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// Geometric Tools, LLC
// Copyright (c) 1998-2014
// Distributed under the Boost Software License, Version 1.0.
// http://www.boost.org/LICENSE_1_0.txt
// http://www.geometrictools.com/License/Boost/LICENSE_1_0.txt
//
// File Version: 5.0.1 (2010/10/01)
#include "Wm5MathematicsPCH.h"
#include "Wm5ApprGreatCircleFit3.h"
#include "Wm5EigenDecomposition.h"
namespace Wm5
{
//----------------------------------------------------------------------------
template <typename Real>
Vector3<Real> GreatCircleFit3 (int numPoints, const Vector3<Real>* points)
{
// Compute the covariance matrix of the vectors.
Real sumXX = (Real)0, sumXY = (Real)0, sumXZ = (Real)0;
Real sumYY = (Real)0, sumYZ = (Real)0, sumZZ = (Real)0;
for (int i = 0; i < numPoints; i++)
{
Vector3<Real> diff = points[i];
sumXX += diff[0]*diff[0];
sumXY += diff[0]*diff[1];
sumXZ += diff[0]*diff[2];
sumYY += diff[1]*diff[1];
sumYZ += diff[1]*diff[2];
sumZZ += diff[2]*diff[2];
}
Real invNumPoints = ((Real)1)/numPoints;
sumXX *= invNumPoints;
sumXY *= invNumPoints;
sumXZ *= invNumPoints;
sumYY *= invNumPoints;
sumYZ *= invNumPoints;
sumZZ *= invNumPoints;
// Set up the eigensolver.
EigenDecomposition<Real> esystem(3);
esystem(0,0) = sumXX;
esystem(0,1) = sumXY;
esystem(0,2) = sumXZ;
esystem(1,0) = esystem(0,1);
esystem(1,1) = sumYY;
esystem(1,2) = sumYZ;
esystem(2,0) = esystem(0,2);
esystem(2,1) = esystem(1,2);
esystem(2,2) = sumZZ;
// Compute eigenstuff; the smallest eigenvalue is in last position.
esystem.Solve(false);
// Unit-length direction for best-fit great circle.
Vector3<Real> normal = esystem.GetEigenvector3(2);
return normal;
}
//----------------------------------------------------------------------------
template <typename Real>
GreatArcFit3<Real>::GreatArcFit3 (int numPoints,
const Vector3<Real>* points, Vector3<Real>& normal,
Vector3<Real>& arcEnd0, Vector3<Real>& arcEnd1)
{
// Get the least-squares great circle for the vectors. The circle is on
// the plane Dot(N,X) = 0.
normal = GreatCircleFit3<Real>(numPoints, points);
// Compute a coordinate system to allow projection of the vectors onto
// the great circle. The coordinates axes have directions U, V, and N.
Vector3<Real> U, V;
Vector3<Real>::GenerateComplementBasis(U, V, normal);
// The vectors are X[i] = u[i]*U + v[i]*V + w[i]*N. The projections
// are P[i] = (u[i]*U + v[i]*V)/sqrt(u[i]*u[i] + v[i]*v[i]). The great
// circle is parameterized by C(t) = cos(t)*U + sin(t)*V. Compute the
// angles t in [-pi,pi] for the projections onto the great circle. It
// is not necesarily to normalize (u[i],v[i]), instead computing
// t = atan2(v[i],u[i]).
std::vector<Item> items(numPoints);
int i;
for (i = 0; i < numPoints; ++i)
{
Item& item = items[i];
item.U = U.Dot(points[i]);
item.V = V.Dot(points[i]);
item.Angle = Math<Real>::ATan2(item.V, item.U);
}
std::sort(items.begin(), items.end());
// Locate the pair of consecutive angles whose difference is a maximum.
// Effectively, we are constructing a cone of minimum angle that contains
// the unit-length vectors.
int numPointsM1 = numPoints - 1;
Real maxDiff = Math<Real>::TWO_PI + items[0].Angle -
items[numPointsM1].Angle;
int end0 = 0, end1 = numPointsM1;
for (int i0 = 0, i1 = 1; i0 < numPointsM1; i0 = i1++)
{
Real diff = items[i1].Angle - items[i0].Angle;
if (diff > maxDiff)
{
maxDiff = diff;
end0 = i1;
end1 = i0;
}
}
arcEnd0 = items[end0].U*U + items[end0].V*V;
arcEnd1 = items[end1].U*U + items[end1].V*V;
arcEnd0.Normalize();
arcEnd1.Normalize();
}
//----------------------------------------------------------------------------
//----------------------------------------------------------------------------
// GreatArcFit3::Item
//----------------------------------------------------------------------------
template <typename Real>
bool GreatArcFit3<Real>::Item::operator< (const Item& item) const
{
return Angle < item.Angle;
}
//----------------------------------------------------------------------------
//----------------------------------------------------------------------------
// Explicit instantiation.
//----------------------------------------------------------------------------
template WM5_MATHEMATICS_ITEM
Vector3<float> GreatCircleFit3<float> (int, const Vector3<float>*);
template WM5_MATHEMATICS_ITEM
class GreatArcFit3<float>;
template WM5_MATHEMATICS_ITEM
Vector3<double> GreatCircleFit3<double> (int, const Vector3<double>*);
template WM5_MATHEMATICS_ITEM
class GreatArcFit3<double>;
//----------------------------------------------------------------------------
}
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