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
|
<!DOCTYPE html>
<html>
<!-- Created by GNU Texinfo 7.1.1, https://www.gnu.org/software/texinfo/ -->
<head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
<title>Vector Rotation Matrices (GNU Octave (version 10.3.0))</title>
<meta name="description" content="Vector Rotation Matrices (GNU Octave (version 10.3.0))">
<meta name="keywords" content="Vector Rotation Matrices (GNU Octave (version 10.3.0))">
<meta name="resource-type" content="document">
<meta name="distribution" content="global">
<meta name="Generator" content="makeinfo">
<meta name="viewport" content="width=device-width,initial-scale=1">
<link href="index.html" rel="start" title="Top">
<link href="Concept-Index.html" rel="index" title="Concept Index">
<link href="index.html#SEC_Contents" rel="contents" title="Table of Contents">
<link href="Geometry.html" rel="up" title="Geometry">
<link href="Interpolation-on-Scattered-Data.html" rel="prev" title="Interpolation on Scattered Data">
<style type="text/css">
<!--
a.copiable-link {visibility: hidden; text-decoration: none; line-height: 0em}
div.example {margin-left: 3.2em}
span:hover a.copiable-link {visibility: visible}
strong.def-name {font-family: monospace; font-weight: bold; font-size: larger}
-->
</style>
<link rel="stylesheet" type="text/css" href="octave.css">
</head>
<body lang="en">
<div class="section-level-extent" id="Vector-Rotation-Matrices">
<div class="nav-panel">
<p>
Previous: <a href="Interpolation-on-Scattered-Data.html" accesskey="p" rel="prev">Interpolation on Scattered Data</a>, Up: <a href="Geometry.html" accesskey="u" rel="up">Geometry</a> [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html" title="Index" rel="index">Index</a>]</p>
</div>
<hr>
<h3 class="section" id="Vector-Rotation-Matrices-1"><span>30.5 Vector Rotation Matrices<a class="copiable-link" href="#Vector-Rotation-Matrices-1"> ¶</a></span></h3>
<p>Also included in Octave’s geometry functions are primitive functions to enable
vector rotations in 3-dimensional space. Separate functions are provided for
rotation about each of the principle axes, <var class="var">x</var>, <var class="var">y</var>, and <var class="var">z</var>.
According to Euler’s rotation theorem, any arbitrary rotation, <var class="var">R</var>, of any
vector, <var class="var">p</var>, can be expressed as a product of the three principle
rotations:
</p>
<div class="example">
<pre class="example-preformatted">p' = Rp = Rz*Ry*Rx*p
</pre></div>
<a class="anchor" id="XREFrotx"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-rotx"><span><code class="def-type"><var class="var">T</var> =</code> <strong class="def-name">rotx</strong> <code class="def-code-arguments">(<var class="var">angle</var>)</code><a class="copiable-link" href="#index-rotx"> ¶</a></span></dt>
<dd>
<p><code class="code">rotx</code> returns the 3x3 transformation matrix corresponding to an active
rotation of a vector about the x-axis by the specified <var class="var">angle</var>, given in
degrees, where a positive angle corresponds to a counterclockwise
rotation when viewing the y-z plane from the positive x side.
</p>
<p>The form of the transformation matrix is:
</p>
<div class="example">
<div class="group"><pre class="example-preformatted"> | 1 0 0 |
T = | 0 cos(<var class="var">angle</var>) -sin(<var class="var">angle</var>) |
| 0 sin(<var class="var">angle</var>) cos(<var class="var">angle</var>) |
</pre></div></div>
<p>This rotation matrix is intended to be used as a left-multiplying matrix
when acting on a column vector, using the notation
<code class="code"><var class="var">v</var> = <var class="var">T</var>*<var class="var">u</var></code>.
For example, a vector, <var class="var">u</var>, pointing along the positive y-axis, rotated
90-degrees about the x-axis, will result in a vector pointing along the
positive z-axis:
</p>
<div class="example">
<div class="group"><pre class="example-preformatted">>> u = [0 1 0]'
u =
0
1
0
>> T = rotx (90)
T =
1.00000 0.00000 0.00000
0.00000 0.00000 -1.00000
0.00000 1.00000 0.00000
>> v = T*u
v =
0.00000
0.00000
1.00000
</pre></div></div>
<p><strong class="strong">See also:</strong> <a class="ref" href="#XREFroty">roty</a>, <a class="ref" href="#XREFrotz">rotz</a>.
</p></dd></dl>
<a class="anchor" id="XREFroty"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-roty"><span><code class="def-type"><var class="var">T</var> =</code> <strong class="def-name">roty</strong> <code class="def-code-arguments">(<var class="var">angle</var>)</code><a class="copiable-link" href="#index-roty"> ¶</a></span></dt>
<dd>
<p><code class="code">roty</code> returns the 3x3 transformation matrix corresponding to an active
rotation of a vector about the y-axis by the specified <var class="var">angle</var>, given in
degrees, where a positive angle corresponds to a counterclockwise
rotation when viewing the z-x plane from the positive y side.
</p>
<p>The form of the transformation matrix is:
</p>
<div class="example">
<div class="group"><pre class="example-preformatted"> | cos(<var class="var">angle</var>) 0 sin(<var class="var">angle</var>) |
T = | 0 1 0 |
| -sin(<var class="var">angle</var>) 0 cos(<var class="var">angle</var>) |
</pre></div></div>
<p>This rotation matrix is intended to be used as a left-multiplying matrix
when acting on a column vector, using the notation
<code class="code"><var class="var">v</var> = <var class="var">T</var>*<var class="var">u</var></code>.
For example, a vector, <var class="var">u</var>, pointing along the positive z-axis, rotated
90-degrees about the y-axis, will result in a vector pointing along the
positive x-axis:
</p>
<div class="example">
<div class="group"><pre class="example-preformatted"> >> u = [0 0 1]'
u =
0
0
1
>> T = roty (90)
T =
0.00000 0.00000 1.00000
0.00000 1.00000 0.00000
-1.00000 0.00000 0.00000
>> v = T*u
v =
1.00000
0.00000
0.00000
</pre></div></div>
<p><strong class="strong">See also:</strong> <a class="ref" href="#XREFrotx">rotx</a>, <a class="ref" href="#XREFrotz">rotz</a>.
</p></dd></dl>
<a class="anchor" id="XREFrotz"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-rotz"><span><code class="def-type"><var class="var">T</var> =</code> <strong class="def-name">rotz</strong> <code class="def-code-arguments">(<var class="var">angle</var>)</code><a class="copiable-link" href="#index-rotz"> ¶</a></span></dt>
<dd>
<p><code class="code">rotz</code> returns the 3x3 transformation matrix corresponding to an active
rotation of a vector about the z-axis by the specified <var class="var">angle</var>, given in
degrees, where a positive angle corresponds to a counterclockwise
rotation when viewing the x-y plane from the positive z side.
</p>
<p>The form of the transformation matrix is:
</p>
<div class="example">
<div class="group"><pre class="example-preformatted"> | cos(<var class="var">angle</var>) -sin(<var class="var">angle</var>) 0 |
T = | sin(<var class="var">angle</var>) cos(<var class="var">angle</var>) 0 |
| 0 0 1 |
</pre></div></div>
<p>This rotation matrix is intended to be used as a left-multiplying matrix
when acting on a column vector, using the notation
<code class="code"><var class="var">v</var> = <var class="var">T</var>*<var class="var">u</var></code>.
For example, a vector, <var class="var">u</var>, pointing along the positive x-axis, rotated
90-degrees about the z-axis, will result in a vector pointing along the
positive y-axis:
</p>
<div class="example">
<div class="group"><pre class="example-preformatted"> >> u = [1 0 0]'
u =
1
0
0
>> T = rotz (90)
T =
0.00000 -1.00000 0.00000
1.00000 0.00000 0.00000
0.00000 0.00000 1.00000
>> v = T*u
v =
0.00000
1.00000
0.00000
</pre></div></div>
<p><strong class="strong">See also:</strong> <a class="ref" href="#XREFrotx">rotx</a>, <a class="ref" href="#XREFroty">roty</a>.
</p></dd></dl>
</div>
<hr>
<div class="nav-panel">
<p>
Previous: <a href="Interpolation-on-Scattered-Data.html">Interpolation on Scattered Data</a>, Up: <a href="Geometry.html">Geometry</a> [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html" title="Index" rel="index">Index</a>]</p>
</div>
</body>
</html>
|
