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3	<title>ADG: Geometry</title>
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28	</td>
29	<td ALIGN="Right">
30	<font SIZE="-1" COLOR="#238E23">
31	<b>Geant4 User's Guide</b>
32	<br>
33	<b>For Application Developers</b>
34	<br>
35	<b>Geometry</b>
36	</font>
37	</td>
38	</tr></table>
39	<br><br>
40
41	<a name="4.1.2">
42	<h2>4.1.2 Solids</h2></a>
43
44	<p>
45	The STEP standard supports multiple solid representations. Constructive
46	Solid Geometry (CSG) representations and Boundary Represented Solids (BREPs)
47	are available. Different representations are suitable for different
48	purposes, applications, required complexity, and levels of detail.
49	CSG representations are easy to use and normally give superior performance,
50	but they cannot reproduce complex solids such as those used in CAD systems.
51	BREP representations can handle more extended topologies and reproduce the
52	most complex solids, thus allowing the exchange of models with CAD systems.
53	<br>
54	All constructed solids can stream out their contents via appropriate methods
55	and streaming operators.
56	</p>
57
58	<p>
59	For all solids it is possible to estimate the geometrical volume and the
60	surface area by invoking the methods:
61	<pre>
62	G4double GetCubicVolume()
63	G4double GetSurfaceArea()
64	</pre>
65	which return an estimate of the solid volume and total area in internal
66	units respectively. For elementary solids the functions compute the exact
67	geometrical quantities, while for composite or complex solids an estimate
68	is made using Monte Carlo techniques.
69	</p>
70
71	<p>
72	For all solids it is also possible to generate pseudo-random points lying
73	on their surfaces, by invoking the method
74	<pre>
75	G4ThreeVector GetPointOnSurface() const
76	</pre>
77	which returns the generated point in local coordinates relative to the solid.
78	</p>
79
80	<a name="4.1.2.1">
81	<H4>4.1.2.1 Constructed Solid Geometry (CSG) Solids</H4></a>
82
83	CSG solids are defined directly as three-dimensional primitives. They are
84	described by a minimal set of parameters necessary to define the shape and
85	size of the solid. CSG solids are Boxes, Tubes and their sections, Cones
86	and their sections, Spheres, Wedges, and Toruses.
87	<P>
88	<HR width=40% align=center noshade>
89	</P>
90	To create a <b>box</b> one can use the constructor:
91
92	<table border="0" width="100%" id="table1">
93	<tr>
94	<td width="480" valign="top">
95	<font face="Courier">
96	G4Box(const G4String& pName,<br>
97
98	G4double  pX,<br>
99
100	G4double  pY,<br>
101
102	G4double  pZ)
103	</font>
104	<P>
105	by giving the box a name and its half-lengths along the
106	X, Y and Z axis:
107	</P>
108	<P>
109	<table border=1 cellpadding=8>
110	<tr>
111	<td><tt>pX</tt><td>half length in X
112	<td><tt>pY</tt><td>half length in Y
113	<td><tt>pZ</tt><td>half length in Z
114	</tr>
115	</table>
116	</P>
117	<P>
118	This will create a box that extends from <tt>-pX</tt> to
119	<tt>+pX</tt> in X, from <tt>-pY</tt> to <tt>+pY</tt> in Y,
120	and from <tt>-pZ</tt> to <tt>+pZ</tt> in Z.
121	</P>
122	<P> </P><P> </P>
123	<P>
124	<div align="right"><font size=-1><I>
125	<U>In the picture</U>: pX = 30, pY = 40, pZ = 60
126	</I></font></div>
127	</P>
128	<P>
129	For example to create a box that is 2 by 6 by 10 centimeters
130	in full length, and called <tt>BoxA</tt> one should use the
131	following code:
132	</P>
133	</td>
134	<td>
135	<a href="javascript:remote_win(1)"
136	onMouseOver="window.status='Get alive picture...'; return true">
137	<img src="geometry.src/aBox.jpg" border=0></a>
138	</td>
139	</tr>
140	</table>
141
142	<PRE>
143	G4Box* aBox = new G4Box("BoxA", 1.0cm, 3.0cm, 5.0*cm);
144	</PRE>
145	</P>
146	<P>
147	<HR width=40% align=center noshade>
148	</P>
149	<P>
150	<table border="0" width="100%" id="table2">
151	<tr>
152	<td width="480" valign="top">
153	Similarly to create a <b>cylindrical section</b> or <b>tube</b>,
154	one would use the constructor:
155	<P>
156	<font face="Courier">
157	G4Tubs(const G4String& pName,<br>
158
159	G4double  pRMin,<br>
160
161	G4double  pRMax,<br>
162
163	G4double  pDz,<br>
164
165	G4double  pSPhi,<br>
166
167	G4double  pDPhi)
168	</font>
169	<P> </P><P> </P><P> </P><P> </P>
170	<P>
171	<div align="right"><font size=-1><I>
172	<U>In the picture</U>: pRMin = 10, pRMax = 15, pDz = 20<br>
173	pSPhi = 0Degree, pDPhi = 90Degree
174	</I></font></div>
175	</P>
176	</td>
177	<td>
178	<a href="javascript:remote_win(2)"
179	onMouseOver="window.status='Get alive picture...'; return true">
180	<img src="geometry.src/aTubs.jpg" border=0 height=380></a>
181	</td>
182	</tr>
183	</table>
184
185	giving its name <tt>pName</tt> and its parameters which are:</P>
186	<p>
187	<table border=1 cellpadding=8>
188	<tr>
189	<td><tt>pRMin</tt> <td>Inner radius
190	<td><tt>pRMax</tt> <td>Outer radius
191	<tr>
192	<td><tt>pDz</tt> <td> half length in z
193	<td><tt>pSPhi</tt> <td>the starting phi angle in radians
194	<tr>
195	<td><tt>pDPhi</tt> <td>the angle of the segment in radians
196	<td> <td>&n
197	</table>
198	</P>
199	<P>
200	<HR width=40% align=center noshade>
201	</P>
202	<P>
203	<table border="0" width="100%" id="table3">
204	<tr>
205	<td width="480" valign="top">
206	Similarly to create a <b>cone</b>, or <b>conical section</b>,
207	one would use the constructor:
208	<P>
209	<font face="Courier">
210	G4Cons(const G4String& pName,<br>
211
212	G4double  pRmin1,<br>
213
214	G4double  pRmax1,<br>
215
216	G4double  pRmin2,<br>
217
218	G4double  pRmax2,<br>
219
220	G4double  pDz,<br>
221
222	G4double  pSPhi,<br>
223
224	G4double  pDPhi)
225	</font>
226	<P>
227	<div align="right"><font size=-1><I>
228	<U>In the picture</U>: pRmin1 = 5, pRmax1 = 10,<br>
229	pRmin2 = 20, pRmax2 = 25<br>
230	pDz = 40, pSPhi = 0, pDPhi = 4/3*Pi
231	</I></font></div>
232	</P>
233	</td>
234	<td><a href="javascript:remote_win(3)"
235	onMouseOver="window.status='Get alive picture...'; return true">
236	<img src="geometry.src/aCons.jpg" border="0"></a>
237	</td>
238	</tr>
239	</table>
240
241	giving its name <tt>pName</tt>, and its parameters which are:</P>
242	<P>
243	<table border=1 cellpadding=8>
244	<tr>
245	<TD><tt>pRmin1 <td>inside radius at <tt>-pDz</tt>
246	<TD><tt>pRmax1 <td>outside radius at <tt>-pDz</tt>
247	<tr>
248	<TD><tt>pRmin2 <td>inside radius at <tt>+pDz</tt>
249	<TD><tt>pRmax2 <td>outside radius at <tt>+pDz</tt>
250	<tr>
251	<TD><tt>pDz </tt> <td> half length in z
252	<TD><tt>pSPhi</tt> <td> starting angle of the segment in radians
253	<tr>
254	<TD><tt>pDPhi</tt> <td> the angle of the segment in radians
255	<td> <td>
256	</table>
257	</P>
258	<P>
259	<HR width=40% align=center noshade>
260	</P>
261	<P>
262	<table border="0" width="100%" id="table4">
263	<tr>
264	<td width="480" valign="top">
265	A <b>parallelepiped</b> is constructed using:
266	<P>
267	<font face="Courier">
268	G4Para(const G4String& pName,<br>
269
270	G4double  dx,<br>
271
272	G4double  dy,<br>
273
274	G4double  dz,<br>
275
276	G4double  alpha,<br>
277
278	G4double  theta,<br>
279
280	G4double  phi)
281	</font>
282	<P>
283	<div align="right"><font size=-1><I>
284	<U>In the picture</U>: dx = 30, dy = 40, dz = 60<br>
285	alpha = 10Degree, theta = 20Degree,<br>
286	phi = 5*Degree
287	</I></font></div>
288	</P>
289	</td>
290	<td><a href="javascript:remote_win(4)"
291	onMouseOver="window.status='Get alive picture...'; return true">
292	<img src="geometry.src/aPara.jpg" border="0"></a>
293	</td>
294	</tr>
295	</table>
296
297	giving its name <tt>pName</tt> and its parameters which are:</P>
298	<P>
299	<table border=1 cellpadding=8>
300	<tr>
301	<TD><tt>dx,dy,dz</tt> <td> Half-length in x,y,z
302	<tr>
303	<TD valign=top><tt>alpha</tt> <td>Angle formed by the y axis and by the
304	plane joining the centre of the faces <i>parallel</i> to
305	the z-x plane at -dy and +dy
306	<tr>
307	<TD valign=top><tt>theta</tt> <td>Polar angle of the line joining the
308	centres of the faces at -dz and +dz in z
309	<tr>
310	<TD valign=top><tt>phi</tt> <td>Azimuthal angle of the line joining the
311	centres of the faces at -dz and +dz in z
312	</table>
313	</P>
314	<P>
315	<HR width=40% align=center noshade>
316	</P>
317	<P>
318	<table border="0" width="100%" id="table5">
319	<tr>
320	<td width="480" valign="top">
321	To construct a <b>trapezoid</b> use:
322	<P>
323	<font face="Courier">
324	G4Trd(const G4String& pName,<br>
325
326	G4double  dx1,<br>
327
328	G4double  dx2,<br>
329
330	G4double  dy1,<br>
331
332	G4double  dy2,<br>
333
334	G4double  dz)
335	</font>
336	<P> </P><P> </P>
337	<P>
338	<div align="right"><font size=-1><I>
339	<U>In the picture</U>: dx1 = 30, dx2 = 10<br>
340	dy1 = 40, dy2 = 15<br>
341	dz = 60
342	</I></font></div>
343	</P>
344	</td>
345	<td><a href="javascript:remote_win(5)"
346	onMouseOver="window.status='Get alive picture...'; return true">
347	<img src="geometry.src/aTrd.jpg" border="0"></a>
348	</td>
349	</tr>
350	</table>
351
352	to obtain a solid with name <tt>pName</tt> and parameters</P>
353	<P>
354	<table border=1 cellpadding=8>
355	<tr>
356	<TD><tt>dx1</tt> <td>Half-length along x at the surface positioned at <tt>-dz</tt>
357	<tr>
358	<TD><tt>dx2</tt> <td>Half-length along x at the surface positioned at <tt>+dz</tt>
359	<tr>
360	<TD><tt>dy1</tt> <td>Half-length along y at the surface positioned at <tt>-dz</tt>
361	<tr>
362	<TD><tt>dy2</tt> <td>Half-length along y at the surface positioned at <tt>+dz</tt>
363	<tr>
364	<TD><tt>dz</tt> <td>Half-length along z axis
365	</table>
366	</P>
367	<P>
368	<HR width=40% align=center noshade>
369	</P>
370	<P>
371	<table border="0" width="100%" id="table6">
372	<tr>
373	<td width="480" valign="top">
374	To build a <b>generic trapezoid</b>, the <tt>G4Trap</tt> class is provided.
375	Here are the two costructors for a Right Angular Wedge and for the general
376	trapezoid for it:
377	<P>
378	<font face="Courier">
379	G4Trap(const G4String& pName,<br>
380
381	G4double   pZ,<br>
382
383	G4double   pY,<br>
384
385	G4double   pX,<br>
386
387	G4double   pLTX)
388	<P></P>
389	G4Trap(const G4String& pName,<br>
390
391	G4double  pDz,
392	G4double  pTheta,<br>
393
394	G4double  pPhi,
395	G4double  pDy1,<br>
396
397	G4double  pDx1,
398	G4double  pDx2,<br>
399
400	G4double  pAlp1,
401	G4double  pDy2,<br>
402
403	G4double  pDx3,
404	G4double  pDx4,<br>
405
406	G4double  pAlp2)
407	</font>
408	<P>
409	<div align="right"><font size=-1><I>
410	<U>In the picture</U>: pDx1 = 30, pDx2 = 40, pDy1 = 40<br>
411	pDx3 = 10, pDx4 = 14, pDy2 = 16<br>
412	pDz = 60, pTheta = 20*Degree<br>
413	pDphi = 5Degree, pAlph1 = pAlph2 = 10Degree
414	</I></font></div>
415	</P>
416	</td>
417	<td><a href="javascript:remote_win(6)"
418	onMouseOver="window.status='Get alive picture...'; return true">
419	<img src="geometry.src/aTrap.jpg" border="0"></a>
420	</td>
421	</tr>
422	</table>
423
424	to obtain a Right Angular Wedge with name <tt>pName</tt> and parameters:</P>
425	<p>
426	<table border=1 cellpadding=8>
427	<tr>
428	<TD><tt>pZ</tt> <td>Length along z
429	<tr>
430	<TD><tt>pY</tt> <td>Length along y
431	<tr>
432	<TD><tt>pX</tt> <td>Length along x at the wider side
433	<tr>
434	<TD><tt>pLTX</tt> <td>Length along x at the narrower side (<tt>plTX<=pX</tt>)
435	</table>
436	</P>
437	<p>
438	or to obtain the general trapezoid (see the Software Reference Manual):
439	</p>
440	<table border=1 cellpadding=8 id="table31">
441	<tr>
442	<td><tt>pDx1</tt><td>Half x length at y=-pDy
443	<tr>
444	<td><tt>pDx2</tt><td>Half x length at y=+pDy
445	<tr>
446	<td><tt>pDy</tt><td>Half y length
447	<tr>
448	<td><tt>pDz</tt><td>Half z length
449	<tr>
450	<td><tt>pTheta</tt><td>Polar angle of the line joining the centres of the faces at -/+pDz
451	<tr>
452	<td><tt>pDy1</tt><td>Half y length at -pDz
453	<tr>
454	<td><tt>pDx1</tt><td>Half x length at -pDz, y=-pDy1
455	<tr>
456	<td><tt>pDx2</tt><td>Half x length at -pDz, y=+pDy1
457	<tr>
458	<td><tt>pDy2</tt><td>Half y length at +pDz
459	<tr>
460	<td><tt>pDx3</tt><td>Half x length at +pDz, y=-pDy2
461	<tr>
462	<td><tt>pDx4</tt><td>Half x length at +pDz, y=+pDy2
463	<tr>
464	<td><tt>pAlph1</tt><td>Angle with respect to the y axis from the centre of the side
465	(lower endcap)</tr>
466	<tr>
467	<td><tt>pAlph2</tt><td>Angle with respect to the y axis from the centre of the side
468	(upper endcap)</table>
469	<P>
470	<B>Note on <tt>pAlph1/2</tt></B>:
471	the two angles have to be the same due to the planarity condition.
472	</P>
473	<P>
474	<HR width=40% align=center noshade>
475	</P>
476	<P>
477	<table border="0" width="100%" id="table7">
478	<tr>
479	<td width="480" valign="top">
480	To build a <b>sphere</b>, or a <b>spherical shell section</b>, use:
481	<P>
482	<font face="Courier">
483	G4Sphere(const G4String& pName,<br>
484
485	G4double  pRmin,<br>
486
487	G4double  pRmax,<br>
488
489	G4double  pSPhi,<br>
490
491	G4double  pDPhi,<br>
492
493	G4double  pSTheta,<br>
494
495	G4double  pDTheta )
496	</font>
497	<P>
498	<div align="right"><font size=-1><I>
499	<U>In the picture</U>: pRmin = 100, pRmax = 120<br>
500	pSPhi = 0Degree, pDPhi = 180Degree<br>
501	pSTheta = 0 Degree, pDTheta = 180*Degree
502	</I></font>
503	</P>
504	</td>
505	<td><a href="javascript:remote_win(7)"
506	onMouseOver="window.status='Get alive picture...'; return true">
507	<img src="geometry.src/aSphere.jpg" border="0"></a>
508	</td>
509	</tr>
510	</table>
511
512	to obtain a solid with name <tt>pName</tt> and parameters:</P>
513	<p>
514	<table border=1 cellpadding=8>
515	<tr>
516	<TD><tt>pRmin</tt> <td>Inner radius
517	<tr>
518	<TD><tt>pRmax</tt> <td>Outer radius
519	<tr>
520	<TD><tt>pSPhi</tt> <td>Starting Phi angle of the segment in radians
521	<tr>
522	<TD><tt>pDPhi</tt> <td>Delta Phi angle of the segment in radians
523	<tr>
524	<TD><tt>pSTheta</tt> <td>Starting Theta angle of the segment in radians
525	<tr>
526	<TD><tt>pDTheta</tt> <td>Delta Theta angle of the segment in radians
527	</table>
528	</p>
529	<P>
530	<HR width=40% align=center noshade>
531	</P>
532	<P>
533	<table border="0" width="100%" id="table29">
534	<tr>
535	<td width="480" valign="top">
536	To build a <b>full solid sphere</b> use:
537	<P>
538	<font face="Courier">
539	G4Orb(const G4String& pName, <br>
540
541	G4double pRmax)
542	</font>
543	<P>
544	<div align="right"><font size=-1><I>
545	<U>In the picture</U>: pRmax = 100
546	</I></font></div>
547	</P>
548	<P>
549	The Orb can be obtained from a Sphere with:<br>
550	<tt>pRmin</tt> = 0, <tt>pSPhi</tt> = 0, <tt>pDPhi</tt> = 2*Pi,
551	<tt>pSTheta</tt> = 0, <tt>pDTheta</tt> = Pi.
552	</p>
553
554	<table border=1 cellpadding=8 id="table30">
555	<tr>
556	<TD><tt>pRmax</tt> <td>Outer radius
557	</tr>
558	</table>
559	</td>
560	<td><a href="javascript:remote_win(8)"
561	onMouseOver="window.status='Get alive picture...'; return true">
562	<img src="geometry.src/aOrb.jpg" border="0"></a>
563
564	</td>
565	</tr>
566	</table>
567	</P>
568	<P>
569	<HR width=40% align=center noshade>
570	</P>
571	<P>
572	<table border="0" width="100%" id="table8">
573	<tr>
574	<td width="480" valign="top">
575	To build a <b>torus</b> use:
576	<P>
577	<font face="Courier">
578	G4Torus(const G4String& pName,<br>
579
580	G4double  pRmin,<br>
581
582	G4double  pRmax,<br>
583
584	G4double  pRtor,<br>
585
586	G4double  pSPhi,<br>
587
588	G4double  pDPhi)
589	</font>
590	<P>
591	<div align="right"><font size=-1><I>
592	<U>In the picture</U>: pRmin = 40, pRmax = 60, pRtor = 200<br>
593	pSPhi = 0, pDPhi = 90*Degree
594	</I></font></div>
595	</P>
596	</td>
597	<td><a href="javascript:remote_win(9)"
598	onMouseOver="window.status='Get alive picture...'; return true">
599	<img src="geometry.src/aTorus.jpg" border="0"></a>
600	</td>
601	</tr>
602	</table>
603
604	to obtain a solid with name <tt>pName</tt> and parameters:</P>
605	<p>
606	<table border=1 cellpadding=8>
607	<tr>
608	<TD><tt>pRmin</tt> <td>Inside radius
609	<tr>
610	<TD><tt>pRmax</tt> <td>Outside radius
611	<tr>
612	<TD><tt>pRtor</tt> <td>Swept radius of torus
613	<tr>
614	<TD><tt>pSPhi</tt> <td>Starting Phi angle in radians
615	(<tt>fSPhi+fDPhi<=2PI</tt>, <tt>fSPhi>-2PI</tt>)
616	<tr>
617	<TD><tt>pDPhi</tt> <td>Delta angle of the segment in radians
618	</table>
619	</P>
620	<P>
621	In addition, the Geant4 Design Documentation shows in the Solids Class Diagram
622	the complete list of CSG classes, and the STEP documentation contains a
623	detailed EXPRESS description of each CSG solid.</P>
624
625	<P></P>
626
627	<b>Specific CSG Solids</b>
628	<P>
629	<b>Polycons</b> (PCON) are implemented in Geant4 through the
630	<tt>G4Polycon</tt> class:
631	</P>
632	<table border="0" width="100%" id="table9">
633	<tr>
634	<td width="480" valign="top">
635	<font face="Courier">
636	G4Polycone(const G4String& pName,<br>
637
638	G4double  phiStart,<br>
639
640	G4double  phiTotal,<br>
641
642	G4int         numZPlanes,<br>
643
644	const G4double  zPlane[],<br>
645
646	const G4double  rInner[],<br>
647
648	const G4double  rOuter[])<br>
649	<br>
650	G4Polycone(const G4String& pName, <br>
651
652	G4double  phiStart,<br>
653
654	G4double  phiTotal,<br>
655
656	G4int         numRZ,<br>
657
658	const G4double  r[],<br>
659
660	const G4double  z[])
661	</font>
662	<P>
663	<div align="right"><font size=-1><I>
664	<U>In the picture</U>: phiStart = 0Degree, phiTotal = 2Pi<br>
665	numZPlanes = 9<br>
666	rInner = { 0, 0, 0, 0, 0, 0, 0, 0, 0}<br>
667	rOuter = { 0, 10, 10, 5 , 5, 10 , 10 , 2, 2}<br>
668	z = { 5, 7, 9, 11, 25, 27, 29, 31, 35 }
669	</I></font></div>
670	</P>
671	</td>
672	<td><a href="javascript:remote_win(10)"
673	onMouseOver="window.status='Get alive picture...'; return true">
674	<img src="geometry.src/aBREPSolidPCone.jpg" border="0"></a>
675	</td>
676	</tr>
677	</table>
678
679	where:
680	<p>
681	<table border=1 cellpadding=8>
682	<tr>
683	<TD><tt>phiStart</tt> <td>Initial Phi starting angle
684	<tr>
685	<TD><tt>phiTotal</tt> <td>Total Phi angle
686	<tr>
687	<TD><tt>numZPlanes</tt> <td>Number of z planes
688	<tr>
689	<TD><tt>numRZ</tt> <td>Number of corners in r,z space
690	<tr>
691	<TD><tt>zPlane</tt> <td>Position of z planes
692	<tr>
693	<TD><tt>rInner</tt> <td>Tangent distance to inner surface
694	<tr>
695	<TD><tt>rOuter</tt> <td>Tangent distance to outer surface
696	<tr>
697	<TD><tt>r</tt> <td>r coordinate of corners
698	<tr>
699	<TD><tt>z</tt> <td>z coordinate of corners
700	</table>
701	</P>
702	<P>
703	<HR width=40% align=center noshade>
704	</P>
705	<P>
706	<b>Polyhedra</b> (PGON) are implemented through <tt>G4Polyhedra</tt>:
707	</P>
708	<table border="0" width="100%" id="table10">
709	<tr>
710	<td width="480" valign="top">
711	<font face="Courier">
712	G4Polyhedra(const G4String& pName,<br>
713
714	G4double   phiStart,<br>
715
716	G4double   phiTotal,<br>
717
718	G4int         numSide,<br>
719
720	G4int         numZPlanes,<br>
721
722	const G4double  zPlane[],<br>
723
724	const G4double  rInner[],<br>
725
726	const G4double  rOuter[] )<br>
727	<br>
728	G4Polyhedra(const G4String& pName,<br>
729
730	G4double   phiStart,<br>
731
732	G4double   phiTotal,<br>
733
734	G4int          numSide,<br>
735
736	G4int          numRZ,<br>
737
738	const G4double   r[],<br>
739
740	const G4double   z[])
741	</font>
742	<P>
743	<div align="right"><font size=-1><I>
744	<U>In the picture</U>: phiStart = 0, phiTotal= 2 Pi<br>
745	numSide = 5, nunZPlanes = 7<br>
746	rInner = { 0, 0, 0, 0, 0, 0, 0 }<br>
747	rOuter = { 0, 15, 15, 4, 4, 10, 10 }<br>
748	z = { 0, 5, 8, 13 , 30, 32, 35 }
749	</I></font></div>
750	</P>
751	</td>
752	<td><a href="javascript:remote_win(11)"
753	onMouseOver="window.status='Get alive picture...'; return true">
754	<img src="geometry.src/aBREPSolidPolyhedra.jpg" border="0"></a>
755	</td>
756	</tr>
757	</table>
758
759	where:
760	<p>
761	<table border=1 cellpadding=8>
762	<tr>
763	<TD><tt>phiStart</tt> <td>Initial Phi starting angle
764	<tr>
765	<TD><tt>phiTotal</tt> <td>Total Phi angle
766	<tr>
767	<TD><tt>numSide</tt> <td>Number of sides
768	<tr>
769	<TD><tt>numZPlanes</tt> <td>Number of z planes
770	<tr>
771	<TD><tt>numRZ</tt> <td>Number of corners in r,z space
772	<tr>
773	<TD><tt>zPlane</tt> <td>Position of z planes
774	<tr>
775	<TD><tt>rInner</tt> <td>Tangent distance to inner surface
776	<tr>
777	<TD><tt>rOuter</tt> <td>Tangent distance to outer surface
778	<tr>
779	<TD><tt>r</tt> <td>r coordinate of corners
780	<tr>
781	<TD><tt>z</tt> <td>z coordinate of corners
782	</table>
783	</P>
784	<P>
785	<HR width=40% align=center noshade>
786	</P>
787	<P>
788
789	<table border="0" width="100%" id="table11">
790	<tr>
791	<td width="480" valign="top">
792	A <b>tube with an elliptical cross section</b> (ELTU) can be defined
793	as follows:
794	<P>
795	<font face="Courier">
796	G4EllipticalTube(const G4String& pName,<br>
797
798	G4double  Dx,<br>
799
800	G4double  Dy,<br>
801
802	G4double  Dz)
803	</font>
804	<P>
805	The equation of the surface in x/y is
806	<tt>1.0 = (x/dx)2 + (y/dy)2</tt>
807	</P>
808	<P>
809	<table border=1 cellpadding=8 width="455" id="table23">
810	<tr>
811	<td><tt>Dx</tt><td>Half length X
812	<td><tt>Dy</tt><td>Half length Y
813	<td><tt>Dz</tt><td>Half length Z
814	</table></P>
815	<P> </P><P> </P><P> </P><P> </P>
816	<div align="right"><font size=-1><I>
817	<U>In the picture</U>: Dx = 5, Dy = 10, Dz = 20
818	</I></font></div>
819	</td>
820	<td><a href="javascript:remote_win(12)"
821	onMouseOver="window.status='Get alive picture...'; return true">
822	<img src="geometry.src/aEllipticalTube.jpg" border="0"></a>
823	</td>
824	</tr>
825	</table>
826	</P>
827	<P>
828	<HR width=40% align=center noshade>
829	</P>
830	<table border="0" width="100%" id="table17">
831	<tr>
832	<td width="480" valign="top">
833	The general <b>ellipsoid</b> with possible cut in <tt>Z</tt> can be
834	defined as follows:
835	<P>
836	<font face="Courier">
837	G4Ellipsoid(const G4String& pName,<br>
838
839	G4double  pxSemiAxis,<br>
840
841	G4double  pySemiAxis,<br>
842
843	G4double  pzSemiAxis,<br>
844
845	G4double  pzBottomCut=0,<br>
846
847	G4double  pzTopCut=0)
848	</font>
849	<P> </P><P> </P><P> </P>
850	<div align="right"><font size=-1><I>
851	<U>In the picture</U>: pxSemiAxis = 10, pySemiAxis = 20, pzSemiAxis = 50<br>
852	pzBottomCut = -10, pzTopCut = 40
853	</I></font></div>
854	</td>
855	<td><a href="javascript:remote_win(13)"
856	onMouseOver="window.status='Get alive picture...'; return true">
857	<img src="geometry.src/aEllipsoid.jpg" border="0"></a>
858	</td>
859	</tr>
860	</table>
861
862	A general (or triaxial) ellipsoid is a quadratic surface which is given in
863	Cartesian coordinates by:
864	<p>
865
866	<tt>1.0 = (x/pxSemiAxis)2 + (y/pySemiAxis)2  +  (z/pzSemiAxis)**2</tt>
867	</p>
868	where:
869	<P>
870	<table border=1 cellpadding=8 id="table26">
871	<tr>
872	<TD><tt>pxSemiAxis</tt><td>Semiaxis in X
873	<tr>
874	<TD><tt>pySemiAxis</tt> <td>Semiaxis in Y
875	<tr>
876	<TD><tt>pzSemiAxis</tt><td>Semiaxis in Z
877	<tr>
878	<TD><tt>pzBottomCut</tt> <td>lower cut plane level, z<tr>
879	<TD><tt>pzTopCut</tt><td>upper cut plane level, z
880	</table>
881	</P>
882	<P>
883	<HR width=40% align=center noshade>
884	</P>
885	<P>
886	A <b>cone with an elliptical cross section</b> can be defined as follows:
887	<P>
888	<table border="0" width="100%" id="table24">
889	<tr>
890	<td width="480" valign="top">
891	<font face="Courier">
892	G4EllipticalCone(const G4String& pName,<br>
893
894	G4double pxSemiAxis,<br>
895
896	G4double pySemiAxis,<br>
897
898	G4double zMax,<br>
899
900	G4double pzTopCut)
901	</font>
902	<P> </P>
903	<div align="right"><font size=-1><I>
904	<U>In the picture</U>: pxSemiAxis = 30, pySemiAxis = 60<br>
905	zMax = 50, pzTopCut = 25
906	</I></font></div>
907	</td>
908	<td><a href="javascript:remote_win(14)"
909	onMouseOver="window.status='Get alive picture...'; return true">
910	<img src="geometry.src/aEllipticalCone.jpg" border="0"></a>
911	</td>
912	</tr>
913	</table>
914	</P>
915
916	where:
917	<P>
918	<table border=1 cellpadding=8 id="table25">
919	<tr>
920	<TD><tt>pxSemiAxis</tt><td>Semiaxis in X
921	<tr>
922	<TD><tt>pySemiAxis</tt> <td>Semiaxis in Y
923	<tr>
924	<TD><tt>zMax</tt> <td>Height  of elliptical cone<tr>
925	<TD><tt>pzTopCut</tt> <td>upper cut plane level
926	</table>
927	</P>
928	<P>
929	An elliptical cone of height <tt>zMax</tt>, semiaxis <tt>pxSemiAxis</tt>,
930	and semiaxis <tt>pySemiAxis</tt> is given by the parametric equations:
931	<P>
932
933	<tt>x = pxSemiAxis * ( zMax - u ) / u  * Cos(v)</tt><br>
934
935	<tt>y = pySemiAxis * ( zMax - u ) / u * Sin(v)</tt><br>
936
937	<tt>z = u</tt>
938	</P>
939	<P>
940	Where <tt>v</tt> is between <tt>0</tt> and <tt>2*Pi</tt>,
941	and <tt>u</tt> between <tt>0</tt> and <tt>h</tt> respectively.
942	</P>
943	<P>
944	<HR width=40% align=center noshade>
945	</P>
946	<table border="0" width="100%" id="table12">
947	<tr>
948	<td width="480" valign="top">
949	A <b>tube with a hyperbolic profile</b> (HYPE) can be defined as follows:
950	<P>
951	<font face="Courier">
952	G4Hype(const G4String& pName,<br>
953
954	G4double  innerRadius,<br>
955
956	G4double  outerRadius,<br>
957
958	G4double  innerStereo,<br>
959
960	G4double  outerStereo,<br>
961
962	G4double  halfLenZ)
963	</font>
964	<div align="right"><font size=-1><I>
965	<U>In the picture</U>: innerStereo = 0.7, outerStereo = 0.7<br>
966	halfLenZ = 50<br>
967	innerRadius = 20, outerRadius = 30
968	</I></font></div>
969	</td>
970	<td><a href="javascript:remote_win(15)"
971	onMouseOver="window.status='Get alive picture...'; return true">
972	<img src="geometry.src/aHyperboloid.jpg" border="0"></a>
973	</td>
974	</tr>
975	</table>
976	<P>
977	<tt>G4Hype</tt> is shaped with curved sides parallel to the <tt>z</tt>-axis,
978	has a specified half-length along the <tt>z</tt> axis about which it is
979	centred, and a given minimum and maximum radius.<BR>
980	A minimum radius of <tt>0</tt> defines a filled Hype (with hyperbolic
981	inner surface), i.e. inner radius = 0 AND inner stereo angle = 0.<BR>
982	The inner and outer hyperbolic surfaces can have different
983	stereo angles. A stereo angle of <tt>0</tt> gives a cylindrical surface:</P>
984	<P>
985	<table border=1 cellpadding=8>
986	<tr>
987	<td><tt>innerRadius</tt><td>Inner radius
988	<tr>
989	<td><tt>outerRadius</tt><td>Outer radius
990	<tr>
991	<td><tt>innerStereo</tt><td>Inner stereo angle in radians
992	<tr>
993	<td><tt>outerStereo</tt><td>Outer stereo angle in radians
994	<tr>
995	<td><tt>halfLenZ</tt><td>Half length in Z
996	</table>
997	</P>
998	<P>
999	<HR width=40% align=center noshade>
1000	</P>
1001	<P>
1002	<table border="0" width="100%" id="table27">
1003	<tr>
1004	<td width="480" valign="top">
1005	A <b>tetrahedra</b> solid can be defined as follows:
1006	<P>
1007	<font face="Courier">
1008	G4Tet(const G4String& pName,<br>
1009
1010	G4ThreeVector anchor,<br>
1011
1012	G4ThreeVector p2,<br>
1013
1014	G4ThreeVector p3,<br>
1015
1016	G4ThreeVector p4,<br>
1017
1018	G4bool *degeneracyFlag=0)
1019	</font>
1020	<div align="right"><font size=-1><I>
1021	<U>In the picture</U>: anchor = {0, 0, sqrt(3)}<br>
1022	p2 = { 0, 2*sqrt(2/3), -1/sqrt(3) }<br>
1023	p3 = { -sqrt(2), -sqrt(2/3),-1/sqrt(3) }<br>
1024	p4 = { sqrt(2), -sqrt(2/3) , -1/sqrt(3) }
1025	</I></font></div>
1026	</td>
1027	<td><a href="javascript:remote_win(16)"
1028	onMouseOver="window.status='Get alive picture...'; return true">
1029	<img src="geometry.src/aTet.jpg" border="0"></a>
1030	</td>
1031	</tr>
1032	</table>
1033
1034	The solid is defined by 4 points in space:
1035	<P>
1036	<table border=1 cellpadding=8 id="table28" width="292">
1037	<tr>
1038	<td><tt>anchor</tt><td>Anchor point
1039	<tr>
1040	<td><tt>p2</tt><td>Point 2
1041	<tr>
1042	<td><tt>p3</tt><td>Point 3<tr>
1043	<td><tt>p4</tt><td>Point 4<tr>
1044	<td><tt>degeneracyFlag</tt><td>Flag indicating degeneracy of points
1045	</table>
1046	</P>
1047	<P>
1048	<HR width=40% align=center noshade>
1049	</P>
1050	<P>
1051	<table border="0" width="100%" id="table13">
1052	<tr>
1053	<td width="480" valign="top">
1054	A <b>box twisted</b> along one axis can be defined as follows:
1055	<P>
1056	<font face="Courier">
1057	G4TwistedBox(const G4String& pName,<br>
1058
1059	G4double  twistedangle,<br>
1060
1061	G4double  pDx,<br>
1062
1063	G4double  pDy,<br>
1064
1065	G4double  pDz)
1066	</font>
1067	<P> </P><P> </P>
1068	<div align="right"><font size=-1><I>
1069	<U>In the picture</U>: twistedangle = 30*Degree<br>
1070	pDx = 30, pDy =40, pDz = 60
1071	</I></font></div>
1072	</td>
1073	<td><a href="javascript:remote_win(17)"
1074	onMouseOver="window.status='Get alive picture...'; return true">
1075	<img src="geometry.src/aTwistedBox.jpg" border="0"></a>
1076	</td>
1077	</tr>
1078	</table>
1079	<P>
1080	<tt>G4TwistedBox</tt> is a box twisted along the z-axis.
1081	The twist angle cannot be greater than 90 degrees:</P>
1082	<P>
1083	<table border=1 cellpadding=8>
1084	<tr>
1085	<td><tt>twistedangle</tt><td>Twist angle
1086	<tr>
1087	<td><tt>pDx</tt><td>Half x length
1088	<tr>
1089	<td><tt>pDy</tt><td>Half y length
1090	<tr>
1091	<td><tt>pDz</tt><td>Half z length
1092	</table>
1093	</P>
1094	<P>
1095	<HR width=40% align=center noshade>
1096	</P>
1097	<p>
1098	A <b>trapezoid twisted</b> along one axis can be defined as follows:
1099	<p>
1100	<table border="0" width="100%" id="table14">
1101	<tr>
1102	<td width="480" valign="top">
1103	<font face="Courier">
1104	G4TwistedTrap(const G4String& pName,<br>
1105
1106	G4double  twistedangle,<br>
1107
1108	G4double  pDxx1, G4double  pDxx2,<br>
1109
1110	G4double  pDy, G4double  pDz)<br>
1111	<br>
1112	G4TwistedTrap(const G4String& pName,<br>
1113
1114	G4double  twistedangle,<br>
1115
1116	G4double  pDz,<br>
1117
1118	G4double  pTheta, G4double  pPhi,<br>
1119
1120	G4double  pDy1, G4double  pDx1,<br>
1121
1122	G4double  pDx2, G4double  pDy2,<br>
1123
1124	G4double  pDx3, G4double  pDx4,<br>
1125
1126	G4double  pAlph)
1127	</font>
1128	<div align="right"><font size=-1><I>
1129	<U>In the picture</U>: pDx1 = 30, pDx2 = 40, pDy1 = 40<br>
1130	pDx3 = 10, pDx4 = 14, pDy2 = 16<br>
1131	pDz = 60<br>
1132	pTheta = 20Degree, pDphi = 5Degree<br>
1133	pAlph = 10Degree, twistedangle = 30Degree
1134	</I></font></div>
1135	</td>
1136	<td><a href="javascript:remote_win(18)"
1137	onMouseOver="window.status='Get alive picture...'; return true">
1138	<img src="geometry.src/aTwistedTrap.jpg" border="0"></a>
1139	</td>
1140	</tr>
1141	</table>
1142	<P>
1143	The first constructor of <tt>G4TwistedTrap</tt> produces a regular trapezoid
1144	twisted along the <tt>z</tt>-axis, where the caps of the trapezoid are of the
1145	same shape and size.<br>
1146	The second constructor produces a generic trapezoid with
1147	polar, azimuthal and tilt angles.<br>
1148	The twist angle cannot be greater than 90 degrees:
1149	</P>
1150	<P>
1151	<table border=1 cellpadding=8>
1152	<tr>
1153	<td><tt>twistedangle</tt><td>Twisted angle
1154	<tr>
1155	<td><tt>pDx1</tt><td>Half x length at y=-pDy
1156	<tr>
1157	<td><tt>pDx2</tt><td>Half x length at y=+pDy
1158	<tr>
1159	<td><tt>pDy</tt><td>Half y length
1160	<tr>
1161	<td><tt>pDz</tt><td>Half z length
1162	<tr>
1163	<td><tt>pTheta</tt><td>Polar angle of the line joining the centres of the faces at -/+pDz
1164	<tr>
1165	<td><tt>pDy1</tt><td>Half y length at -pDz
1166	<tr>
1167	<td><tt>pDx1</tt><td>Half x length at -pDz, y=-pDy1
1168	<tr>
1169	<td><tt>pDx2</tt><td>Half x length at -pDz, y=+pDy1
1170	<tr>
1171	<td><tt>pDy2</tt><td>Half y length at +pDz
1172	<tr>
1173	<td><tt>pDx3</tt><td>Half x length at +pDz, y=-pDy2
1174	<tr>
1175	<td><tt>pDx4</tt><td>Half x length at +pDz, y=+pDy2
1176	<tr>
1177	<td><tt>pAlph</tt><td>Angle with respect to the y axis from the centre of the side
1178	</table>
1179	</P>
1180	<P>
1181	<HR width=40% align=center noshade>
1182	</P>
1183	<p>
1184	<table border="0" width="100%" id="table15">
1185	<tr>
1186	<td width="480" valign="top">
1187	A <b>twisted trapezoid</b> with the <tt>x</tt> and </tt>y</tt> dimensions
1188	<b>varying along <tt>z</tt></b> can be defined as follows:
1189	<P>
1190	<font face="Courier">
1191	G4TwistedTrd(const G4String& pName,<br>
1192
1193	G4double  pDx1,<br>
1194
1195	G4double  pDx2,<br>
1196
1197	G4double  pDy1,<br>
1198
1199	G4double  pDy2,<br>
1200
1201	G4double  pDz,<br>
1202
1203	G4double  twistedangle)
1204	</font>
1205	<div align="right"><font size=-1><I>
1206	<U>In the picture</U>: dx1 = 30, dx2 = 10<br>
1207	dy1 = 40, dy2 = 15<br>
1208	dz = 60<br>
1209	twistedangle = 30*Degree
1210	</I></font></div>
1211	</td>
1212	<td><a href="javascript:remote_win(19)"
1213	onMouseOver="window.status='Get alive picture...'; return true">
1214	<img src="geometry.src/aTwistedTrd.jpg" border="0"></a>
1215	</td>
1216	</tr>
1217	</table>
1218	</p>
1219	where:
1220	<p>
1221	<table border=1 cellpadding=8>
1222	<tr>
1223	<td><tt>pDx1</tt><td>Half x length at the surface positioned at -dz
1224	<tr>
1225	<td><tt>pDx2</tt><td>Half x length at the surface positioned at +dz
1226	<tr>
1227	<td><tt>pDy1</tt><td>Half y length at the surface positioned at -dz
1228	<tr>
1229	<td><tt>pDy2</tt><td>Half y length at the surface positioned at +dz
1230	<tr>
1231	<td><tt>pDz</tt><td>Half z length
1232	<tr>
1233	<td><tt>twistedangle</tt><td>Twisted angle
1234	</table>
1235	</P>
1236	<P>
1237	<HR width=40% align=center noshade>
1238	</P>
1239	<P>
1240	<table border="0" width="100%" id="table16">
1241	<tr>
1242	<td width="480" valign="top">
1243	A <b>tube section twisted</b> along its axis can be defined as follows:
1244	<P>
1245	<font face="Courier">
1246	G4TwistedTubs(const G4String& pName,<br>
1247
1248	G4double  twistedangle,<br>
1249
1250	G4double  endinnerrad,<br>
1251
1252	G4double  endouterrad,<br>
1253
1254	G4double  halfzlen,<br>
1255
1256	G4double  dphi)
1257	</font>
1258	<P> </P><P> </P><P> </P>
1259	<div align="right"><font size=-1><I>
1260	<U>In the picture</U>: endinnerrad = 10, endouterrad = 15<br>
1261	halfzlen = 20, dphi = 90*Degree<br>
1262	twistedangle = 60*Degree
1263	</I></font></div>
1264	</td>
1265	<td><a href="javascript:remote_win(20)"
1266	onMouseOver="window.status='Get alive picture...'; return true">
1267	<img src="geometry.src/aTwistedTubs.jpg" border="0"></a>
1268	</td>
1269	</tr>
1270	</table>
1271	</P>
1272	<P>
1273	<tt>G4TwistedTubs</tt> is a sort of twisted cylinder which, placed along
1274	the <tt>z</tt>-axis and divided into <tt>phi</tt>-segments is shaped like an
1275	hyperboloid, where each of its segmented pieces can be tilted with a stereo
1276	angle.<br>
1277	It can have inner and outer surfaces with the same stereo angle:
1278	</P>
1279	<P>
1280	<table border=1 cellpadding=8>
1281	<tr>
1282	<td><tt>twistedangle</tt><td>Twisted angle
1283	<tr>
1284	<td><tt>endinnerrad</tt><td>Inner radius at endcap
1285	<tr>
1286	<td><tt>endouterrad</tt><td>Outer radius at endcap
1287	<tr>
1288	<td><tt>halfzlen</tt><td>Half z length
1289	<tr>
1290	<td><tt>dphi</tt><td>Phi angle of a segment
1291	</table>
1292	</P>
1293	<P>
1294	Additional constructors are provided, allowing the shape to be specified
1295	either as:
1296	<UL>
1297	<LI>the number of segments in <tt>phi</tt> and the total angle for all
1298	segments, or</LI>
1299	<LI>a combination of the above constructors providing instead the inner and
1300	outer radii at <TT>z=0</TT> with different <tt>z</tt>-lengths along
1301	negative and positive <tt>z</tt>-axis.</LI>
1302	</UL>
1303	</P>
1304
1305	<P> </P>
1306
1307	<a name="4.1.2.2">
1308	<H4>4.1.2.2 Solids made by Boolean operations</H4></a>
1309
1310	Simple solids can be combined using Boolean operations.
1311	For example, a cylinder and a half-sphere can be combined with the
1312	union Boolean operation.
1313	<P>
1314	Creating such a new <i>Boolean</i> solid, requires:
1315	<UL>
1316	<LI>Two solids
1317	<LI>A Boolean operation: union, intersection or subtraction.
1318	<LI>Optionally a transformation for the second solid.
1319	</UL></P>
1320	<P>
1321	The solids used should be either CSG solids (for examples a box, a
1322	spherical shell, or a tube) or another Boolean solid: the product of a
1323	previous Boolean operation.
1324	An important purpose of Boolean solids is to allow the description of
1325	solids with peculiar shapes in a simple and intuitive way, still allowing
1326	an efficient geometrical navigation inside them.</P>
1327	<P>
1328	Note: The solids used can actually be of any type. However, in order to
1329	fully support the export of a Geant4 solid model via STEP to CAD
1330	systems, we restrict the use of Boolean operations to this subset of
1331	solids. But this subset contains all the most interesting use cases.</P>
1332	<P>
1333	Note: The tracking cost for navigating in a Boolean solid in the
1334	current implementation, is proportional to the number of constituent
1335	solids. So care must be taken to avoid extensive, unecessary use of
1336	Boolean solids in performance-critical areas of a geometry description,
1337	where each solid is created from Boolean combinations of many other
1338	solids.</P>
1339	<P>
1340	Examples of the creation of the simplest Boolean solids are given below:
1341
1342	<PRE>
1343	G4Box* box =
1344	new G4Box("Box",20mm,30mm,40*mm);
1345	G4Tubs* cyl =
1346	new G4Tubs("Cylinder",0,50mm,50mm,0,twopi); // r: 0 mm -> 50 mm
1347	// z: -50 mm -> 50 mm
1348	// phi: 0 -> 2 pi
1349	G4UnionSolid* union =
1350	new G4UnionSolid("Box+Cylinder", box, cyl);
1351	G4IntersectionSolid* intersection =
1352	new G4IntersectionSolid("Box*Cylinder", box, cyl);
1353	G4SubtractionSolid* subtraction =
1354	new G4SubtractionSolid("Box-Cylinder", box, cyl);
1355	</PRE>
1356
1357	where the union, intersection and subtraction of a box and cylinder are
1358	constructed.</P>
1359	<P>
1360	The more useful case where one of the solids is displaced from the
1361	origin of coordinates also exists. In this case the second solid is
1362	positioned relative to the coordinate system (and thus relative to the
1363	first). This can be done in two ways:
1364	<UL>
1365	<LI>Either by giving a rotation matrix and translation vector that
1366	are used to transform the coordinate system of the second solid to the
1367	coordinate system of the first solid. This is called the <I>passive</I>
1368	method.
1369	<LI>Or by creating a transformation that moves the second solid from
1370	its desired position to its standard position, e.g., a box's standard
1371	position is with its centre at the origin and sides parallel to the
1372	three axes. This is called the <I>active</I> method.
1373	</UL></P>
1374	<P>
1375	In the first case, the translation is applied first to move the origin
1376	of coordinates. Then the rotation is used to rotate the
1377	coordinate system of the second solid to the coordinate system of the
1378	first.</P>
1379	<P>
1380	<PRE>
1381	G4RotationMatrix* yRot = new G4RotationMatrix; // Rotates X and Z axes only
1382	yRot->rotateY(M_PI/4.*rad); // Rotates 45 degrees
1383	G4ThreeVector zTrans(0, 0, 50);
1384
1385	G4UnionSolid* unionMoved =
1386	new G4UnionSolid("Box+CylinderMoved", box, cyl, yRot, zTrans);
1387	//
1388	// The new coordinate system of the cylinder is translated so that
1389	// its centre is at +50 on the original Z axis, and it is rotated
1390	// with its X axis halfway between the original X and Z axes.
1391
1392	// Now we build the same solid using the alternative method
1393	//
1394	G4RotationMatrix invRot = *(yRot->invert());
1395	G4Transform3D transform(invRot, zTrans);
1396	G4UnionSolid* unionMoved =
1397	new G4UnionSolid("Box+CylinderMoved", box, cyl, transform);
1398	</PRE>
1399
1400	Note that the first constructor that takes a pointer to the rotation-matrix
1401	(<tt>G4RotationMatrix*</tt>), does NOT copy it.
1402	Therefore once used a rotation-matrix to construct a Boolean solid, it must
1403	NOT be modified.<BR>
1404	In contrast, with the alternative method shown, a <tt>G4Transform3D</tt> is
1405	provided to the constructor by value, and its transformation is stored by the
1406	Boolean solid. The user may modify the <tt>G4Transform3D</tt> and eventually
1407	use it again.</P>
1408	<P>
1409	When positioning a volume associated to a Boolean solid, the relative center
1410	of coordinates considered for the positioning is the one related to the
1411	<i>first</i> of the two constituent solids.</P>
1412
1413	<P> </P>
1414
1415	<a name="4.1.2.3">
1416	<H4>4.1.2.3 Boundary Represented (BREPS) Solids</H4></a>
1417
1418	BREP solids are defined via the description of their boundaries. The
1419	boundaries can be made of planar and second order surfaces.
1420	Eventually these can be trimmed and have holes.
1421	The resulting solids, such as polygonal, polyconical solids
1422	are known as Elementary BREPS.
1423	<P>
1424	In addition, the boundary surfaces can be made of Bezier surfaces and B-Splines,
1425	or of NURBS (Non-Uniform-Rational-B-Splines) surfaces.
1426	The resulting solids are Advanced BREPS.<BR>
1427	<b>Note</b> - <i>Currently, the implementation for surfaces generated by Beziers, B-Splines
1428	or NURBS is only at the level of prototype and not fully functional</i>.<BR>
1429	Extensions are foreseen in the near future, also to allow exchange of geometrical
1430	models with CAD systems.</P>
1431	<P>
1432	We have defined a few simple Elementary BREPS, that can be instantiated
1433	simply by a user in a manner similar to the construction of Constructed Solids
1434	(CSGs). We summarize their capabilities in the following section.</P>
1435	<P>
1436	However most BREPS Solids are defined by creating each surface separately
1437	and tying them together. Though it is possible to do this using code, it is
1438	potentially error prone. So it is generally much more productive to utilize a tool
1439	to create these volumes, and the tools of choice are CAD systems. In future, it will
1440	be possible to import/export models created in CAD systems, utilizing the STEP standard.</P>
1441
1442	<P> </P>
1443
1444	<b>Specific BREP Solids</b>
1445	<P>
1446	We have defined one polygonal and one polyconical shape using BREPS.
1447	The polycone provides a shape defined by a series of conical sections with the
1448	same axis, contiguous along it.</P>
1449	<P>
1450	The polyconical solid <tt>G4BREPSolidPCone</tt> is a shape defined by a set
1451	of inner and outer conical or cylindrical surface sections and
1452	two planes perpendicular to the Z axis. Each conical surface is
1453	defined by its radius at two different
1454	planes perpendicular to the Z-axis. Inner and outer conical surfaces are
1455	defined using common Z planes.</P>
1456	<P>
1457	<PRE>
1458	G4BREPSolidPCone( const G4String& pName,
1459	G4double start_angle,
1460	G4double opening_angle,
1461	G4int num_z_planes, // sections,
1462	G4double z_start,
1463	const G4double z_values[],
1464	const G4double RMIN[],
1465	const G4double RMAX[] )
1466	</PRE>
1467	The conical sections do not need to fill 360 degrees, but can have a common
1468	start and opening angle.</P>
1469	<P>
1470	<table border=1 cellpadding=8>
1471	<tr>
1472	<td><tt>start_angle</tt> <td>starting angle
1473	<tr>
1474	<td><tt>opening_angle</tt> <td>opening angle
1475	<tr>
1476	<td><tt>num_z_planes</tt> <td>number of planes perpendicular to the z-axis used.
1477	<tr>
1478	<td><tt>z_start</tt> <td>starting value of z
1479	<tr>
1480	<td><tt>z_values</tt> <td>z coordinates of each plane
1481	<tr>
1482	<td><tt>RMIN</tt> <td>radius of inner cone at each plane
1483	<tr>
1484	<td><tt>RMAX</tt> <td>radius of outer cone at each plane
1485	</table></P>
1486	<P>
1487	The polygonal solid <tt>G4BREPSolidPolyhedra</tt> is a shape defined by an
1488	inner and outer polygonal surface and two planes perpendicular to the Z axis.
1489	Each polygonal surface is created by linking a series of polygons created at
1490	different planes perpendicular to the Z-axis. All these polygons all have the
1491	same number of sides (<tt>sides</tt>) and are defined at the same Z planes for
1492	both inner and outer polygonal surfaces.</P>
1493	<P>
1494	The polygons do not need to fill 360 degrees, but have a start and
1495	opening angle.</P>
1496	<P>
1497	The constructor takes the following parameters:
1498	<PRE>
1499	G4BREPSolidPolyhedra( const G4String& pName,
1500	G4double start_angle,
1501	G4double opening_angle,
1502	G4int sides,
1503	G4int num_z_planes,
1504	G4double z_start,
1505	const G4double z_values[],
1506	const G4double RMIN[],
1507	const G4double RMAX[] )
1508	</PRE>
1509	which in addition to its name have the following meaning:</P>
1510	<p>
1511	<table border=1 cellpadding=8>
1512	<tr>
1513	<td><tt>start_angle</tt> <td>starting angle
1514	<tr>
1515	<TD><tt>opening_angle</tt> <td>opening angle
1516	<tr>
1517	<TD><tt>sides</tt> <td>number of sides of each polygon in the x-y plane
1518	<tr>
1519	<TD><tt>num_z_planes</tt> <td>number of planes perpendicular to the z-axis used.
1520	<tr>
1521	<TD><tt>z_start</tt> <td>starting value of z
1522	<tr>
1523	<TD><tt>z_values</tt> <td>z coordinates of each plane
1524	<tr>
1525	<TD><tt>RMIN</tt> <td>radius of inner polygon at each corner
1526	<tr>
1527	<TD><tt>RMAX</tt> <td>radius of outer polygon at each corner
1528	</table></P>
1529	<P>
1530	the shape is defined by the number of sides <tt>sides</tt> of the polygon
1531	in the plane perpendicular to the z-axis. </P>
1532
1533	<a name="4.1.2.4">
1534	<H4>4.1.2.4 Tessellated Solids</H4></a>
1535
1536	In Geant4 it is also implemented a class <tt>G4TessellatedSolid</tt> which
1537	can be used to generate a generic solid defined by a number of facets
1538	(<tt>G4VFacet</tt>). Such constructs are especially important for conversion
1539	of complex geometrical shapes imported from CAD systems bounded with generic
1540	surfaces into an approximate description with facets of defined dimension
1541	(see figure 4.1.1).
1542	<P>
1543	<center>
1544	<table BORDER=1 CELLPADDING=8>
1545	<tr>
1546	<td><IMG SRC="geometry.src/cad-tess1.jpg"
1547	ALT="Tessellated imported geometry - 1" height=350 width=420>
1548	<IMG SRC="geometry.src/cad-tess2.jpg"
1549	ALT="Tessellated imported geometry - 2" height=350 width=420></td>
1550	<tr>
1551	<td ALIGN=center>
1552	Figure 4.1.1<br>
1553	Example of geometries imported from CAD system and converted
1554	to tessellated solids.</td>
1555	</tr>
1556	</table>
1557	</center>
1558	</P>
1559	<P>
1560	They can also be used to generate a solid bounded with a generic surface made
1561	of planar facets. It is important that the supplied facets shall form a fully
1562	enclose space to represent the solid.<BR>
1563	Two types of facet can be used for the construction of a
1564	<tt>G4TessellatedSolid</tt>: a triangular facet (<tt>G4TriangularFacet</tt>)
1565	and a quadrangular facet (<tt>G4QuadrangularFacet</tt>).</P>
1566	<P>
1567	An example on how to generate a simple tessellated shape is given below.</P>
1568	<P>
1569	Example:
1570	<center><table border=1 cellpadding=8>
1571	<tr><td>
1572	<PRE>
1573	// First declare a tessellated solid
1574	//
1575	G4TessellatedSolid solidTarget = new G4TessellatedSolid("Solid_name");
1576
1577	// Define the facets which form the solid
1578	//
1579	G4double targetSize = 10*cm ;
1580	G4TriangularFacet *facet1 = new
1581	G4TriangularFacet (G4ThreeVector(-targetSize,-targetSize, 0.0),
1582	G4ThreeVector(+targetSize,-targetSize, 0.0),
1583	G4ThreeVector( 0.0, 0.0,+targetSize),
1584	ABSOLUTE);
1585	G4TriangularFacet *facet2 = new
1586	G4TriangularFacet (G4ThreeVector(+targetSize,-targetSize, 0.0),
1587	G4ThreeVector(+targetSize,+targetSize, 0.0),
1588	G4ThreeVector( 0.0, 0.0,+targetSize),
1589	ABSOLUTE);
1590	G4TriangularFacet *facet3 = new
1591	G4TriangularFacet (G4ThreeVector(+targetSize,+targetSize, 0.0),
1592	G4ThreeVector(-targetSize,+targetSize, 0.0),
1593	G4ThreeVector( 0.0, 0.0,+targetSize),
1594	ABSOLUTE);
1595	G4TriangularFacet *facet4 = new
1596	G4TriangularFacet (G4ThreeVector(-targetSize,+targetSize, 0.0),
1597	G4ThreeVector(-targetSize,-targetSize, 0.0),
1598	G4ThreeVector( 0.0, 0.0,+targetSize),
1599	ABSOLUTE);
1600	G4QuadrangularFacet *facet5 = new
1601	G4QuadrangularFacet (G4ThreeVector(-targetSize,-targetSize, 0.0),
1602	G4ThreeVector(-targetSize,+targetSize, 0.0),
1603	G4ThreeVector(+targetSize,+targetSize, 0.0),
1604	G4ThreeVector(+targetSize,-targetSize, 0.0),
1605	ABSOLUTE);
1606
1607	// Now add the facets to the solid
1608	//
1609	solidTarget->AddFacet((G4VFacet*) facet1);
1610	solidTarget->AddFacet((G4VFacet*) facet2);
1611	solidTarget->AddFacet((G4VFacet*) facet3);
1612	solidTarget->AddFacet((G4VFacet*) facet4);
1613	solidTarget->AddFacet((G4VFacet*) facet5);
1614
1615	Finally declare the solid is complete
1616	//
1617	solidTarget->SetSolidClosed(true);
1618	</PRE>
1619	<tr>
1620	<td align=center>
1621	Source listing 4.1.1<BR>
1622	An example of a simple tessellated solid with <tt>G4TessellatedSolid</tt>.
1623	</table></center></P>
1624	<P>
1625	The <tt>G4TriangularFacet</tt> class is used for the contruction of
1626	<tt>G4TessellatedSolid</tt>. It is defined by three vertices, which shall be
1627	supplied in <I>anti-clockwise order</I> looking from the outside of the solid
1628	where it belongs. Its constructor looks like:</P>
1629	<P>
1630	<PRE>
1631	G4TriangularFacet ( const G4ThreeVector Pt0,
1632	const G4ThreeVector vt1,
1633	const G4ThreeVector vt2,
1634	G4FacetVertexType fType )
1635	</PRE>
1636	i.e., it takes 4 parameters to define the three vertices:</P>
1637	<P>
1638	<table border=1 cellpadding=8>
1639	<tr>
1640	<TD><tt>G4FacetVertexType</tt> <td><tt>ABSOLUTE</tt> in which case <tt>Pt0</tt>,
1641	<tt>vt1</tt> and <tt>vt2</tt> are the three vertices in anti-clockwise
1642	order looking from the outside.
1643	<tr>
1644	<TD><tt>G4FacetVertexType</tt> <td><tt>RELATIVE</tt> in which case the first
1645	vertex is <tt>Pt0</tt>, the second vertex is <tt>Pt0+vt1</tt> and the
1646	third vertex is <tt>Pt0+vt2</tt>, all in anti-clockwise order when
1647	looking from the outside.
1648	</table></P>
1649	<P>
1650	The <tt>G4QuadrangularFacet</tt> class can be used for the contruction of
1651	<tt>G4TessellatedSolid</tt> as well. It is defined by four vertices, which
1652	shall be in the same plane and be supplied in <I>anti-clockwise order</I>
1653	looking from the outside of the solid where it belongs. Its constructor
1654	looks like:</P>
1655	<P>
1656	<PRE>
1657	G4QuadrangularFacet ( const G4ThreeVector Pt0,
1658	const G4ThreeVector vt1,
1659	const G4ThreeVector vt2,
1660	const G4ThreeVector vt3,
1661	G4FacetVertexType fType )
1662	</PRE>
1663	i.e., it takes 5 parameters to define the four vertices:</P>
1664	<P>
1665	<table border=1 cellpadding=8>
1666	<tr>
1667	<TD><tt>G4FacetVertexType</tt> <td><tt>ABSOLUTE</tt> in which case <tt>Pt0</tt>,
1668	<tt>vt1</tt>, <tt>vt2</tt> and <tt>vt3</tt> are the four vertices required
1669	in anti-clockwise order when looking from the outside.
1670	<tr>
1671	<TD><tt>G4FacetVertexType</tt> <td><tt>RELATIVE</tt> in which case the first
1672	vertex is <tt>Pt0</tt>, the second vertex is <tt>Pt0+vt</tt>, the third
1673	vertex is <tt>Pt0+vt2</tt> and the fourth vertex is <tt>Pt0+vt3</tt>, in
1674	anti-clockwise order when looking from the outside.
1675	</table>
1676	</P>
1677
1678	<hr><a href="../../../../Authors/html/subjectsToAuthors.html">
1679	<i>About the authors</a></i> </P>
1680
1681	</body>
1682	</html>

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