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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 "Wm5ContEllipse2.h"
#include "Wm5ApprGaussPointsFit2.h"
namespace Wm5
{
//----------------------------------------------------------------------------
template <typename Real>
Ellipse2<Real> ContEllipse (int numPoints, const Vector2<Real>* points)
{
// Fit the points with a Gaussian distribution. The covariance matrix
// is M = D[0]*U[0]*U[0]^T+D[1]*U[1]*U[1]^T, where D[0] and D[1] are the
// eigenvalues and U[0] and U[1] are corresponding unit-length
// eigenvectors.
Box2<Real> box = GaussPointsFit2<Real>(numPoints, points);
Real* diag = box.Extent;
Vector2<Real>* axis = box.Axis;
// If either eigenvalue is nonpositive, adjust the D[] values so that
// we actually build an ellipse.
int i;
for (i = 0; i < 2; ++i)
{
if (diag[i] < (Real)0)
{
diag[i] = -diag[i];
}
if (diag[i] < Math<Real>::ZERO_TOLERANCE)
{
diag[i] = Math<Real>::ZERO_TOLERANCE;
}
}
// Grow the ellipse, while retaining its shape determined by the
// covariance matrix, to enclose all the input points. The quadratic form
// that is used for the ellipse construction is
//
// Q(X) = (X-C)^T*M*(X-C)
// = (X-C)^T*(U[0]*U[0]^T/D[0]+U[1]*U[1]^T/D[1])*(X-C)
// = Dot(U[0],X-C)^2/D[0] + Dot(U[1],X-C)^2/D[1]
//
// If the maximum value of Q(X[i]) for all input points is V^2, then a
// bounding ellipse is Q(X) = V^2 since Q(X[i]) <= V^2 for all i.
Real maxValue = (Real)0;
for (i = 0; i < numPoints; ++i)
{
Vector2<Real> diff = points[i] - box.Center;
Real dot[2] = { axis[0].Dot(diff), axis[1].Dot(diff) };
Real value = dot[0]*dot[0]/diag[0] + dot[1]*dot[1]/diag[1];
if (value > maxValue)
{
maxValue = value;
}
}
// Arrange for quadratic to satisfy Q(X) <= 1.
for (i = 0; i < 2; ++i)
{
diag[i] *= maxValue;
}
Ellipse2<Real> ellipse;
ellipse.Center = box.Center;
for (i = 0; i < 2; ++i)
{
ellipse.Axis[i] = axis[i];
ellipse.Extent[i] = Math<Real>::Sqrt(diag[i]);
}
return ellipse;
}
//----------------------------------------------------------------------------
template <typename Real>
void ProjectEllipse (const Ellipse2<Real>& ellipse, const Line2<Real>& line,
Real& smin, Real& smax)
{
// Center of projection interval.
Real center = line.Direction.Dot(ellipse.Center - line.Origin);
// Radius of projection interval.
Real tmp[2] =
{
ellipse.Extent[0]*(line.Direction.Dot(ellipse.Axis[0])),
ellipse.Extent[1]*(line.Direction.Dot(ellipse.Axis[1]))
};
Real rSqr = tmp[0]*tmp[0] + tmp[1]*tmp[1];
Real radius = Math<Real>::Sqrt(rSqr);
smin = center - radius;
smax = center + radius;
}
//----------------------------------------------------------------------------
template <typename Real>
const Ellipse2<Real> MergeEllipses (const Ellipse2<Real>& ellipse0,
const Ellipse2<Real>& ellipse1)
{
Ellipse2<Real> merge;
// Compute the average of the input centers.
merge.Center = ((Real)0.5)*(ellipse0.Center + ellipse1.Center);
// Bounding ellipse orientation is average of input orientations.
if (ellipse0.Axis[0].Dot(ellipse1.Axis[0]) >= (Real)0)
{
merge.Axis[0] = ((Real)0.5)*(ellipse0.Axis[0] + ellipse1.Axis[0]);
merge.Axis[0].Normalize();
}
else
{
merge.Axis[0] = ((Real)0.5)*(ellipse0.Axis[0] - ellipse1.Axis[0]);
merge.Axis[0].Normalize();
}
merge.Axis[1] = -merge.Axis[0].Perp();
// Project the input ellipses onto the axes obtained by the average
// of the orientations and that go through the center obtained by the
// average of the centers.
for (int i = 0; i < 2; ++i)
{
// Projection axis.
Line2<Real> line(merge.Center, merge.Axis[i]);
// Project ellipsoids onto the axis.
Real min0, max0, min1, max1;
ProjectEllipse(ellipse0, line, min0, max0);
ProjectEllipse(ellipse1, line, min1, max1);
// Determine the smallest interval containing the projected
// intervals.
Real maxIntr = (max0 >= max1 ? max0 : max1);
Real minIntr = (min0 <= min1 ? min0 : min1);
// Update the average center to be the center of the bounding box
// defined by the projected intervals.
merge.Center += line.Direction*(((Real)0.5)*(minIntr + maxIntr));
// Compute the extents of the box based on the new center.
merge.Extent[i] = ((Real)0.5)*(maxIntr - minIntr);
}
return merge;
}
//----------------------------------------------------------------------------
//----------------------------------------------------------------------------
// Explicit instantiation.
//----------------------------------------------------------------------------
template WM5_MATHEMATICS_ITEM
Ellipse2<float> ContEllipse<float> (int, const Vector2<float>*);
template WM5_MATHEMATICS_ITEM
void ProjectEllipse<float> (const Ellipse2<float>&, const Line2<float>&,
float&, float&);
template WM5_MATHEMATICS_ITEM
const Ellipse2<float> MergeEllipses<float> (const Ellipse2<float>&,
const Ellipse2<float>&);
template WM5_MATHEMATICS_ITEM
Ellipse2<double> ContEllipse<double> (int, const Vector2<double>*);
template WM5_MATHEMATICS_ITEM
void ProjectEllipse<double> (const Ellipse2<double>&, const Line2<double>&,
double&, double&);
template WM5_MATHEMATICS_ITEM
const Ellipse2<double> MergeEllipses<double> (const Ellipse2<double>&,
const Ellipse2<double>&);
//----------------------------------------------------------------------------
}
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