0 is inside)
var lyz_x2 = 0, lyz_y2 = 0, lyz_xy=0; // the lyz will be [x2 xy; xy y2]
for (var i = n-2, j = 0; i >= 0; i-=2, j = i+2) {
d = Math.abs(determinant(poly[i],poly[i+1],poly[j],poly[j+1]));
if (d==0) return [0,0,0,0];
a += d;
lyz_x2 += (poly[i+1]-poly[j+1])*(poly[i+1]-poly[j+1])/d;
lyz_y2 += (poly[i]-poly[j])*(poly[i]-poly[j])/d;
lyz_xy += (poly[i]-poly[j])*(poly[j+1]-poly[i+1])/d;
}
a = a/2;
return [lyz_x2/a,lyz_xy/a,lyz_xy/a,lyz_y2/a];
}
// -------------------------------------------------
// calculate the product of two matrices
// [a0,a2] [b0,b2]
// [a1,a3] [b1,b3]
function matrixByMatrix(a,b){
return [
a[0]*b[0]+a[2]*b[1],
a[1]*b[0]+a[3]*b[1],
a[0]*b[2]+a[2]*b[3],
a[1]*b[2]+a[3]*b[3]
];
}
// -------------------------------------------------
// calculate the product of matrix by scalar
function matrixByScalar(k,a){
return [k*a[0],k*a[1],k*a[2],k*a[3]];
}
// -------------------------------------------------
// calculate from quadratic form matrix the ellipse pareters
// and return {rx,ry,theta} where theta (in radians) is the angle with rx
function matrix2ellipse(q,c){
var e = [q[0]-c[0]*c[0],q[1]-c[0]*c[1],q[2]-c[1]*c[0],q[3]-c[1]*c[1]];
// if diagonal matrix
if (e[1] == 0){
return {rx:Math.sqrt(e[0]),ry:Math.sqrt(e[3]),theta:0};
}
var tr = e[0]+e[3];
var det = e[0]*e[3]-e[1]*e[1];
var delta = Math.sqrt(Math.abs(tr*tr-4*det)); // abs for preventing precision errors
var lx = (tr+delta)/2;
var ly = (tr-delta)/2;
var theta = Math.atan((lx-e[0])/e[1]);
return {rx:Math.sqrt(lx),ry:Math.sqrt(ly),theta:theta};
}