From c3e758476ec10723ccb81a7dd91967d2996665b9 Mon Sep 17 00:00:00 2001
From: Ian Jackson <ian@davenant.relativity.greenend.org.uk>
Date: Wed, 16 Jan 2008 18:07:02 +0000
Subject: [PATCH] before abandon normalisation effort

---
 energy.c | 68 +++++++++++++++++++++++++++++++++++++++++++++++---------
 1 file changed, 58 insertions(+), 10 deletions(-)

diff --git a/energy.c b/energy.c
index db13fbf..1d0360f 100644
--- a/energy.c
+++ b/energy.c
@@ -138,24 +138,64 @@ void compute_vertex_areas(const Vertices vertices) {
    *      is nonnegative.  We add epsilon to |AxB| to avoid division
    *      by zero.
    *
+   *                     r
+   *      cost    = delta
+   *          Q,e
+   *
    * Normalisation:
    *
-   *    We want the answer to remain unchanged when the vertices lie
-   *    on circles.  Interposing M and N so that we have P-M-Q-N-R
+   *    We want the minimum energy to remain unchanged with changes in
+   *    triangle densitiy, when the vertices lie evenly spaced on
+   *    circles, and we do this by normalising the force ie the
+   *    derivative of the energy with respect to linear motions of the
+   *    vertices.
+   *
+   *    We consider only the force on Q due to PQR, wlog.  (Forces on
+   *    P qnd R due to PQR are equal and opposite so normalising
+   *    forces on Q will normalise them too.)
+   *
+   *    Force on Q is in the plnae PQR and normal to PR, so we can
+   *    consider it only linearly in that dimension.  WLOG let that be
+   *    the x dimension.  So with f' representing df'/dx_Q:
+   *
+   *                  ,         d
+   *      F    =  cost      =  --
+   *       Q,e         Q,e           err looks like we can only do
+   *                                 this if we make some kind of
+   *                                 assumption about delta or
+   *                                 something   give up
+   *
+   *
+   *    Interposing M and N so that we have P-M-Q-N-R
    *    generates half as much delta for each vertex.  So
    *
-   *           ,       -1
+
+   In that case the force on Q
+   *    due to PQR
+   *
+   *Normalising for equal linear
+   *    forces:
+   *
+   *                                      d
+   *      linear force on Q due to e  =  -------  cost
+   *                                     d coord       Q,e
+   *                                            Q
+   *
+   *       (we will consider only one e and one coord and hope
+   *        that doesn't lead us astray.)
+   *
+   *
+   *           ,       -r
    *       cost    =  D   . cost
    *           Q,e              Q,e
    *
    *                           where D is the linear density.
    *
-   *                ,              -1   
+   *                ,              -r
    *     Sigma  cost      =   N . D  .  Sigma  cost
    *      Q,e       Q,e                  Q,e       Q,e
    *
-   *
-   */
+   * */
 
 double line_bending_adjcost(const Vertices vertices) {
   static const double axb_epsilon= 1e-6;
@@ -186,13 +226,21 @@ double line_bending_adjcost(const Vertices vertices) {
 
 /*---------- edge length variation ----------*/
 
+  /*
+   * Definition:
+   *
+   *    See the diagram above.
+   *
+   *       cost    =
+   *           Q,e
+
 double edge_length_variation_cost(const Vertices vertices) {
   double diff, cost= 0;
-  int v0, efwd,vfwd, eback;
+  int q, e,r, eback;
 
-  FOR_EDGE(v0,efwd,vfwd) {
-    eback= edge_reverse(v0,efwd);
-    diff= edge_lengths[v0][efwd] - edge_lengths[v0][eback];
+  FOR_EDGE(q,e,r) {
+    eback= edge_reverse(q,e);
+    diff= edge_lengths[q][e] - edge_lengths[q][eback];
     cost += diff*diff;
   }
   return cost;
-- 
2.30.2