#ifndef OctfExtensions_H
#define OctfExtensions_H

/* Last edited on 2009-02-09 18:26:38 by stolfi */

/* Extensions to the facet-edge data strcuture. */
   
#define OctfExtensions_H_copyright \
  "Copyright © 1998 Universidade Estadual de Campinas (UNICAMP)"
   
/* ELEMENT ORIENTATIONS
  
  The /canonical {d}-dimensional simplex/ {K^d} is the regular
  tetrahedron in {R^{d+1}} whose vertices are the unit points on each
  axis, {u[d,0] = (1,0,...,0), u[d,1] = (0,1,...,0), ..., u[d,d] = (0,0,...,1)}.
  
  Let $p$ be a place in either map, and $h$ be any homeomorphism from
  {K^3} to {p.s$} that takes the corner {u[3,i]} of {K^3} to knot
  {p.knt[i]} of {p.s}, for all {i} in {0..3}. The /orientation defined
  by {p} on {p.s$}/ is the class of homeomorphisms of {K^3} to {p.s$}
  that is smoothly deformable to {h}.
    
  The orientation defined by {s} on {s$} induces internal orientations for the
  elements of {s} . An orientation of a node is just a sign {+} or
  {-}. An internal orientation on an edge allows us to distinguish {+}
  from {-} continuous travel along it. An internal orientation of a
  wall {f} allows us to distinguish {+} and {-} traversal of any
  Jordan curve on {f}.
  
  So, for instance, the {+} travel along {e.elm[1]} is the one that
  traverses the link {r} of {s} with {M}-type {{0,1}} in the direction
  from {(s,M).knt[0]} to {(s,M).knt[1]}. Observe that the knots
  {(s,M).knt[i]} and {(s,M).knt[1]} are always distinct, whereas the
  two endpoints of {e} may be the same node of {M}. Similarly,
  the {+} orientation on {s.elm[2]} is that which agrees 
  with the oriented triangle {s.knt[0],s.knt[1],s.knt[2]}.
  
   even then, {s} and {t} will
  imply different orientations for at least one element of that 
  element sequence.
  
  In particular, each chip {s} defines
  
  The orientation provided by {s} on its elements makes it possible to
  move around {M} without ambiguity, e.g. by defining what is the
  `origin' of the edge {e = (s,M).elm[1]}; the `next edge' after {e}
  around wall [f = (s,M).elm[2]}; and so forth. */
 
/* ORIENTED DATA POINTERS
  
  The value of each slot {p.data[i]} is an /oriented data pointer/,
  which consists of a /base data pointer/
  combined with /data orientation bits/.  These bits depend on the
  internal and external orientation of the element {p.el[i]}, and which 
  of the two maps {p} belongs to. 
  
  The operations {iflip}, {oflip} and {dual} apply to oriented data
  pointers. They preserve the data pointer and change the 
  orientation bits: {iflip} and {oflip} reverse the internal and external
  orientations, respectively, and {dual} exchanges the underlying map.
  The three operators satisfy the identities {iflip.oflip == oflip.iflip},
  {iflip.dual == dual.oflip}, {oflip.dual == dual.iflip},
  {oflip.oflip == iflip.iflip == dual.dual == nop}.
  Moreover, for any oriented data pointer {t}, all eight oriented
  pointers {t.oflip^o.iflip^i.dual^d} are distinct.
  
  The data structure enforces the identities 
    
    {p.flp[0].data[1] == p.data[1].iflip}
    {p.flp[0].data[2] == p.data[2].iflip}
    
    {p.flp[3].data[1] == p.data[1].oflip}
    {p.flp[3].data[2] == p.data[2].oflip}
    
    {p.dual.data[i] == p.data[3-i].dual}
    
  */
  

#define OctfExtensions_H_author \
  "Created by J. Stolfi, UNICAMP in jan/2007."

#endif