bool GetTangentsToCircles ( const Circle2<Real>& circle0,
const Circle2<Real>& circle1, Line2<Real> line[4] )
{
Vector2<Real> W = circle1.Center - circle0.Center;
Real wLenSqr = W.SquaredLength();
Real rSum = circle0.Radius + circle1.Radius;
if ( wLenSqr <= rSum * rSum )
{
// Circles are either intersecting or nested.
return false;
}
Real r0Sqr = circle0.Radius * circle0.Radius;
Real tmp, a;
Real rDiff = circle1.Radius - circle0.Radius;
if ( Math<Real>::FAbs( rDiff ) >= Math<Real>::ZERO_TOLERANCE )
{
// Solve (R1^2-R0^2)*s^2 + 2*R0^2*s - R0^2 = 0..
Real r1Sqr = circle1.Radius * circle1.Radius;
Real c0 = -r0Sqr;
Real c1 = ( ( Real )2 ) * r0Sqr;
Real c2 = circle1.Radius * circle1.Radius - r0Sqr;
Real minusHalfInvC2 = ( ( Real ) - 0.5 ) / c2;
Real discr = Math<Real>::FAbs( c1 * c1 - ( ( Real )4 ) * c0 * c2 );
Real root = Math<Real>::Sqrt( discr );
Real s, oneMinusS;
// Get the first root.
s = ( c1 + root ) * minusHalfInvC2;
line[0].Origin = circle0.Center + s * W;
line[1].Origin = line[0].Origin;
if ( s >= ( Real )0.5 )
{
tmp = Math<Real>::FAbs( wLenSqr - r0Sqr / ( s * s ) );
a = Math<Real>::Sqrt( tmp );
}
else
{
oneMinusS = ( Real )1 - s;
tmp = Math<Real>::FAbs( wLenSqr - r1Sqr / ( oneMinusS * oneMinusS ) );
a = Math<Real>::Sqrt( tmp );
}
GetDirections( W, a, line[0].Direction, line[1].Direction );
// Get the second root.
s = ( c1 - root ) * minusHalfInvC2;
line[2].Origin = circle0.Center + s * W;
line[3].Origin = line[2].Origin;
if ( s >= ( Real )0.5 )
{
tmp = Math<Real>::FAbs( wLenSqr - r0Sqr / ( s * s ) );
a = Math<Real>::Sqrt( tmp );
}
else
{
oneMinusS = ( Real )1 - s;
tmp = Math<Real>::FAbs( wLenSqr - r1Sqr / ( oneMinusS * oneMinusS ) );
a = Math<Real>::Sqrt( tmp );
}
GetDirections( W, a, line[2].Direction, line[3].Direction );
}
else
{
// Circles effectively have same radius.
// Midpoint of circle centers.
Vector2<Real> mid = ( ( Real )0.5 ) * ( circle0.Center + circle1.Center );
// Tangent lines passing through midpoint.
tmp = Math<Real>::FAbs( wLenSqr - ( ( Real )4 ) * r0Sqr );
a = Math<Real>::Sqrt( tmp );
GetDirections( W, a, line[0].Direction, line[1].Direction );
line[0].Origin = mid;
line[1].Origin = mid;
// Normalize W.
W /= Math<Real>::Sqrt( wLenSqr );
// Tangent lines parallel to normalized W:
// 1. D = W
// 2. a. P = mid+R0*perp(W), perp(a,b) = (b,-a)
// b. P = mid-R0*perp(W)
line[2].Origin.X() = mid.X() + circle0.Radius * W.Y();
line[2].Origin.Y() = mid.Y() - circle0.Radius * W.X();
line[2].Direction = W;
line[3].Origin.X() = mid.X() - circle0.Radius * W.Y();
line[3].Origin.Y() = mid.Y() + circle0.Radius * W.X();
line[3].Direction = W;
}
return true;
}
bool GetTangentsToCircles (const Circle2<Real>& rkCircle0,
const Circle2<Real>& rkCircle1, Line2<Real> akLine[4])
{
Vector2<Real> kW = rkCircle1.Center - rkCircle0.Center;
Real fWLenSqr = kW.SquaredLength();
Real fRSum = rkCircle0.Radius + rkCircle1.Radius;
if (fWLenSqr <= fRSum*fRSum)
{
// circles are either intersecting or nested
return false;
}
Real fR0Sqr = rkCircle0.Radius*rkCircle0.Radius;
Real fTmp, fA;
Real fRDiff = rkCircle1.Radius - rkCircle0.Radius;
if (Math<Real>::FAbs(fRDiff) >= Math<Real>::ZERO_TOLERANCE)
{
// solve (R1^2-R0^2)*s^2 + 2*R0^2*s - R0^2 = 0.
Real fR1Sqr = rkCircle1.Radius*rkCircle1.Radius;
Real fC0 = -fR0Sqr;
Real fC1 = ((Real)2.0)*fR0Sqr;
Real fC2 = rkCircle1.Radius*rkCircle1.Radius - fR0Sqr;
Real fMHalfInvC2 = ((Real)-0.5)/fC2;
Real fDiscr = Math<Real>::FAbs(fC1*fC1 - ((Real)4.0)*fC0*fC2);
Real fRoot = Math<Real>::Sqrt(fDiscr);
Real fS, fOmS;
// first root
fS = (fC1 + fRoot)*fMHalfInvC2;
akLine[0].Origin = rkCircle0.Center + fS*kW;
akLine[1].Origin = akLine[0].Origin;
if (fS >= (Real)0.5)
{
fTmp = Math<Real>::FAbs(fWLenSqr - fR0Sqr/(fS*fS));
fA = Math<Real>::Sqrt(fTmp);
}
else
{
fOmS = (Real)1.0 - fS;
fTmp = Math<Real>::FAbs(fWLenSqr - fR1Sqr/(fOmS*fOmS));
fA = Math<Real>::Sqrt(fTmp);
}
GetDirections(kW,fA,akLine[0].Direction,akLine[1].Direction);
// second root
fS = (fC1 - fRoot)*fMHalfInvC2;
akLine[2].Origin = rkCircle0.Center + fS*kW;
akLine[3].Origin = akLine[2].Origin;
if (fS >= (Real)0.5)
{
fTmp = Math<Real>::FAbs(fWLenSqr - fR0Sqr/(fS*fS));
fA = Math<Real>::Sqrt(fTmp);
}
else
{
fOmS = (Real)1.0 - fS;
fTmp = Math<Real>::FAbs(fWLenSqr - fR1Sqr/(fOmS*fOmS));
fA = Math<Real>::Sqrt(fTmp);
}
GetDirections(kW,fA,akLine[2].Direction,akLine[3].Direction);
}
else
{
// circles effectively have same radius
// midpoint of circle centers
Vector2<Real> kMid = ((Real)0.5)*(rkCircle0.Center+rkCircle1.Center);
// tangent lines passing through midpoint
fTmp = Math<Real>::FAbs(fWLenSqr - ((Real)4.0)*fR0Sqr);
fA = Math<Real>::Sqrt(fTmp);
GetDirections(kW,fA,akLine[0].Direction,akLine[1].Direction);
akLine[0].Origin = kMid;
akLine[1].Origin = kMid;
// normalize W
kW /= Math<Real>::Sqrt(fWLenSqr);
// tangent lines parallel to normalized W
// 1. D = W
// 2. a. P = mid+R0*perp(W), perp(a,b) = (b,-a)
// b. P = mid-R0*perp(W)
akLine[2].Origin.X() = kMid.X() + rkCircle0.Radius*kW.Y();
akLine[2].Origin.Y() = kMid.Y() - rkCircle0.Radius*kW.X();
akLine[2].Direction = kW;
akLine[3].Origin.X() = kMid.X() - rkCircle0.Radius*kW.Y();
akLine[3].Origin.Y() = kMid.Y() + rkCircle0.Radius*kW.X();
akLine[3].Direction = kW;
}
return true;
}
