RS_VectorSolutions RS_Information::getIntersectionEllipseEllipse(RS_Ellipse* e1, RS_Ellipse* e2) {
RS_VectorSolutions ret;
if (e1==NULL || e2==NULL ) {
return ret;
}
if (
(e1->getCenter() - e2 -> getCenter() ).magnitude() < RS_TOLERANCE &&
( e1->getMajorP() - e2 ->getMajorP()).magnitude() < RS_TOLERANCE &&
fabs(e1->getMajorRadius() - e2 ->getMajorRadius()) < RS_TOLERANCE &&
fabs(e1->getMinorRadius() - e2 ->getMinorRadius()) < RS_TOLERANCE
) { // overlapped ellipses, do not do overlap
return ret;
}
RS_Ellipse *e01= ( RS_Ellipse *) e1->clone();
if( e01->getMajorRadius() < e01->getMinorRadius() ) e01->switchMajorMinor();
RS_Ellipse *e02= ( RS_Ellipse *) e2->clone();
if( e02->getMajorRadius() < e02->getMinorRadius() ) e02->switchMajorMinor();
//transform ellipse2 to ellipse1's coordinates
RS_Vector shiftc1=- e01->getCenter();
double shifta1=-e01->getAngle();
e02->move(shiftc1);
e02->rotate(shifta1);
RS_Vector majorP2=e02->getMajorP();
double a1=e01->getMajorRadius();
double b1=e01->getMinorRadius();
double x2=e02->getCenter().x,
y2=e02->getCenter().y;
double a2=e02->getMajorRadius();
double b2=e02->getMinorRadius();
if( e01->getMinorRadius() < RS_TOLERANCE || e01 -> getRatio()< RS_TOLERANCE) {
// treate e01 as a line
RS_LineData ldata0(RS_Vector(-a1,0.),RS_Vector(a1,0.));
RS_Line *l0=new RS_Line(e1->getParent(),ldata0);
ret= getIntersectionLineEllipse(l0, e02);
ret.rotate(-shifta1);
ret.move(-shiftc1);
return ret;
}
if( e02->getMinorRadius() < RS_TOLERANCE || e02 -> getRatio()< RS_TOLERANCE) {
// treate e02 as a line
RS_LineData ldata0(RS_Vector(-a2,0.),RS_Vector(a2,0.));
RS_Line *l0=new RS_Line(e1->getParent(),ldata0);
l0->rotate(RS_Vector(0.,0.),e02->getAngle());
l0->move(e02->getCenter());
ret= getIntersectionLineEllipse(l0, e01);
ret.rotate(-shifta1);
ret.move(-shiftc1);
return ret;
}
//ellipse01 equation:
// x^2/(a1^2) + y^2/(b1^2) - 1 =0
double t2= - e02->getAngle();
//ellipse2 equation:
// ( (x - u) cos(t) - (y - v) sin(t))^2/a^2 + ( (x - u) sin(t) + (y-v) cos(t))^2/b^2 =1
// ( cos^2/a^2 + sin^2/b^2) x^2 +
// ( sin^2/a^2 + cos^2/b^2) y^2 +
// 2 sin cos (1/b^2 - 1/a^2) x y +
// ( ( 2 v sin cos - 2 u cos^2)/a^2 - ( 2v sin cos + 2 u sin^2)/b^2) x +
// ( ( 2 u sin cos - 2 v sin^2)/a^2 - ( 2u sin cos + 2 v cos^2)/b^2) y +
// (u cos - v sin)^2/a^2 + (u sin + v cos)^2/b^2 -1 =0
// detect whether any ellipse radius is zero
double cs=cos(t2),si=sin(t2);
double ucs=x2*cs,usi=x2*si,
vcs=y2*cs,vsi=y2*si;
double cs2=cs*cs,si2=1-cs2;
double tcssi=2.*cs*si;
double ia2=1./(a2*a2),ib2=1./(b2*b2);
// std::cout<<"e1: x^2/("<<a1<<")^2+y^2/("<<b1<<")^2-1 =0\n";
// std::cout<<"e2: ( (x-("<<x2<<"))*("<<cs<<")-(y-("<<y2<<"))*("<<si<<"))^2/"<<a2<<"^2+( ( x - ("<<x2<<"))*("<<si<<")+(y-("<<y2<<"))*("<<cs<<"))^2/"<<b2<<"^2 -1 =0\n";
double mc1=(ucs - vsi)*(ucs-vsi)*ia2+(usi+vcs)*(usi+vcs)*ib2 -1.;
double mb10= ( y2*tcssi - 2.*x2*cs2)*ia2 - ( y2*tcssi+2*x2*si2)*ib2; //x
double mb11= ( x2*tcssi - 2.*y2*si2)*ia2 - ( x2*tcssi+2*y2*cs2)*ib2; //y
double ma100= cs2*ia2 + si2*ib2; // x^2
double ma101= cs*si*(ib2 - ia2); // xy term is 2*ma101*x*y
double ma111= si2*ia2 + cs2*ib2; // y^2
double ma000= 1./(a1*a1),ma011=1./(b1*b1);
// std::cout<<"simplified e1: "<<ma000<<"*x^2 + "<<ma011<<"*y^2 -1 =0\n";
// std::cout<<"simplified e2: "<<ma100<<"*x^2 + 2*("<<ma101<<")*x*y + "<<ma111<<"*y^2 "<<" + ("<<mb10<<")*x + ("<<mb11<<")*y + ("<<mc1<<") =0\n";
// construct the Bezout determinant
double v0=2.*ma000*ma101;
double v2=ma000*mb10;
double v3=ma000*mb11;
double v4=ma000*mc1+ma100;
//double v5= 2.*ma101*ma011;
//double v6= ma000*ma111;
//double v7= 2.*ma101;
double v8= 2.*ma011*mb10;
//double v9= ma100*ma011;
double v1=ma000*ma111-ma100*ma011;
//double v1= v6 - v9;
double u0 = v4*v4-v2*mb10;
double u1 = 2.*(v3*v4-v0*mb10);
double u2 = 2.*(v4*v1-ma101*v0)+v3*v3+0.5*v2*v8;
double u3 = v0*v8+2.*v3*v1;
double u4 = v1*v1+2.*ma101*ma011*v0;
//std::cout<<"u0="<<u0<<"\tu1="<<u1<<"\tu2="<<u2<<"\tu3="<<u3<<"\tu4="<<u4<<std::endl;
//std::cout<<"("<<u4<<")*x^4+("<<u3<<")*x^3+("<<u2<<")*x^2+("<<u1<<")*x+("<<u0<<")=0\n";
double ce[4];
double roots[4];
unsigned int counts=0;
if ( fabs(u4) < 1.0e-75) { // this should not happen
if ( fabs(u3) < 1.0e-75) { // this should not happen
if ( fabs(u2) < 1.0e-75) { // this should not happen
if( fabs(u1) > 1.0e-75) {
counts=1;
roots[0]=-u0/u1;
} else { // can not determine y. this means overlapped, but overlap should have been detected before, therefore return empty set
return ret;
}
} else {
ce[0]=u1/u2;
ce[1]=u0/u2;
//std::cout<<"ce[2]={ "<<ce[0]<<' '<<ce[1]<<" }\n";
counts=RS_Math::quadraticSolver(ce,roots);
}
} else {
ce[0]=u2/u3;
ce[1]=u1/u3;
ce[2]=u0/u3;
//std::cout<<"ce[3]={ "<<ce[0]<<' '<<ce[1]<<' '<<ce[2]<<" }\n";
counts=RS_Math::cubicSolver(ce,roots);
}
} else {
ce[0]=u3/u4;
ce[1]=u2/u4;
ce[2]=u1/u4;
ce[3]=u0/u4;
//std::cout<<"ce[4]={ "<<ce[0]<<' '<<ce[1]<<' '<<ce[2]<<' '<<ce[3]<<" }\n";
counts=RS_Math::quarticSolver(ce,roots);
}
// std::cout<<"Equation for y: y^4";
// for(int i=3; i>=0; i--) {
// std::cout<<"+("<<ce[3-i]<<")";
// if ( i ) {
// std::cout<<"*y^"<<i;
// }else {
// std::cout<<" ==0\n";
// }
// }
if (! counts ) { // no intersection found
return ret;
}
// std::cout<<"counts="<<counts<<": ";
// for(unsigned int i=0;i<counts;i++){
// std::cout<<roots[i]<<" ";
// }
// std::cout<<std::endl;
RS_VectorSolutions vs0(8);
unsigned int ivs0=0;
for(unsigned int i=0; i<counts; i++) {
double y=roots[i];
//double x=(ma100*(ma011*y*y-1.)-ma000*(ma111*y*y+mb11*y+mc1))/(ma000*(2.*ma101*y+mb11));
double x,d=v0*y+v2;
// std::cout<<"d= "<<d<<std::endl;
if( fabs(d)>RS_TOLERANCE*sqrt(RS_TOLERANCE)) {//whether there's x^1 term in bezout determinant
x=-((v1*y+v3)*y+v4 )/d;
if(vs0.getClosestDistance(RS_Vector(x,y),ivs0)>RS_TOLERANCE)
vs0.set(ivs0++, RS_Vector(x,y));
} else { // no x^1 term, have to use x^2 term, then, have to check plus/minus sqrt
x=a1*sqrt(1-y*y*ma011);
if(vs0.getClosestDistance(RS_Vector(x,y),ivs0)>RS_TOLERANCE)
vs0.set(ivs0++, RS_Vector(x,y));
x=-x;
if(vs0.getClosestDistance(RS_Vector(x,y),ivs0)>RS_TOLERANCE)
vs0.set(ivs0++, RS_Vector(x,y));
}
//std::cout<<"eq1="<<ma000*x*x+ma011*y*y-1.<<std::endl;
//std::cout<<"eq2="<<ma100*x*x + 2.*ma101*x*y+ma111*y*y+mb10*x+mb11*y+mc1<<std::endl;
// if (
// fabs(ma100*x*x + 2.*ma101*x*y+ma111*y*y+mb10*x+mb11*y+mc1)< RS_TOLERANCE
// ) {//found
// vs0.set(ivs0++, RS_Vector(x,y));
// }
}
// for(unsigned int j=0; j<vs0.getNumber(); j++) {
// std::cout<<" ( "<<vs0.get(j).x<<" , "<<vs0.get(j).y<<" ) ";
// }
// std::cout<<std::endl;
// std::cout<<"counts= "<<counts<<"\tFound "<<ivs0<<" EllipseEllipse intersections\n";
ret.alloc(ivs0);
for(unsigned i=0; i<ivs0; i++) {
RS_Vector vp=vs0.get(i);
vp.rotate(-shifta1);
vp.move(-shiftc1);
ret.set(i,vp);
}
return ret;
}
RS_VectorSolutions RS_Information::getIntersectionEllipseEllipse(RS_Ellipse* e1, RS_Ellipse* e2) {
RS_VectorSolutions ret;
if (e1==NULL || e2==NULL ) {
return ret;
}
if (
(e1->getCenter() - e2 ->getCenter()).squared() < RS_TOLERANCE2 &&
(e1->getMajorP() - e2 ->getMajorP()).squared() < RS_TOLERANCE2 &&
fabs(e1->getRatio() - e2 ->getRatio()) < RS_TOLERANCE
) { // overlapped ellipses, do not do overlap
return ret;
}
RS_Ellipse ellipse01(NULL,e1->getData());
RS_Ellipse *e01= & ellipse01;
if( e01->getMajorRadius() < e01->getMinorRadius() ) e01->switchMajorMinor();
RS_Ellipse ellipse02(NULL,e2->getData());
RS_Ellipse *e02= &ellipse02;
if( e02->getMajorRadius() < e02->getMinorRadius() ) e02->switchMajorMinor();
//transform ellipse2 to ellipse1's coordinates
RS_Vector shiftc1=- e01->getCenter();
double shifta1=-e01->getAngle();
e02->move(shiftc1);
e02->rotate(shifta1);
RS_Vector majorP2=e02->getMajorP();
double a1=e01->getMajorRadius();
double b1=e01->getMinorRadius();
double x2=e02->getCenter().x,
y2=e02->getCenter().y;
double a2=e02->getMajorRadius();
double b2=e02->getMinorRadius();
if( e01->getMinorRadius() < RS_TOLERANCE || e01 -> getRatio()< RS_TOLERANCE) {
// treate e01 as a line
RS_LineData ldata0(RS_Vector(-a1,0.),RS_Vector(a1,0.));
RS_Line *l0=new RS_Line(e1->getParent(),ldata0);
ret= getIntersectionEllipseLine(l0, e02);
ret.rotate(-shifta1);
ret.move(-shiftc1);
return ret;
}
if( e02->getMinorRadius() < RS_TOLERANCE || e02 -> getRatio()< RS_TOLERANCE) {
// treate e02 as a line
RS_LineData ldata0(RS_Vector(-a2,0.),RS_Vector(a2,0.));
RS_Line *l0=new RS_Line(e1->getParent(),ldata0);
l0->rotate(RS_Vector(0.,0.),e02->getAngle());
l0->move(e02->getCenter());
ret= getIntersectionEllipseLine(l0, e01);
ret.rotate(-shifta1);
ret.move(-shiftc1);
return ret;
}
//ellipse01 equation:
// x^2/(a1^2) + y^2/(b1^2) - 1 =0
double t2= - e02->getAngle();
//ellipse2 equation:
// ( (x - u) cos(t) - (y - v) sin(t))^2/a^2 + ( (x - u) sin(t) + (y-v) cos(t))^2/b^2 =1
// ( cos^2/a^2 + sin^2/b^2) x^2 +
// ( sin^2/a^2 + cos^2/b^2) y^2 +
// 2 sin cos (1/b^2 - 1/a^2) x y +
// ( ( 2 v sin cos - 2 u cos^2)/a^2 - ( 2v sin cos + 2 u sin^2)/b^2) x +
// ( ( 2 u sin cos - 2 v sin^2)/a^2 - ( 2u sin cos + 2 v cos^2)/b^2) y +
// (u cos - v sin)^2/a^2 + (u sin + v cos)^2/b^2 -1 =0
// detect whether any ellipse radius is zero
double cs=cos(t2),si=sin(t2);
double ucs=x2*cs,usi=x2*si,
vcs=y2*cs,vsi=y2*si;
double cs2=cs*cs,si2=1-cs2;
double tcssi=2.*cs*si;
double ia2=1./(a2*a2),ib2=1./(b2*b2);
std::vector<double> m(0,0.);
m.push_back( 1./(a1*a1)); //ma000
m.push_back( 1./(b1*b1)); //ma011
m.push_back(cs2*ia2 + si2*ib2); //ma100
m.push_back(cs*si*(ib2 - ia2)); //ma101
m.push_back(si2*ia2 + cs2*ib2); //ma111
m.push_back(( y2*tcssi - 2.*x2*cs2)*ia2 - ( y2*tcssi+2*x2*si2)*ib2); //mb10
m.push_back( ( x2*tcssi - 2.*y2*si2)*ia2 - ( x2*tcssi+2*y2*cs2)*ib2); //mb11
m.push_back((ucs - vsi)*(ucs-vsi)*ia2+(usi+vcs)*(usi+vcs)*ib2 -1.); //mc1
auto&& vs0=RS_Math::simultaneousQuadraticSolver(m);
shifta1 = - shifta1;
shiftc1 = - shiftc1;
for(int i=0; i<vs0.getNumber(); i++) {
RS_Vector vp=vs0.get(i);
vp.rotate(shifta1);
vp.move(shiftc1);
ret.push_back(vp);
}
return ret;
}
