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
|
/*
* Licensed to the Apache Software Foundation (ASF) under one or more
* contributor license agreements. See the NOTICE file distributed with
* this work for additional information regarding copyright ownership.
* The ASF licenses this file to You under the Apache License, Version 2.0
* (the "License"); you may not use this file except in compliance with
* the License. You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*/
package org.apache.commons.math3.geometry.euclidean.threed;
import java.util.Arrays;
import java.util.List;
import org.apache.commons.math3.fraction.BigFraction;
import org.apache.commons.math3.geometry.enclosing.EnclosingBall;
import org.apache.commons.math3.geometry.enclosing.SupportBallGenerator;
import org.apache.commons.math3.geometry.euclidean.twod.DiskGenerator;
import org.apache.commons.math3.geometry.euclidean.twod.Euclidean2D;
import org.apache.commons.math3.geometry.euclidean.twod.Vector2D;
import org.apache.commons.math3.util.FastMath;
/** Class generating an enclosing ball from its support points.
* @since 3.3
*/
public class SphereGenerator implements SupportBallGenerator<Euclidean3D, Vector3D> {
/** {@inheritDoc} */
public EnclosingBall<Euclidean3D, Vector3D> ballOnSupport(final List<Vector3D> support) {
if (support.size() < 1) {
return new EnclosingBall<Euclidean3D, Vector3D>(Vector3D.ZERO, Double.NEGATIVE_INFINITY);
} else {
final Vector3D vA = support.get(0);
if (support.size() < 2) {
return new EnclosingBall<Euclidean3D, Vector3D>(vA, 0, vA);
} else {
final Vector3D vB = support.get(1);
if (support.size() < 3) {
return new EnclosingBall<Euclidean3D, Vector3D>(new Vector3D(0.5, vA, 0.5, vB),
0.5 * vA.distance(vB),
vA, vB);
} else {
final Vector3D vC = support.get(2);
if (support.size() < 4) {
// delegate to 2D disk generator
final Plane p = new Plane(vA, vB, vC,
1.0e-10 * (vA.getNorm1() + vB.getNorm1() + vC.getNorm1()));
final EnclosingBall<Euclidean2D, Vector2D> disk =
new DiskGenerator().ballOnSupport(Arrays.asList(p.toSubSpace(vA),
p.toSubSpace(vB),
p.toSubSpace(vC)));
// convert back to 3D
return new EnclosingBall<Euclidean3D, Vector3D>(p.toSpace(disk.getCenter()),
disk.getRadius(), vA, vB, vC);
} else {
final Vector3D vD = support.get(3);
// a sphere is 3D can be defined as:
// (1) (x - x_0)^2 + (y - y_0)^2 + (z - z_0)^2 = r^2
// which can be written:
// (2) (x^2 + y^2 + z^2) - 2 x_0 x - 2 y_0 y - 2 z_0 z + (x_0^2 + y_0^2 + z_0^2 - r^2) = 0
// or simply:
// (3) (x^2 + y^2 + z^2) + a x + b y + c z + d = 0
// with sphere center coordinates -a/2, -b/2, -c/2
// If the sphere exists, a b, c and d are a non zero solution to
// [ (x^2 + y^2 + z^2) x y z 1 ] [ 1 ] [ 0 ]
// [ (xA^2 + yA^2 + zA^2) xA yA zA 1 ] [ a ] [ 0 ]
// [ (xB^2 + yB^2 + zB^2) xB yB zB 1 ] * [ b ] = [ 0 ]
// [ (xC^2 + yC^2 + zC^2) xC yC zC 1 ] [ c ] [ 0 ]
// [ (xD^2 + yD^2 + zD^2) xD yD zD 1 ] [ d ] [ 0 ]
// So the determinant of the matrix is zero. Computing this determinant
// by expanding it using the minors m_ij of first row leads to
// (4) m_11 (x^2 + y^2 + z^2) - m_12 x + m_13 y - m_14 z + m_15 = 0
// So by identifying equations (2) and (4) we get the coordinates
// of center as:
// x_0 = +m_12 / (2 m_11)
// y_0 = -m_13 / (2 m_11)
// z_0 = +m_14 / (2 m_11)
// Note that the minors m_11, m_12, m_13 and m_14 all have the last column
// filled with 1.0, hence simplifying the computation
final BigFraction[] c2 = new BigFraction[] {
new BigFraction(vA.getX()), new BigFraction(vB.getX()),
new BigFraction(vC.getX()), new BigFraction(vD.getX())
};
final BigFraction[] c3 = new BigFraction[] {
new BigFraction(vA.getY()), new BigFraction(vB.getY()),
new BigFraction(vC.getY()), new BigFraction(vD.getY())
};
final BigFraction[] c4 = new BigFraction[] {
new BigFraction(vA.getZ()), new BigFraction(vB.getZ()),
new BigFraction(vC.getZ()), new BigFraction(vD.getZ())
};
final BigFraction[] c1 = new BigFraction[] {
c2[0].multiply(c2[0]).add(c3[0].multiply(c3[0])).add(c4[0].multiply(c4[0])),
c2[1].multiply(c2[1]).add(c3[1].multiply(c3[1])).add(c4[1].multiply(c4[1])),
c2[2].multiply(c2[2]).add(c3[2].multiply(c3[2])).add(c4[2].multiply(c4[2])),
c2[3].multiply(c2[3]).add(c3[3].multiply(c3[3])).add(c4[3].multiply(c4[3]))
};
final BigFraction twoM11 = minor(c2, c3, c4).multiply(2);
final BigFraction m12 = minor(c1, c3, c4);
final BigFraction m13 = minor(c1, c2, c4);
final BigFraction m14 = minor(c1, c2, c3);
final BigFraction centerX = m12.divide(twoM11);
final BigFraction centerY = m13.divide(twoM11).negate();
final BigFraction centerZ = m14.divide(twoM11);
final BigFraction dx = c2[0].subtract(centerX);
final BigFraction dy = c3[0].subtract(centerY);
final BigFraction dz = c4[0].subtract(centerZ);
final BigFraction r2 = dx.multiply(dx).add(dy.multiply(dy)).add(dz.multiply(dz));
return new EnclosingBall<Euclidean3D, Vector3D>(new Vector3D(centerX.doubleValue(),
centerY.doubleValue(),
centerZ.doubleValue()),
FastMath.sqrt(r2.doubleValue()),
vA, vB, vC, vD);
}
}
}
}
}
/** Compute a dimension 4 minor, when 4<sup>th</sup> column is known to be filled with 1.0.
* @param c1 first column
* @param c2 second column
* @param c3 third column
* @return value of the minor computed has an exact fraction
*/
private BigFraction minor(final BigFraction[] c1, final BigFraction[] c2, final BigFraction[] c3) {
return c2[0].multiply(c3[1]).multiply(c1[2].subtract(c1[3])).
add(c2[0].multiply(c3[2]).multiply(c1[3].subtract(c1[1]))).
add(c2[0].multiply(c3[3]).multiply(c1[1].subtract(c1[2]))).
add(c2[1].multiply(c3[0]).multiply(c1[3].subtract(c1[2]))).
add(c2[1].multiply(c3[2]).multiply(c1[0].subtract(c1[3]))).
add(c2[1].multiply(c3[3]).multiply(c1[2].subtract(c1[0]))).
add(c2[2].multiply(c3[0]).multiply(c1[1].subtract(c1[3]))).
add(c2[2].multiply(c3[1]).multiply(c1[3].subtract(c1[0]))).
add(c2[2].multiply(c3[3]).multiply(c1[0].subtract(c1[1]))).
add(c2[3].multiply(c3[0]).multiply(c1[2].subtract(c1[1]))).
add(c2[3].multiply(c3[1]).multiply(c1[0].subtract(c1[2]))).
add(c2[3].multiply(c3[2]).multiply(c1[1].subtract(c1[0])));
}
}
|
