# ======================================================================== #
#
# Copyright 2012, Gabor Hegedus, Josef Schicho, and Hans-Peter Schroecker
#
# This program is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
#
# This program is distributed in the hope that it will be useful, but
# WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
# General Public License for more details.
#
# You should have received a copy of the GNU General Public License
# along with this program. If not, see <http://www.gnu.org/licenses/>.
#
# ======================================================================== #

read("bonds.mpl"):

# define a Bennett linkage
# ========================
h1 := [0, 1, 0, 0]:
h2 := [0, 9*e, 1, -9*e]:
h3 := [0, -1/3-4*e, -2/3+4*e, 2/3+2*e]:
h4 := [0, 2/3+5*e, 1/3+4*e, 2/3-7*e]:

# sanity checks of input data
for i from 1 to 4 do
  if Mul(h||i, Cj(h||i)) <> [1, 0, 0, 0] then
    print("Error: Input quaternions are not normalized!"):
    stop:
  end if:
  if h||i + Cj(h||i) <> [0, 0, 0, 0] then
    print("Error: Input quaternions are not rotations quaternions!"):
    stop:
  end if:
od:

# Ideal of configuration curve
# ============================

for i from 1 to 4 do
  u||i := [t||i, 0, 0, 0] - h||i:
od:
eqs := RVec(Mulall(u1, u2, u3, u4)):

es := {op(eqs[2..-1]), eqs[1]*u-1}:
vars := [t1, t2, u, t3, t4]:
Gb1 := Groebner[Basis]([op(es)], tdeg(t1, t2, u, t3, t4)):
egb := select(T->not (u in indets(T)), Gb1):
egb := Groebner[Basis]({op(egb)}, plex(t4, t3, t2, t1)):
configcurve := solve(egb, [t1, t2, t3, t4]);

# parametrized equation of configuration curve
C := subs(t4 = -t, [rhs(configcurve[1][1]),
                    rhs(configcurve[1][2]),
                    rhs(configcurve[1][3]),
                    rhs(configcurve[1][4])]);

for i from 1 to 4 do
  U||i := [C[i], 0, 0, 0] - h||i:
od:
factor(Mulall(U1, U2, U3, U4)[1]);
sol := [solve(%, t)];

bonds := [seq(subs(t = sol[i], C), i=1..4)];

# Verify vanishing of certain coupler polynomials:
u5 := u1: u6 := u2: u7 := u3: u8 := u4:
for j from 1 to nops(bonds) do
  # assign(bonds[j]):
  t1 := bonds[j][1]:
  t2 := bonds[j][2]:
  t3 := bonds[j][3]:
  t4 := bonds[j][4]:
  i := 1:
  while (t||i^2 <> -1) do i := i + 1: od:
  printf("	F_{%d,%d}(%a, %a, %a, %a) = %a\n\n", i, cc(i+2, 4), t1, t2, t3, t4, Mulall(seq(u||k, k=i..i+2))):
  printf("	F_{%d,%d}(%a, %a, %a, %a) = %a\n\n", cc(i+2, 4), cc(i+4, 4), t1, t2, t3, t4, Mulall(seq(u||k, k=i+2..i+4))):
  unassign('t1', 't2', 't3', 't4'):
od:

# Compute all local distances
# ===========================
for j from 1 to nops(bonds) do
  t1 := bonds[j][1]:
  t2 := bonds[j][2]:
  t3 := bonds[j][3]:
  t4 := bonds[j][4]:
  t0 := -t4:
  printf("\n		%a\n", map(simplify, subs(t = t0, C))):
  for k from 1 to 4 do
    for l from k+1  to 4 do
      F||k||l := map(simplify, Mulall(seq([C[m], 0, 0, 0] - h||m, m=k+1..l))):
      v||k||l := min(map(vanishingOrder, F||k||l, t, t0)):
      d||k||l := 1/2*vanishingOrder(simplify(pNorm(F||k||l)), t, t0) - v||k||l:
      printf("		d(%d,%d) = %a	(v_{%d,%d}=%a)\n", k, l, d||k||l, k, l, v||k||l):
    od:
  od:
  unassign('t0', 't1', 't2', 't3', 't4'):
od:

# Details of computation of d13 and bond (-1-I, -I, -1-I, I)
# ==========================================================
t0 := -I:
for i from 1 to 4 do
  t||i := subs(t = t0, C[i]):
od:

F13 := map(factor, Mulall([C[2], 0, 0, 0] - h2, [C[3], 0, 0, 0] - h3));
map(simplify, subs(t = t0, F13));
factor(pNorm(F13));
v13 := min(map(vanishingOrder, F13, t, t0));
d13 := 1/2*vanishingOrder(simplify(pNorm(F13)), t, t0);
d13 := d13 - v13;