-
Notifications
You must be signed in to change notification settings - Fork 2
/
Copy pathclassCGAL_1_1Aff__transformation__2.html
559 lines (545 loc) · 44.6 KB
/
classCGAL_1_1Aff__transformation__2.html
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
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "https://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
<html xmlns="http://www.w3.org/1999/xhtml">
<head>
<link rel="canonical" href="https://doc.cgal.org/latest/Kernel_23/classCGAL_1_1Aff__transformation__2.html"/>
<link rel="icon" type="image/png" href="../Manual/g-196x196-doc.png">
<meta http-equiv="Content-Type" content="text/xhtml;charset=UTF-8">
<meta http-equiv="X-UA-Compatible" content="IE=11">
<meta name="generator" content="Doxygen 1.9.6">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>CGAL 6.1 - 2D and 3D Linear Geometry Kernel: CGAL::Aff_transformation_2< Kernel > Class Template Reference</title>
<!-- <link href="../Manual/tabs.css" rel="stylesheet" type="text/css"/> -->
<script type="text/javascript" src="../Manual/jquery.js"></script>
<script type="text/javascript" src="../Manual/dynsections.js"></script>
<script src="../Manual/hacks.js" type="text/javascript"></script>
<!-- Manually include treeview and search to avoid bloat and to fix
paths to the directory Manual . -->
<!-- $.treeview -->
<!-- $.search -->
<link href="navtree.css" rel="stylesheet" type="text/css">
<script type="text/javascript" src="../Manual/resize.js"></script>
<script type="text/javascript" src="navtreedata.js"></script>
<script type="text/javascript" src="navtree.js"></script>
<script type="text/javascript">
$(document).ready(initResizable);
</script>
<link href="../Manual/search/search.css" rel="stylesheet" type="text/css">
<script type="text/javascript" src="../Manual/search/searchdata.js"></script>
<script type="text/javascript" src="../Manual/search/search.js"></script>
<script type="text/javascript">
$(document).ready(function() { init_search(); });
</script>
<link href="../Manual/search/search.css" rel="stylesheet" type="text/css">
<script type="text/javascript" src="../Manual/search/search.js"></script>
<!-- Manually done below. -->
<link href="../Manual/doxygen.css" rel="stylesheet" type="text/css">
<link href="../Manual/cgal_stylesheet.css" rel="stylesheet" type="text/css">
<script type="text/x-mathjax-config">
MathJax.Hub.Config({
extensions: ["tex2jax.js", "TeX/AMSmath.js", "TeX/AMSsymbols.js"],
jax: ["input/TeX","output/HTML-CSS"],
});
//<![CDATA[
MathJax.Hub.Config(
{
TeX: {
Macros: {
qprel: [ "{\\gtreqless}", 0],
qpx: [ "{\\mathbf{x}}", 0],
qpl: [ "{\\mathbf{l}}", 0],
qpu: [ "{\\mathbf{u}}", 0],
qpc: [ "{\\mathbf{c}}", 0],
qpb: [ "{\\mathbf{b}}", 0],
qpy: [ "{\\mathbf{y}}", 0],
qpw: [ "{\\mathbf{w}}", 0],
qplambda: [ "{\\mathbf{\\lambda}}", 0],
ssWpoint: [ "{\\bf #1}", 1],
ssWeight: [ "{w_{#1}}", 1],
dabs: [ "{\\parallel\\! #1 \\!\\parallel}", 1],
E: [ "{\\mathrm{E}}", 0],
A: [ "{\\mathrm{A}}", 0],
R: [ "{\\mathrm{R}}", 0],
N: [ "{\\mathrm{N}}", 0],
Q: [ "{\\mathrm{Q}}", 0],
Z: [ "{\\mathrm{Z}}", 0],
ccSum: [ "{\\sum_{#1}^{#2}{#3}}", 3],
ccProd: [ "{\\prod_{#1}^{#2}{#3}}", 3],
pyr: [ "{\\operatorname{Pyr}}", 0],
aff: [ "{\\operatorname{aff}}", 0],
Ac: [ "{\\cal A}", 0],
Sc: [ "{\\cal S}", 0],
},
equationNumbers: { autoNumber: "AMS" }
}
}
);
MathJax.Hub.Register.StartupHook("TeX Jax Ready",function () {
var PARSE = MathJax.InputJax.TeX.Parse,
TEXT = PARSE.prototype.InternalText;
PARSE.Augment({
InternalText: function (text,def) {
text = text.replace(/\\/g,"");
return TEXT.call(this,text,def);
}
});
});
//]]>
</script>
<script type="text/javascript" async="async" src="https://cdn.jsdelivr.net/npm/mathjax@2/MathJax.js"></script>
<script src="modules.js" type="text/javascript"></script>
</head>
<body>
<div id="top"><!-- do not remove this div, it is closed by doxygen! -->
<div id="back-nav">
<ul>
<li><a href="https://www.cgal.org/">cgal.org</a></li>
<li><a href="../Manual/index.html">Top</a></li>
<li><a href="../Manual/general_intro.html">Getting Started</a></li>
<li><a href="../Manual/tutorials.html">Tutorials</a></li>
<li><a href="../Manual/packages.html">Package Overview</a></li>
<li><a href="../Manual/how_to_cite_cgal.html">Acknowledging CGAL</a></li>
</ul>
<!-- In a package SEARCHENGINE = false, so we cannot use
insertion. That's why we have to do it manually here. Notice
that we also take pngs from the Manual. -->
<div id="MSearchBox" class="MSearchBoxInactive">
<span class="left">
<span id="MSearchSelect" onmouseover="return searchBox.OnSearchSelectShow()" onmouseout="return searchBox.OnSearchSelectHide()">
</span>
<input type="text" id="MSearchField" value="" placeholder="Search" accesskey="S" onfocus="searchBox.OnSearchFieldFocus(true)" onblur="searchBox.OnSearchFieldFocus(false)" onkeyup="searchBox.OnSearchFieldChange(event)">
</span>
<span class="right">
<a id="MSearchClose" href="javascript:searchBox.CloseResultsWindow()"><img id="MSearchCloseImg" border="0" src="../Manual/search/close.svg" alt=""></a>
</span>
</div>
</div>
<div id="titlearea">
<table cellspacing="0" cellpadding="0">
<tbody>
<tr id="projectrow">
<td id="projectalign">
<div id="projectname">CGAL 6.1 - 2D and 3D Linear Geometry Kernel
</div>
</td>
</tr>
</tbody>
</table>
</div>
<!-- Code below is usually inserted by doxygen when SEARCHENGINE =
true. Notice that the path to the search directory is adjusted to
the top-level.-->
<script type="text/javascript">
var searchBox = new SearchBox("searchBox", "../Manual/search/",'.html');
</script>
<!-- window showing the filter options -->
<div id="MSearchSelectWindow" onmouseover="return searchBox.OnSearchSelectShow()" onmouseout="return searchBox.OnSearchSelectHide()" onkeydown="return searchBox.OnSearchSelectKey(event)">
</div>
<!-- iframe showing the search results (closed by default) -->
<div id="MSearchResultsWindow">
<div id="MSearchResults">
<div class="SRPage">
<div id="SRIndex">
<div id="SRResults"></div>
<div class="SRStatus" id="Loading">Loading...</div>
<div class="SRStatus" id="Searching">Searching...</div>
<div class="SRStatus" id="NoMatches">No Matches</div>
</div>
</div>
</div>
</div>
<!-- end header part -->
<!-- Generated by Doxygen 1.9.6 -->
</div><!-- top -->
<div id="side-nav" class="ui-resizable side-nav-resizable">
<div id="nav-tree">
<div id="nav-tree-contents">
<div id="nav-sync" class="sync" style="display: none"></div>
</div>
</div>
<div id="splitbar" style="-moz-user-select:none;" class="ui-resizable-handle">
</div>
</div>
<script type="text/javascript">
/* @license magnet:?xt=urn:btih:d3d9a9a6595521f9666a5e94cc830dab83b65699&dn=expat.txt MIT */
$(document).ready(function(){initNavTree('classCGAL_1_1Aff__transformation__2.html',''); initResizable(); });
/* @license-end */
</script>
<div id="doc-content">
<div class="header">
<div class="summary">
<a href="classCGAL_1_1Aff__transformation__2-members.html">List of all members</a> </div>
<div class="headertitle"><div class="title">CGAL::Aff_transformation_2< Kernel > Class Template Reference<div class="ingroups"><a class="el" href="group__PkgKernel23Ref.html">2D and 3D Linear Geometry Kernel Reference</a> » <a class="el" href="group__kernel__classes2.html">2D Kernel Objects</a></div></div></div>
</div><!--header-->
<div class="contents">
<p><code>#include <CGAL/Aff_transformation_2.h></code></p>
<a name="details" id="details"></a><h2 class="groupheader">Definition</h2>
<div class="textblock"><div class="compoundTemplParams">template<typename <a class="el" href="classKernel.html">Kernel</a>><br>
class CGAL::Aff_transformation_2< Kernel ></div><p>The class <code><a class="el" href="classCGAL_1_1Aff__transformation__2.html" title="The class Aff_transformation_2 represents two-dimensional affine transformations.">Aff_transformation_2</a></code> represents two-dimensional affine transformations. </p>
<p>The general form of an affine transformation is based on a homogeneous representation of points. Thereby all transformations can be realized by matrix multiplications.</p>
<p>Multiplying the transformation matrix by a scalar does not change the represented transformation. Therefore, any transformation represented by a matrix with rational entries can be represented by a transformation matrix with integer entries as well. (Multiply the matrix with the common denominator of the rational entries.) Hence, it is sufficient to use the number type <code>Kernel::RT</code> to represent the entries of the transformation matrix.</p>
<p>CGAL offers several specialized affine transformations. Different constructors are provided to create them. They are parameterized with a symbolic name to denote the transformation type, followed by additional parameters. The symbolic name tags solve ambiguities in the function overloading and they make the code more readable, i.e., what type of transformation is created.</p>
<p>Since two-dimensional points have three homogeneous coordinates, we have a \( 3\times 3\) matrix \( {(m_{ij})}_{i,\,j=0\ldots 2}\).</p>
<p>If the homogeneous representations are normalized (the homogenizing coordinate is 1), then the upper left \( 2\times 2\) matrix realizes linear transformations. In the matrix form of a translation, the translation vector \( (v_0,\,v_1,\,1)\) appears in the last column of the matrix. The entries \( m_{20}\) and \( m_{21}\) are always zero and therefore do not appear in the constructors.</p>
<dl>
<dt>Is model of</dt>
<dd><code><a class="elRef" href="../STL_Extension/classHashable.html">Hashable</a></code> if <code><a class="el" href="classKernel.html" title="The concept of a kernel is defined by a set of requirements on the provision of certain types and acc...">Kernel</a></code> is a cartesian kernel and if <code>Kernel::FT</code> is <code><a class="elRef" href="../STL_Extension/classHashable.html">Hashable</a></code></dd>
</dl>
<dl class="section see"><dt>See also</dt><dd><code><a class="el" href="classCGAL_1_1Identity__transformation.html" title="Tag class for affine transformations.">Identity_transformation</a></code> </dd>
<dd>
<code><a class="el" href="classCGAL_1_1Rotation.html" title="Tag class for affine transformations.">Rotation</a></code> </dd>
<dd>
<code><a class="el" href="classCGAL_1_1Scaling.html" title="Tag class for affine transformations.">Scaling</a></code> </dd>
<dd>
<code><a class="el" href="classCGAL_1_1Translation.html" title="Tag class for affine transformations.">Translation</a></code> </dd>
<dd>
<code><a class="el" href="classCGAL_1_1Reflection.html" title="Tag class for affine transformations.">Reflection</a></code> </dd>
<dd>
<code><a class="el" href="group__rational__rotation__approximation__grp.html">CGAL::rational_rotation_approximation()</a></code></dd></dl>
<p><b>Example</b><br>
</p>
<div class="fragment"><div class="line"><span class="keyword">typedef</span> <a class="code hl_struct" href="structCGAL_1_1Simple__cartesian.html">CGAL::Simple_cartesian<double></a> K;</div>
<div class="line"><span class="keyword">typedef</span> <a class="code hl_class" href="classCGAL_1_1Aff__transformation__2.html">CGAL::Aff_transformation_2<K></a> Transformation;</div>
<div class="line"><span class="keyword">typedef</span> <a class="code hl_class" href="classCGAL_1_1Point__2.html">CGAL::Point_2<K></a> Point;</div>
<div class="line"><span class="keyword">typedef</span> <a class="code hl_class" href="classCGAL_1_1Vector__2.html">CGAL::Vector_2<K></a> <a class="code hl_classRef" href="../Kernel_d/classVector.html">Vector</a>;</div>
<div class="line"><span class="keyword">typedef</span> <a class="code hl_class" href="classCGAL_1_1Direction__2.html">CGAL::Direction_2<K></a> Direction;</div>
<div class="line"> </div>
<div class="line">Transformation rotate(CGAL::ROTATION, sin(pi), cos(pi));</div>
<div class="line">Transformation rational_rotate(CGAL::ROTATION,Direction(1,1), 1, 100);</div>
<div class="line">Transformation translate(CGAL::TRANSLATION, <a class="code hl_classRef" href="../Kernel_d/classVector.html">Vector</a>(-2, 0));</div>
<div class="line">Transformation scale(CGAL::SCALING, 3);</div>
<div class="line"> </div>
<div class="line">Point q(0, 1);</div>
<div class="line">q = rational_rotate(q);</div>
<div class="line"> </div>
<div class="line">Point p(1, 1);</div>
<div class="line"> </div>
<div class="line">p = rotate(p);</div>
<div class="line"> </div>
<div class="line">p = translate(p);</div>
<div class="line"> </div>
<div class="line">p = scale(p);</div>
<div class="ttc" id="aclassCGAL_1_1Aff__transformation__2_html"><div class="ttname"><a href="classCGAL_1_1Aff__transformation__2.html">CGAL::Aff_transformation_2</a></div><div class="ttdoc">The class Aff_transformation_2 represents two-dimensional affine transformations.</div><div class="ttdef"><b>Definition:</b> Aff_transformation_2.h:87</div></div>
<div class="ttc" id="aclassCGAL_1_1Direction__2_html"><div class="ttname"><a href="classCGAL_1_1Direction__2.html">CGAL::Direction_2</a></div><div class="ttdoc">An object d of the class Direction_2 is a vector in the two-dimensional vector space where we forget...</div><div class="ttdef"><b>Definition:</b> Direction_2.h:21</div></div>
<div class="ttc" id="aclassCGAL_1_1Point__2_html"><div class="ttname"><a href="classCGAL_1_1Point__2.html">CGAL::Point_2</a></div><div class="ttdoc">An object p of the class Point_2 is a point in the two-dimensional Euclidean plane .</div><div class="ttdef"><b>Definition:</b> Point_2.h:37</div></div>
<div class="ttc" id="aclassCGAL_1_1Vector__2_html"><div class="ttname"><a href="classCGAL_1_1Vector__2.html">CGAL::Vector_2</a></div><div class="ttdoc">An object v of the class Vector_2 is a vector in the two-dimensional vector space .</div><div class="ttdef"><b>Definition:</b> Vector_2.h:19</div></div>
<div class="ttc" id="aclassVector_html"><div class="ttname"><a href="../Kernel_d/classVector.html">Vector</a></div></div>
<div class="ttc" id="astructCGAL_1_1Simple__cartesian_html"><div class="ttname"><a href="structCGAL_1_1Simple__cartesian.html">CGAL::Simple_cartesian</a></div><div class="ttdoc">A model for a Kernel using Cartesian coordinates to represent the geometric objects.</div><div class="ttdef"><b>Definition:</b> Simple_cartesian.h:31</div></div>
</div><!-- fragment --><p>The same would have been achieved with</p>
<div class="fragment"><div class="line">Transformation <a class="code hl_function" href="classCGAL_1_1Aff__transformation__2.html#a1991354dac5ce209d71ec82b19187374">transform</a> = scale * (translate * rotate);</div>
<div class="line">p = <a class="code hl_function" href="classCGAL_1_1Aff__transformation__2.html#a1991354dac5ce209d71ec82b19187374">transform</a>(Point(1.0, 1.0));</div>
<div class="ttc" id="aclassCGAL_1_1Aff__transformation__2_html_a1991354dac5ce209d71ec82b19187374"><div class="ttname"><a href="classCGAL_1_1Aff__transformation__2.html#a1991354dac5ce209d71ec82b19187374">CGAL::Aff_transformation_2::transform</a></div><div class="ttdeci">Point_2< Kernel > transform(const Point_2< Kernel > &p) const</div></div>
</div><!-- fragment --><dl class="section see"><dt>See also</dt><dd><code><a class="el" href="classCGAL_1_1Aff__transformation__3.html" title="The class Aff_transformation_3 represents three-dimensional affine transformations.">CGAL::Aff_transformation_3</a><<a class="el" href="classKernel.html" title="The concept of a kernel is defined by a set of requirements on the provision of certain types and acc...">Kernel</a>></code> </dd>
<dd>
<code><a class="el" href="classCGAL_1_1Identity__transformation.html" title="Tag class for affine transformations.">CGAL::Identity_transformation</a></code> </dd>
<dd>
<code><a class="el" href="classCGAL_1_1Reflection.html" title="Tag class for affine transformations.">CGAL::Reflection</a></code> </dd>
<dd>
<code><a class="el" href="classCGAL_1_1Rotation.html" title="Tag class for affine transformations.">CGAL::Rotation</a></code> </dd>
<dd>
<code><a class="el" href="classCGAL_1_1Scaling.html" title="Tag class for affine transformations.">CGAL::Scaling</a></code> </dd>
<dd>
<code><a class="el" href="classCGAL_1_1Translation.html" title="Tag class for affine transformations.">CGAL::Translation</a></code> </dd></dl>
</div><table class="memberdecls">
<tr class="heading"><td colspan="2"><h2 class="groupheader">Creation</h2></td></tr>
<tr class="memitem:a47f305715cb75778048ea7329b5c02dc"><td class="memItemLeft" align="right" valign="top"><a id="a47f305715cb75778048ea7329b5c02dc" name="a47f305715cb75778048ea7329b5c02dc"></a>
</td><td class="memItemRight" valign="bottom"><b>Aff_transformation_2</b> (const <a class="el" href="classCGAL_1_1Identity__transformation.html">Identity_transformation</a> &)</td></tr>
<tr class="memdesc:a47f305715cb75778048ea7329b5c02dc"><td class="mdescLeft"> </td><td class="mdescRight">introduces an identity transformation. <br></td></tr>
<tr class="separator:a47f305715cb75778048ea7329b5c02dc"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:ab97331ea55f685c17b4447a4184835f0"><td class="memItemLeft" align="right" valign="top"><a id="ab97331ea55f685c17b4447a4184835f0" name="ab97331ea55f685c17b4447a4184835f0"></a>
</td><td class="memItemRight" valign="bottom"><b>Aff_transformation_2</b> (const <a class="el" href="classCGAL_1_1Translation.html">Translation</a>, const <a class="el" href="classCGAL_1_1Vector__2.html">Vector_2</a>< <a class="el" href="classKernel.html">Kernel</a> > &v)</td></tr>
<tr class="memdesc:ab97331ea55f685c17b4447a4184835f0"><td class="mdescLeft"> </td><td class="mdescRight">introduces a translation by a vector <code>v</code>. <br></td></tr>
<tr class="separator:ab97331ea55f685c17b4447a4184835f0"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:a5cc6631b0ed023470ccfc9e37e5272fb"><td class="memItemLeft" align="right" valign="top"> </td><td class="memItemRight" valign="bottom"><a class="el" href="classCGAL_1_1Aff__transformation__2.html#a5cc6631b0ed023470ccfc9e37e5272fb">Aff_transformation_2</a> (const <a class="el" href="classCGAL_1_1Rotation.html">Rotation</a>, const <a class="el" href="classCGAL_1_1Direction__2.html">Direction_2</a>< <a class="el" href="classKernel.html">Kernel</a> > &d, const Kernel::RT &num, const Kernel::RT &den=RT(1))</td></tr>
<tr class="memdesc:a5cc6631b0ed023470ccfc9e37e5272fb"><td class="mdescLeft"> </td><td class="mdescRight">approximates the rotation over the angle indicated by direction <code>d</code>, such that the differences between the sines and cosines of the rotation given by d and the approximating rotation are at most \( num/den\) each. <br></td></tr>
<tr class="separator:a5cc6631b0ed023470ccfc9e37e5272fb"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:a56a137e059bc71662485093c1bcd04c8"><td class="memItemLeft" align="right" valign="top"> </td><td class="memItemRight" valign="bottom"><a class="el" href="classCGAL_1_1Aff__transformation__2.html#a56a137e059bc71662485093c1bcd04c8">Aff_transformation_2</a> (const <a class="el" href="classCGAL_1_1Rotation.html">Rotation</a>, const Kernel::RT &sine_rho, const Kernel::RT &cosine_rho, const Kernel::RT &hw=RT(1))</td></tr>
<tr class="memdesc:a56a137e059bc71662485093c1bcd04c8"><td class="mdescLeft"> </td><td class="mdescRight">introduces a rotation by the angle <code>rho</code>. <br></td></tr>
<tr class="separator:a56a137e059bc71662485093c1bcd04c8"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:adcdac26dc822f15053218abd7f706c1d"><td class="memItemLeft" align="right" valign="top"><a id="adcdac26dc822f15053218abd7f706c1d" name="adcdac26dc822f15053218abd7f706c1d"></a>
</td><td class="memItemRight" valign="bottom"><b>Aff_transformation_2</b> (const <a class="el" href="classCGAL_1_1Scaling.html">Scaling</a>, const Kernel::RT &s, const Kernel::RT &hw=RT(1))</td></tr>
<tr class="memdesc:adcdac26dc822f15053218abd7f706c1d"><td class="mdescLeft"> </td><td class="mdescRight">introduces a scaling by a scale factor \( s/hw\). <br></td></tr>
<tr class="separator:adcdac26dc822f15053218abd7f706c1d"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:a4911d35529487e8b95f4950e56de329e"><td class="memItemLeft" align="right" valign="top"><a id="a4911d35529487e8b95f4950e56de329e" name="a4911d35529487e8b95f4950e56de329e"></a>
</td><td class="memItemRight" valign="bottom"><b>Aff_transformation_2</b> (const <a class="el" href="classCGAL_1_1Reflection.html">Reflection</a>, const <a class="el" href="classCGAL_1_1Line__2.html">Line_2</a>< <a class="el" href="classKernel.html">Kernel</a> > &l)</td></tr>
<tr class="memdesc:a4911d35529487e8b95f4950e56de329e"><td class="mdescLeft"> </td><td class="mdescRight">introduces a reflection by a line <code>l</code>. <br></td></tr>
<tr class="separator:a4911d35529487e8b95f4950e56de329e"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:ad0a7c3e145a0f9e073c0fb5a1ee9973c"><td class="memItemLeft" align="right" valign="top"> </td><td class="memItemRight" valign="bottom"><a class="el" href="classCGAL_1_1Aff__transformation__2.html#ad0a7c3e145a0f9e073c0fb5a1ee9973c">Aff_transformation_2</a> (const Kernel::RT &m00, const Kernel::RT &m01, const Kernel::RT &m02, const Kernel::RT &m10, const Kernel::RT &m11, const Kernel::RT &m12, const Kernel::RT &hw=RT(1))</td></tr>
<tr class="memdesc:ad0a7c3e145a0f9e073c0fb5a1ee9973c"><td class="mdescLeft"> </td><td class="mdescRight">introduces a general affine transformation in the \(3 \times 3\) matrix form \(
\small \mbox{\( \left(\begin{array}{ccc}
m_{00} & m_{01} & m_{02}\\
m_{10} & m_{11} & m_{12}\\
0 & 0 & hw
\end{array}\right) \)}
\). <br></td></tr>
<tr class="separator:ad0a7c3e145a0f9e073c0fb5a1ee9973c"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:a6381959cec20a2652bf8f7bdf8a29295"><td class="memItemLeft" align="right" valign="top"><a id="a6381959cec20a2652bf8f7bdf8a29295" name="a6381959cec20a2652bf8f7bdf8a29295"></a>
</td><td class="memItemRight" valign="bottom"><b>Aff_transformation_2</b> (const Kernel::RT &m00, const Kernel::RT &m01, const Kernel::RT &m10, const Kernel::RT &m11, const Kernel::RT &hw=RT(1))</td></tr>
<tr class="memdesc:a6381959cec20a2652bf8f7bdf8a29295"><td class="mdescLeft"> </td><td class="mdescRight">introduces a general linear transformation \(\small \mbox{\(\left(\begin{array}{ccc}
m_{00} & m_{01} & 0\\
m_{10} & m_{11} & 0\\
0 & 0 & hw
\end{array}\right)\)}\) i.e. there is no translational part. <br></td></tr>
<tr class="separator:a6381959cec20a2652bf8f7bdf8a29295"><td class="memSeparator" colspan="2"> </td></tr>
</table><table class="memberdecls">
<tr class="heading"><td colspan="2"><h2 class="groupheader">Operations</h2></td></tr>
<tr><td class="ititle" colspan="2"><p>The main thing to do with transformations is to apply them on geometric objects.</p>
<p>Each class <code>Class_2<<a class="el" href="classKernel.html" title="The concept of a kernel is defined by a set of requirements on the provision of certain types and acc...">Kernel</a>></code> representing a geometric object has a member function: </p><div class="fragment"><div class="line">Class_2<Kernel> <a class="code hl_function" href="classCGAL_1_1Aff__transformation__2.html#a1991354dac5ce209d71ec82b19187374">transform</a>(<a class="code hl_class" href="classCGAL_1_1Aff__transformation__2.html">Aff_transformation_2<Kernel></a> t).</div>
</div><!-- fragment --><p>The transformation classes provide a member function <code><a class="el" href="classCGAL_1_1Aff__transformation__2.html#a1991354dac5ce209d71ec82b19187374">transform()</a></code> for points, vectors, directions, and lines. The same functionality is also available through <code>operator()</code> overloads. </p>
</td></tr>
<tr class="memitem:a1991354dac5ce209d71ec82b19187374"><td class="memItemLeft" align="right" valign="top"><a id="a1991354dac5ce209d71ec82b19187374" name="a1991354dac5ce209d71ec82b19187374"></a>
<a class="el" href="classCGAL_1_1Point__2.html">Point_2</a>< <a class="el" href="classKernel.html">Kernel</a> > </td><td class="memItemRight" valign="bottom"><b>transform</b> (const <a class="el" href="classCGAL_1_1Point__2.html">Point_2</a>< <a class="el" href="classKernel.html">Kernel</a> > &p) const</td></tr>
<tr class="separator:a1991354dac5ce209d71ec82b19187374"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:a7fb066d9e40bb5c7c6f5b10e1a4bbe4a"><td class="memItemLeft" align="right" valign="top"><a id="a7fb066d9e40bb5c7c6f5b10e1a4bbe4a" name="a7fb066d9e40bb5c7c6f5b10e1a4bbe4a"></a>
<a class="el" href="classCGAL_1_1Vector__2.html">Vector_2</a>< <a class="el" href="classKernel.html">Kernel</a> > </td><td class="memItemRight" valign="bottom"><b>transform</b> (const <a class="el" href="classCGAL_1_1Vector__2.html">Vector_2</a>< <a class="el" href="classKernel.html">Kernel</a> > &p) const</td></tr>
<tr class="separator:a7fb066d9e40bb5c7c6f5b10e1a4bbe4a"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:a6f37a9eaae4b5c78f887bdee65246654"><td class="memItemLeft" align="right" valign="top"><a id="a6f37a9eaae4b5c78f887bdee65246654" name="a6f37a9eaae4b5c78f887bdee65246654"></a>
<a class="el" href="classCGAL_1_1Direction__2.html">Direction_2</a>< <a class="el" href="classKernel.html">Kernel</a> > </td><td class="memItemRight" valign="bottom"><b>transform</b> (const <a class="el" href="classCGAL_1_1Direction__2.html">Direction_2</a>< <a class="el" href="classKernel.html">Kernel</a> > &p) const</td></tr>
<tr class="separator:a6f37a9eaae4b5c78f887bdee65246654"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:a61410eb728dcc3f988f9c0ac40e38c83"><td class="memItemLeft" align="right" valign="top"><a id="a61410eb728dcc3f988f9c0ac40e38c83" name="a61410eb728dcc3f988f9c0ac40e38c83"></a>
<a class="el" href="classCGAL_1_1Line__2.html">Line_2</a>< <a class="el" href="classKernel.html">Kernel</a> > </td><td class="memItemRight" valign="bottom"><b>transform</b> (const <a class="el" href="classCGAL_1_1Line__2.html">Line_2</a>< <a class="el" href="classKernel.html">Kernel</a> > &p) const</td></tr>
<tr class="separator:a61410eb728dcc3f988f9c0ac40e38c83"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:ab8417aef169f1e1e89e76af89f93df47"><td class="memItemLeft" align="right" valign="top"><a id="ab8417aef169f1e1e89e76af89f93df47" name="ab8417aef169f1e1e89e76af89f93df47"></a>
<a class="el" href="classCGAL_1_1Point__2.html">Point_2</a>< <a class="el" href="classKernel.html">Kernel</a> > </td><td class="memItemRight" valign="bottom"><b>operator()</b> (const <a class="el" href="classCGAL_1_1Point__2.html">Point_2</a>< <a class="el" href="classKernel.html">Kernel</a> > &p) const</td></tr>
<tr class="separator:ab8417aef169f1e1e89e76af89f93df47"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:a6e358f0d7a9d7a0cf36ef1931c57250d"><td class="memItemLeft" align="right" valign="top"><a id="a6e358f0d7a9d7a0cf36ef1931c57250d" name="a6e358f0d7a9d7a0cf36ef1931c57250d"></a>
<a class="el" href="classCGAL_1_1Vector__2.html">Vector_2</a>< <a class="el" href="classKernel.html">Kernel</a> > </td><td class="memItemRight" valign="bottom"><b>operator()</b> (const <a class="el" href="classCGAL_1_1Vector__2.html">Vector_2</a>< <a class="el" href="classKernel.html">Kernel</a> > &p) const</td></tr>
<tr class="separator:a6e358f0d7a9d7a0cf36ef1931c57250d"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:a07c796aa54eb8a7e53ddc05f4c7a9d6e"><td class="memItemLeft" align="right" valign="top"><a id="a07c796aa54eb8a7e53ddc05f4c7a9d6e" name="a07c796aa54eb8a7e53ddc05f4c7a9d6e"></a>
<a class="el" href="classCGAL_1_1Direction__2.html">Direction_2</a>< <a class="el" href="classKernel.html">Kernel</a> > </td><td class="memItemRight" valign="bottom"><b>operator()</b> (const <a class="el" href="classCGAL_1_1Direction__2.html">Direction_2</a>< <a class="el" href="classKernel.html">Kernel</a> > &p) const</td></tr>
<tr class="separator:a07c796aa54eb8a7e53ddc05f4c7a9d6e"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:aef621cbf03c8391eed7582877af065e3"><td class="memItemLeft" align="right" valign="top"><a id="aef621cbf03c8391eed7582877af065e3" name="aef621cbf03c8391eed7582877af065e3"></a>
<a class="el" href="classCGAL_1_1Line__2.html">Line_2</a>< <a class="el" href="classKernel.html">Kernel</a> > </td><td class="memItemRight" valign="bottom"><b>operator()</b> (const <a class="el" href="classCGAL_1_1Line__2.html">Line_2</a>< <a class="el" href="classKernel.html">Kernel</a> > &p) const</td></tr>
<tr class="separator:aef621cbf03c8391eed7582877af065e3"><td class="memSeparator" colspan="2"> </td></tr>
</table><table class="memberdecls">
<tr class="heading"><td colspan="2"><h2 class="groupheader">Miscellaneous</h2></td></tr>
<tr class="memitem:aef43784e10b4468777efa7fe43b9e8c0"><td class="memItemLeft" align="right" valign="top"><a id="aef43784e10b4468777efa7fe43b9e8c0" name="aef43784e10b4468777efa7fe43b9e8c0"></a>
<a class="el" href="classCGAL_1_1Aff__transformation__2.html">Aff_transformation_2</a>< <a class="el" href="classKernel.html">Kernel</a> > </td><td class="memItemRight" valign="bottom"><b>operator*</b> (const <a class="el" href="classCGAL_1_1Aff__transformation__2.html">Aff_transformation_2</a>< <a class="el" href="classKernel.html">Kernel</a> > &s) const</td></tr>
<tr class="memdesc:aef43784e10b4468777efa7fe43b9e8c0"><td class="mdescLeft"> </td><td class="mdescRight">composes two affine transformations. <br></td></tr>
<tr class="separator:aef43784e10b4468777efa7fe43b9e8c0"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:aec1c5ee29e92a8a98224200f9fd2f4d3"><td class="memItemLeft" align="right" valign="top"><a id="aec1c5ee29e92a8a98224200f9fd2f4d3" name="aec1c5ee29e92a8a98224200f9fd2f4d3"></a>
<a class="el" href="classCGAL_1_1Aff__transformation__2.html">Aff_transformation_2</a>< <a class="el" href="classKernel.html">Kernel</a> > </td><td class="memItemRight" valign="bottom"><b>inverse</b> () const</td></tr>
<tr class="memdesc:aec1c5ee29e92a8a98224200f9fd2f4d3"><td class="mdescLeft"> </td><td class="mdescRight">gives the inverse transformation. <br></td></tr>
<tr class="separator:aec1c5ee29e92a8a98224200f9fd2f4d3"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:a74c439908100dd7754a649ef631c7f74"><td class="memItemLeft" align="right" valign="top"><a id="a74c439908100dd7754a649ef631c7f74" name="a74c439908100dd7754a649ef631c7f74"></a>
bool </td><td class="memItemRight" valign="bottom"><b>operator==</b> (const <a class="el" href="classCGAL_1_1Aff__transformation__2.html">Aff_transformation_2</a>< <a class="el" href="classKernel.html">Kernel</a> > &s) const</td></tr>
<tr class="memdesc:a74c439908100dd7754a649ef631c7f74"><td class="mdescLeft"> </td><td class="mdescRight">compares two affine transformations. <br></td></tr>
<tr class="separator:a74c439908100dd7754a649ef631c7f74"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:aed27613015c192b02f52237d1c3bb699"><td class="memItemLeft" align="right" valign="top"><a id="aed27613015c192b02f52237d1c3bb699" name="aed27613015c192b02f52237d1c3bb699"></a>
bool </td><td class="memItemRight" valign="bottom"><b>is_even</b> () const</td></tr>
<tr class="memdesc:aed27613015c192b02f52237d1c3bb699"><td class="mdescLeft"> </td><td class="mdescRight">returns <code>true</code>, if the transformation is not reflecting, i.e. the determinant of the involved linear transformation is non-negative. <br></td></tr>
<tr class="separator:aed27613015c192b02f52237d1c3bb699"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:a25e1947fe04411ddf9ecf71197879cc5"><td class="memItemLeft" align="right" valign="top"><a id="a25e1947fe04411ddf9ecf71197879cc5" name="a25e1947fe04411ddf9ecf71197879cc5"></a>
bool </td><td class="memItemRight" valign="bottom"><b>is_odd</b> () const</td></tr>
<tr class="memdesc:a25e1947fe04411ddf9ecf71197879cc5"><td class="mdescLeft"> </td><td class="mdescRight">returns <code>true</code>, if the transformation is reflecting. <br></td></tr>
<tr class="separator:a25e1947fe04411ddf9ecf71197879cc5"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:a733212c6de7458cab118e29fad67fc13"><td class="memItemLeft" align="right" valign="top"><a id="a733212c6de7458cab118e29fad67fc13" name="a733212c6de7458cab118e29fad67fc13"></a>
bool </td><td class="memItemRight" valign="bottom"><b>is_scaling</b> () const</td></tr>
<tr class="memdesc:a733212c6de7458cab118e29fad67fc13"><td class="mdescLeft"> </td><td class="mdescRight">returns <code>true</code>, if the object was constructed using the tag <code><a class="el" href="classCGAL_1_1Scaling.html" title="Tag class for affine transformations.">CGAL::Scaling</a></code>, or is the result of the composition of only such scaling transformation objects. <br></td></tr>
<tr class="separator:a733212c6de7458cab118e29fad67fc13"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:a0ef5b8d0ac49c93f32317dc75c4e9fae"><td class="memItemLeft" align="right" valign="top"><a id="a0ef5b8d0ac49c93f32317dc75c4e9fae" name="a0ef5b8d0ac49c93f32317dc75c4e9fae"></a>
bool </td><td class="memItemRight" valign="bottom"><b>is_translation</b> () const</td></tr>
<tr class="memdesc:a0ef5b8d0ac49c93f32317dc75c4e9fae"><td class="mdescLeft"> </td><td class="mdescRight">returns <code>true</code>, if the object was constructed using the tag <code><a class="el" href="classCGAL_1_1Translation.html" title="Tag class for affine transformations.">CGAL::Translation</a></code>, or is the result of the composition of only such translation transformation objects. <br></td></tr>
<tr class="separator:a0ef5b8d0ac49c93f32317dc75c4e9fae"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:a4175aa498e33a022cd3c40ae08f449a6"><td class="memItemLeft" align="right" valign="top"><a id="a4175aa498e33a022cd3c40ae08f449a6" name="a4175aa498e33a022cd3c40ae08f449a6"></a>
bool </td><td class="memItemRight" valign="bottom"><b>is_rotation</b> () const</td></tr>
<tr class="memdesc:a4175aa498e33a022cd3c40ae08f449a6"><td class="mdescLeft"> </td><td class="mdescRight">returns <code>true</code>, if the object was constructed using the tag <code><a class="el" href="classCGAL_1_1Rotation.html" title="Tag class for affine transformations.">CGAL::Rotation</a></code>, or is the result of the composition of only such rotation transformation objects. <br></td></tr>
<tr class="separator:a4175aa498e33a022cd3c40ae08f449a6"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:aacb1a9b98140591547f0c91b06b64e87"><td class="memItemLeft" align="right" valign="top"><a id="aacb1a9b98140591547f0c91b06b64e87" name="aacb1a9b98140591547f0c91b06b64e87"></a>
bool </td><td class="memItemRight" valign="bottom"><b>is_reflection</b> () const</td></tr>
<tr class="memdesc:aacb1a9b98140591547f0c91b06b64e87"><td class="mdescLeft"> </td><td class="mdescRight">returns <code>true</code>, if the object was constructed using the tag <code><a class="el" href="classCGAL_1_1Reflection.html" title="Tag class for affine transformations.">CGAL::Reflection</a></code>, or is the result of the composition of only such reflection transformation objects. <br></td></tr>
<tr class="separator:aacb1a9b98140591547f0c91b06b64e87"><td class="memSeparator" colspan="2"> </td></tr>
</table><table class="memberdecls">
<tr class="heading"><td colspan="2"><h2 class="groupheader">Matrix Entry Access</h2></td></tr>
<tr class="memitem:a954879511402f4e462ff1f9c0f646f6a"><td class="memItemLeft" align="right" valign="top"><a id="a954879511402f4e462ff1f9c0f646f6a" name="a954879511402f4e462ff1f9c0f646f6a"></a>
Kernel::FT </td><td class="memItemRight" valign="bottom"><b>cartesian</b> (int i, int j) const</td></tr>
<tr class="separator:a954879511402f4e462ff1f9c0f646f6a"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:a7073d7799b1d81f5562b95312d10e0c9"><td class="memItemLeft" align="right" valign="top"><a id="a7073d7799b1d81f5562b95312d10e0c9" name="a7073d7799b1d81f5562b95312d10e0c9"></a>
Kernel::FT </td><td class="memItemRight" valign="bottom"><b>m</b> (int i, int j) const</td></tr>
<tr class="memdesc:a7073d7799b1d81f5562b95312d10e0c9"><td class="mdescLeft"> </td><td class="mdescRight">returns entry \( m_{ij}\) in a matrix representation in which \( m_{22}\) is 1. <br></td></tr>
<tr class="separator:a7073d7799b1d81f5562b95312d10e0c9"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:ac4aabbbdb97f2cd2aefb5a3fd750c13b"><td class="memItemLeft" align="right" valign="top"><a id="ac4aabbbdb97f2cd2aefb5a3fd750c13b" name="ac4aabbbdb97f2cd2aefb5a3fd750c13b"></a>
Kernel::RT </td><td class="memItemRight" valign="bottom"><b>homogeneous</b> (int i, int j) const</td></tr>
<tr class="separator:ac4aabbbdb97f2cd2aefb5a3fd750c13b"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:a00485abf34793a3afc991d0d98f03762"><td class="memItemLeft" align="right" valign="top"><a id="a00485abf34793a3afc991d0d98f03762" name="a00485abf34793a3afc991d0d98f03762"></a>
Kernel::RT </td><td class="memItemRight" valign="bottom"><b>hm</b> (int i, int j) const</td></tr>
<tr class="memdesc:a00485abf34793a3afc991d0d98f03762"><td class="mdescLeft"> </td><td class="mdescRight">returns entry \( m_{ij}\) in some fixed matrix representation. <br></td></tr>
<tr class="separator:a00485abf34793a3afc991d0d98f03762"><td class="memSeparator" colspan="2"> </td></tr>
</table>
<h2 class="groupheader">Constructor & Destructor Documentation</h2>
<a id="a5cc6631b0ed023470ccfc9e37e5272fb" name="a5cc6631b0ed023470ccfc9e37e5272fb"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a5cc6631b0ed023470ccfc9e37e5272fb">◆ </a></span>Aff_transformation_2() <span class="overload">[1/3]</span></h2>
<div class="memitem">
<div class="memproto">
<div class="memtemplate">
template<typename <a class="el" href="classKernel.html">Kernel</a> > </div>
<table class="memname">
<tr>
<td class="memname"><a class="el" href="classCGAL_1_1Aff__transformation__2.html">CGAL::Aff_transformation_2</a>< <a class="el" href="classKernel.html">Kernel</a> >::Aff_transformation_2 </td>
<td>(</td>
<td class="paramtype">const </td>
<td class="paramname"><em>Rotation</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">const <a class="el" href="classCGAL_1_1Direction__2.html">Direction_2</a>< <a class="el" href="classKernel.html">Kernel</a> > & </td>
<td class="paramname"><em>d</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">const Kernel::RT & </td>
<td class="paramname"><em>num</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">const Kernel::RT & </td>
<td class="paramname"><em>den</em> = <code>RT(1)</code> </td>
</tr>
<tr>
<td></td>
<td>)</td>
<td></td><td></td>
</tr>
</table>
</div><div class="memdoc">
<p>approximates the rotation over the angle indicated by direction <code>d</code>, such that the differences between the sines and cosines of the rotation given by d and the approximating rotation are at most \( num/den\) each. </p>
<dl class="section pre"><dt>Precondition</dt><dd><code>num/den > 0</code> and <code>d != 0</code>. </dd></dl>
</div>
</div>
<a id="a56a137e059bc71662485093c1bcd04c8" name="a56a137e059bc71662485093c1bcd04c8"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a56a137e059bc71662485093c1bcd04c8">◆ </a></span>Aff_transformation_2() <span class="overload">[2/3]</span></h2>
<div class="memitem">
<div class="memproto">
<div class="memtemplate">
template<typename <a class="el" href="classKernel.html">Kernel</a> > </div>
<table class="memname">
<tr>
<td class="memname"><a class="el" href="classCGAL_1_1Aff__transformation__2.html">CGAL::Aff_transformation_2</a>< <a class="el" href="classKernel.html">Kernel</a> >::Aff_transformation_2 </td>
<td>(</td>
<td class="paramtype">const </td>
<td class="paramname"><em>Rotation</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">const Kernel::RT & </td>
<td class="paramname"><em>sine_rho</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">const Kernel::RT & </td>
<td class="paramname"><em>cosine_rho</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">const Kernel::RT & </td>
<td class="paramname"><em>hw</em> = <code>RT(1)</code> </td>
</tr>
<tr>
<td></td>
<td>)</td>
<td></td><td></td>
</tr>
</table>
</div><div class="memdoc">
<p>introduces a rotation by the angle <code>rho</code>. </p>
<dl class="section pre"><dt>Precondition</dt><dd><code>sine_rho<sup>2</sup> + cosine_rho<sup>2</sup> == hw<sup>2</sup></code>. </dd></dl>
</div>
</div>
<a id="ad0a7c3e145a0f9e073c0fb5a1ee9973c" name="ad0a7c3e145a0f9e073c0fb5a1ee9973c"></a>
<h2 class="memtitle"><span class="permalink"><a href="#ad0a7c3e145a0f9e073c0fb5a1ee9973c">◆ </a></span>Aff_transformation_2() <span class="overload">[3/3]</span></h2>
<div class="memitem">
<div class="memproto">
<div class="memtemplate">
template<typename <a class="el" href="classKernel.html">Kernel</a> > </div>
<table class="memname">
<tr>
<td class="memname"><a class="el" href="classCGAL_1_1Aff__transformation__2.html">CGAL::Aff_transformation_2</a>< <a class="el" href="classKernel.html">Kernel</a> >::Aff_transformation_2 </td>
<td>(</td>
<td class="paramtype">const Kernel::RT & </td>
<td class="paramname"><em>m00</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">const Kernel::RT & </td>
<td class="paramname"><em>m01</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">const Kernel::RT & </td>
<td class="paramname"><em>m02</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">const Kernel::RT & </td>
<td class="paramname"><em>m10</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">const Kernel::RT & </td>
<td class="paramname"><em>m11</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">const Kernel::RT & </td>
<td class="paramname"><em>m12</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">const Kernel::RT & </td>
<td class="paramname"><em>hw</em> = <code>RT(1)</code> </td>
</tr>
<tr>
<td></td>
<td>)</td>
<td></td><td></td>
</tr>
</table>
</div><div class="memdoc">
<p>introduces a general affine transformation in the \(3 \times 3\) matrix form \(
\small \mbox{\( \left(\begin{array}{ccc}
m_{00} & m_{01} & m_{02}\\
m_{10} & m_{11} & m_{12}\\
0 & 0 & hw
\end{array}\right) \)}
\). </p>
<p>The sub-matrix \(1\over hw\) \(\small \mbox{\( \left(\begin{array}{cc}
m_{00} & m_{01}\\
m_{10} & m_{11}
\end{array}\right) \) }
\) contains the scaling and rotation information, the vector \( \small
\left(
\begin{array}{c}
m_{02}\\
m_{12}
\end{array}
\right)
\) contains the translational part of the transformation. </p>
</div>
</div>
</div><!-- contents -->
</div><!-- doc-content -->
<!-- HTML footer for doxygen 1.9.6-->
<!-- start footer part -->
<!-- The footer div is not part of the default but we require it to
move the footer to the bottom of the page. -->
<div id="footer">
<div id="nav-path" class="navpath"><!-- id is needed for treeview function! -->
<ul>
<li class="navelem"><b>CGAL</b></li><li class="navelem"><a class="el" href="classCGAL_1_1Aff__transformation__2.html">Aff_transformation_2</a></li>
<li class="footer">Generated by <a href="https://www.doxygen.org/index.html"><img class="footer" src="doxygen.svg" width="104" height="31" alt="doxygen"></a> 1.9.6 </li>
</ul>
</div>
</div>
</body>
</html>