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
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
|
/* from
* @article{MollerHughes99,
author = "Tomas Mller and John F. Hughes",
title = "Efficiently Building a Matrix to Rotate One Vector to Another",
journal = "journal of graphics tools",
volume = "4",
number = "4",
pages = "1-4",
year = "1999",
}
http://jgt.akpeters.com/papers/MollerHughes99/code.html
*/
#include <cmath>
#include <itkVector.h>
#include <itkMatrix.h>
template <typename VectorType, typename MatrixType>
MatrixType
RotationMatrixFromVectors(VectorType from, VectorType to)
{
double EPSILON = 0.000001;
typename VectorType::ValueType e, h, f;
MatrixType mtx;
VectorType v = CrossProduct(from, to);
e = from * to;
f = (e < 0) ? -e : e;
if (f > 1.0 - EPSILON) /* "from" and "to"-vector almost parallel */
{
VectorType ut, vt; /* temporary storage vectors */
VectorType x; /* vector most nearly orthogonal to "from" */
float c1, c2, c3; /* coefficients for later use */
int i, j;
x[0] = (from[0] > 0.0) ? from[0] : -from[0];
x[1] = (from[1] > 0.0) ? from[1] : -from[1];
x[2] = (from[2] > 0.0) ? from[2] : -from[2];
if (x[0] < x[1])
{
if (x[0] < x[2])
{
x[0] = 1.0;
x[1] = x[2] = 0.0;
}
else
{
x[2] = 1.0;
x[0] = x[1] = 0.0;
}
}
else
{
if (x[1] < x[2])
{
x[1] = 1.0;
x[0] = x[2] = 0.0;
}
else
{
x[2] = 1.0;
x[0] = x[1] = 0.0;
}
}
ut[0] = x[0] - from[0];
ut[1] = x[1] - from[1];
ut[2] = x[2] - from[2];
vt[0] = x[0] - to[0];
vt[1] = x[1] - to[1];
vt[2] = x[2] - to[2];
c1 = 2.0 / (ut * ut);
c2 = 2.0 / (vt * vt);
c3 = c1 * c2 * (ut * vt);
for (i = 0; i < 3; i++)
{
for (j = 0; j < 3; j++)
{
mtx[i][j] = -c1 * ut[i] * ut[j] - c2 * vt[i] * vt[j] + c3 * vt[i] * ut[j];
}
mtx[i][i] += 1.0;
}
}
else /* the most common case, unless "from"="to", or "from"=-"to" */
{
/* ...otherwise use this hand optimized version (9 mults less) */
float hvx, hvz, hvxy, hvxz, hvyz;
/* h = (1.0 - e)/DOT(v, v); old code */
h = 1.0 / (1.0 + e); /* optimization by Gottfried Chen */
hvx = h * v[0];
hvz = h * v[2];
hvxy = hvx * v[1];
hvxz = hvx * v[2];
hvyz = hvz * v[1];
mtx[0][0] = e + hvx * v[0];
mtx[0][1] = hvxy - v[2];
mtx[0][2] = hvxz + v[1];
mtx[1][0] = hvxy + v[2];
mtx[1][1] = e + h * v[1] * v[1];
mtx[1][2] = hvyz - v[0];
mtx[2][0] = hvxz - v[1];
mtx[2][1] = hvyz + v[0];
mtx[2][2] = e + hvz * v[2];
}
return mtx;
}
// /*
// * A function for creating a rotation matrix that rotates a vector called
// * "from" into another vector called "to".
// * Input : from[3], to[3] which both must be *normalized* non-zero vectors
// * Output: mtx[3][3] -- a 3x3 matrix in colum-major form
// * Authors: Tomas Möller, John Hughes
// * "Efficiently Building a Matrix to Rotate One Vector to Another"
// * Journal of Graphics Tools, 4(4):1-4, 1999
// */
// void fromToRotation(float from[3], float to[3], float mtx[3][3]) {
// float v[3];
// float e, h, f;
// CROSS(v, from, to);
// e = DOT(from, to);
// f = (e < 0)? -e:e;
// if (f > 1.0 - EPSILON) /* "from" and "to"-vector almost parallel */
// {
// float u[3], v[3]; /* temporary storage vectors */
// float x[3]; /* vector most nearly orthogonal to "from" */
// float c1, c2, c3; /* coefficients for later use */
// int i, j;
// x[0] = (from[0] > 0.0)? from[0] : -from[0];
// x[1] = (from[1] > 0.0)? from[1] : -from[1];
// x[2] = (from[2] > 0.0)? from[2] : -from[2];
// if (x[0] < x[1])
// {
// if (x[0] < x[2])
// {
// x[0] = 1.0; x[1] = x[2] = 0.0;
// }
// else
// {
// x[2] = 1.0; x[0] = x[1] = 0.0;
// }
// }
// else
// {
// if (x[1] < x[2])
// {
// x[1] = 1.0; x[0] = x[2] = 0.0;
// }
// else
// {
// x[2] = 1.0; x[0] = x[1] = 0.0;
// }
// }
// u[0] = x[0] - from[0]; u[1] = x[1] - from[1]; u[2] = x[2] - from[2];
// v[0] = x[0] - to[0]; v[1] = x[1] - to[1]; v[2] = x[2] - to[2];
// c1 = 2.0 / DOT(u, u);
// c2 = 2.0 / DOT(v, v);
// c3 = c1 * c2 * DOT(u, v);
// for (i = 0; i < 3; i++) {
// for (j = 0; j < 3; j++) {
// mtx[i][j] = - c1 * u[i] * u[j]
// - c2 * v[i] * v[j]
// + c3 * v[i] * u[j];
// }
// mtx[i][i] += 1.0;
// }
// }
// else /* the most common case, unless "from"="to", or "from"=-"to" */
// {
// #if 0
// /* unoptimized version - a good compiler will optimize this. */
// /* h = (1.0 - e)/DOT(v, v); old code */
// h = 1.0/(1.0 + e); /* optimization by Gottfried Chen */
// mtx[0][0] = e + h * v[0] * v[0];
// mtx[0][1] = h * v[0] * v[1] - v[2];
// mtx[0][2] = h * v[0] * v[2] + v[1];
// mtx[1][0] = h * v[0] * v[1] + v[2];
// mtx[1][1] = e + h * v[1] * v[1];
// mtx[1][2] = h * v[1] * v[2] - v[0];
// mtx[2][0] = h * v[0] * v[2] - v[1];
// mtx[2][1] = h * v[1] * v[2] + v[0];
// mtx[2][2] = e + h * v[2] * v[2];
// #else
// /* ...otherwise use this hand optimized version (9 mults less) */
// float hvx, hvz, hvxy, hvxz, hvyz;
// /* h = (1.0 - e)/DOT(v, v); old code */
// h = 1.0/(1.0 + e); /* optimization by Gottfried Chen */
// hvx = h * v[0];
// hvz = h * v[2];
// hvxy = hvx * v[1];
// hvxz = hvx * v[2];
// hvyz = hvz * v[1];
// mtx[0][0] = e + hvx * v[0];
// mtx[0][1] = hvxy - v[2];
// mtx[0][2] = hvxz + v[1];
// mtx[1][0] = hvxy + v[2];
// mtx[1][1] = e + h * v[1] * v[1];
// mtx[1][2] = hvyz - v[0];
// mtx[2][0] = hvxz - v[1];
// mtx[2][1] = hvyz + v[0];
// mtx[2][2] = e + hvz * v[2];
// #endif
// }
// }
|
