1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
|
// 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 "Wm5ContEllipsoid3.h"
#include "Wm5ApprGaussPointsFit3.h"
#include "Wm5Quaternion.h"
namespace Wm5
{
//----------------------------------------------------------------------------
template <typename Real>
Ellipsoid3<Real> ContEllipsoid (int numPoints, const Vector3<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+D[2]*U[2]*U[2]^T where D[0],
// D[1], and D[2] are the eigenvalues and U[0], U[1], and U[2] are
// corresponding unit-length eigenvectors.
Box3<Real> box = GaussPointsFit3(numPoints, points);
Real* diag = box.Extent;
Vector3<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 < 3; ++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 ellipsoid, while retaining its shape determined by the
// covariance matrix, to enclose all the input points. The quadratic form
// that is used for the ellipsoid construction is
//
// Q(X) = (X-C)^T*M*(X-C)
// = (X-C)^T*(sum_{j=0}^2 U[j]*U[j]^T/D[j])*(X-C)
// = sum_{j=0}^2 Dot(U[i],X-C)^2/D[j]
//
// If the maximum value of Q(X[i]) for all input points is V^2, then a
// bounding ellipsoid is Q(X) = V^2 since Q(X[i]) <= V^2 for all i.
Real maxValue = (Real)0;
for (i = 0; i < numPoints; ++i)
{
Vector3<Real> diff = points[i] - box.Center;
Real dot[3] =
{
axis[0].Dot(diff),
axis[1].Dot(diff),
axis[2].Dot(diff)
};
Real value = dot[0]*dot[0]/diag[0] + dot[1]*dot[1]/diag[1] +
dot[2]*dot[2]/diag[2];
if (value > maxValue)
{
maxValue = value;
}
}
// Arrange for quadratic to satisfy Q(X) <= 1.
for (i = 0; i < 3; ++i)
{
diag[i] *= maxValue;
}
Ellipsoid3<Real> ellipsoid;
ellipsoid.Center = box.Center;
for (i = 0; i < 3; ++i)
{
ellipsoid.Axis[i] = axis[i];
ellipsoid.Extent[i] = Math<Real>::Sqrt(diag[i]);
}
return ellipsoid;
}
//----------------------------------------------------------------------------
template <typename Real>
void ProjectEllipsoid (const Ellipsoid3<Real>& ellipsoid,
const Line3<Real>& line, Real& smin, Real& smax)
{
// Center of projection interval.
Real center = line.Direction.Dot(ellipsoid.Center - line.Origin);
// Radius of projection interval.
Real tmp[3] =
{
ellipsoid.Extent[0]*(line.Direction.Dot(ellipsoid.Axis[0])),
ellipsoid.Extent[1]*(line.Direction.Dot(ellipsoid.Axis[1])),
ellipsoid.Extent[2]*(line.Direction.Dot(ellipsoid.Axis[2]))
};
Real rSqr = tmp[0]*tmp[0] + tmp[1]*tmp[1] + tmp[2]*tmp[2];
Real radius = Math<Real>::Sqrt(rSqr);
smin = center - radius;
smax = center + radius;
}
//----------------------------------------------------------------------------
template <typename Real>
const Ellipsoid3<Real> MergeEllipsoids (const Ellipsoid3<Real>& ellipsoid0,
const Ellipsoid3<Real>& ellipsoid1)
{
Ellipsoid3<Real> merge;
// compute the average of the input centers
merge.Center = ((Real)0.5)*(ellipsoid0.Center + ellipsoid1.Center);
// bounding ellipsoid orientation is average of input orientations
Quaternion<Real> q0(ellipsoid0.Axis), q1(ellipsoid1.Axis);
if (q0.Dot(q1) < (Real)0)
{
q1 = -q1;
}
Quaternion<Real> q = q0 + q1;
q = Math<Real>::InvSqrt(q.Dot(q))*q;
q.ToRotationMatrix(merge.Axis);
// Project the input ellipsoids 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 < 3; ++i)
{
// Projection axis.
Line3<Real> line(merge.Center, merge.Axis[i]);
// Project ellipsoids onto the axis.
Real min0, max0, min1, max1;
ProjectEllipsoid(ellipsoid0, line, min0, max0);
ProjectEllipsoid(ellipsoid1, 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
Ellipsoid3<float> ContEllipsoid<float> (int, const Vector3<float>*);
template WM5_MATHEMATICS_ITEM
void ProjectEllipsoid<float> (const Ellipsoid3<float>&,
const Line3<float>&, float&, float&);
template WM5_MATHEMATICS_ITEM
const Ellipsoid3<float> MergeEllipsoids<float> (const Ellipsoid3<float>&,
const Ellipsoid3<float>&);
template WM5_MATHEMATICS_ITEM
Ellipsoid3<double> ContEllipsoid<double> (int, const Vector3<double>*);
template WM5_MATHEMATICS_ITEM
void ProjectEllipsoid<double> (const Ellipsoid3<double>&,
const Line3<double>&, double&, double&);
template WM5_MATHEMATICS_ITEM
const Ellipsoid3<double> MergeEllipsoids<double> (const Ellipsoid3<double>&,
const Ellipsoid3<double>&);
//----------------------------------------------------------------------------
}
|
