Updated doc page for triclinic cell

git-svn-id: svn://svn.icms.temple.edu/lammps-ro/trunk@8004 f3b2605a-c512-4ea7-a41b-209d697bcdaa

doc/Section_howto.txt

@@ -760,11 +760,11 @@ See the "dump"_dump.html command for more information on XTC files.
 
 By default, LAMMPS uses an orthogonal simulation box to encompass the
 particles.  The "boundary"_boundary.html command sets the boundary
-conditions of the box (periodic, non-,periodic, etc).  The orthogonal
+conditions of the box (periodic, non-periodic, etc).  The orthogonal
 box has its "origin" at (xlo,ylo,zlo) and is defined by 3 edge vectors
 starting from the origin given by [a] = (xhi-xlo,0,0); [b] =
 (0,yhi-ylo,0); [c] = (0,0,zhi-zlo).  The 6 parameters
-(xlo,xhi,ylo,yhi,zlo,zhi) are defined at the time the simluation box
+(xlo,xhi,ylo,yhi,zlo,zhi) are defined at the time the simulation box
 is created, e.g. by the "create_box"_create_box.html or
 "read_data"_read_data.html or "read_restart"_read_restart.html
 commands.  Additionally, LAMMPS defines box size parameters lx,ly,lz
@@ -772,30 +772,53 @@ where lx = xhi-xlo, and similarly in the y and z dimensions.  The 6
 parameters, as well as lx,ly,lz, can be output via the "thermo_style
 custom"_thermo_style.html command.
 
-LAMMPS also allows simulations to be perfored in triclinic
+LAMMPS also allows simulations to be performed in triclinic
 (non-orthogonal) simulation boxes shaped as a parallelepiped with
 triclinic symmetry.  The parallelepiped has its "origin" at
 (xlo,ylo,zlo) and is defined by 3 edge vectors starting from the
 origin given by [a] = (xhi-xlo,0,0); [b] = (xy,yhi-ylo,0); [c] =
-(xz,yz,zhi-zlo).  {Xy,xz,yz} can be 0.0 or positive or negative values
+(xz,yz,zhi-zlo).  {xy,xz,yz} can be 0.0 or positive or negative values
 and are called "tilt factors" because they are the amount of
 displacement applied to faces of an originally orthogonal box to
-transform it into the parallelepiped.  Note that in LAMMPS the
+transform it into the parallelepiped.  In LAMMPS the
 triclinic simulation box edge vectors [a], [b], and [c] cannot be
-arbitrary vectors.  As indicated, [a] must be aligned with the x axis,
-[b] must be in the xy plane, and [c] is arbitrary.  However, this is
-not a restriction since it is possible to rotate any set of 3 crystal
-basis vectors so that they meet this restriction.
+arbitrary vectors.  As indicated, [a] must lie on the positive x axis.
+[b] must lie in the xy plane, with strictly positive y component. [c] may 
+have any orientation with strictly positive z component.
+The requirement that [a], [b], and [c] have strictly positive x, y,
+and z components, respectively, ensures
+that [a], [b], and [c] form a complete right-handed basis.
+These restrictions impose
+no loss of generality, since it is possible to rotate/invert 
+any set of 3 crystal basis vectors so that they conform to the restrictions.
 
 For example, assume that the 3 vectors [A],[B],[C] are the edge
-vectors of a general parallelipied, where there is no directional
-requirements [A],[B],[C] other than they are not co-planar and that
-[C] dotted into ([A] x [B]) be > 0, i.e.  the vectors are ordered to
-satisfy a right-hand rule.  The equivalent LAMMPS [a],[b],[c] vectors
-can be computed as follows where A = |[A]| = scalar length of [A].
+vectors of a general parallelepiped, where there is no restriction
+on [A],[B],[C] other than they form a complete right-handed basis i.e.
+[A] x [B] . [C] > 0.
+The equivalent LAMMPS [a],[b],[c] are a linear rotation of [A], [B], and
+[C] and can be computed as follows:
 
 :c,image(Eqs/transform.jpg)
 
+where A = |[A]| indicates the scalar length of [A]. The ^ hat symbol
+indicates the corresponding unit vector. beta and gamma are angles
+between the vectors described below. The same rotation must also
+be applied to atom positions, velocities, and any other vector quantities.
+This can be done by first converting to fractional coordinates in the
+old basis and then converting to distance coordinates in the new basis.
+The transformation is given by the following equation:
+
+:c,image(Eqs/rotate.jpg)
+
+where V is the volume of the box, [X] is the original vector quantity and 
+[x] is the vector in the LAMMPS basis.
+
+If it should happen that
+[A], [B], and [C] form a left-handed basis, then it is necessary
+to first apply an inversion in addition to rotation. This can be achieved
+by interchanging two of the basis vectors or changing the sign of one of them.
+
 There is no requirement that a triclinic box be periodic in any
 dimension, though it typically should be in at least the 2nd dimension
 of the tilt (y in xy) if you want to enforce a shift in periodic