#construction.maple
#
# Frank Sottile
# 27 November 2004
# Mexico City
#
###############################################################################
#
#    This file contains the construction of four triangles with 62 common
# tangents
#
################################################################################
interface(quiet=true):
read("procedures"):
################################################################################
Digits :=80:
################################################################################
#
#   We begin with four lines l1, l2, l3, l4 having two common transversals, 
# x and y, which meet the line li at points xi and yi.  For each i=1..4, 
# we choose a plane Pi through li whose intersection with the hyperboloid 
# Hi through the other three lines is `convex'.  The we let m1 be the line
# in P1 tangent to H1 at the point x1, m2 the line in P2 tangent to H2 at the
# point y2, and m3 the line in P3 tangent to H3 at the point x3.
#
#  These lines {l1,m1}, {l2, m2}, {l3, m3} define 8 hyperboloids, and we look 
# at their intersections with the plane P4, as well as the line m4 in P4 tangent 
# to H4 at the point y4 (or x4, there is a choice here that we may exploit).
#
#   Our construction of 88 lines tangent to four triangles involves perturbing
# this configuration, doubling each line li to get (rather thin) triangles.
#
############################################################################
#
#    Lines are represented by two points on them, and hyperboloids by a
# regulus (family of lines) on them.
#
############################################################################
#
#   Step 0: The hyperbolic paraboloid, H1
#
############################################################################
H := (t, a) ->  [t, a, t*a]:
l1:=[H(-1/2,1),  H(1/2,1)]:
l2:=[H(-1/2,-1), H(1/2,-1)]:
l3:=[[1/2,1/2,1/4], [-1/2,-1/2,1/4]]:
l4:=[H(-1/2,0),  H(1/2,0)]:
#
#
lx := subs(t=0, MakeRegulus(l1, l4, l2, t)):
ly := subs(t=1, MakeRegulus(l1, l4, l2, t)):
##############################################################################
#read("plotConics.maple");

#read("plotH1.maple");
##############################################################################
#
#  The planes through the lines l1, l2, l3, l4 are parametrized by the points of
# intersection with either of two lines, which we call lv and lh.
#
lh:=t -> ParameterizeLine([ParameterizeLine(lx,4),ParameterizeLine(ly,4)],t):
lv:=t -> ParameterizeLine([ParameterizeLine(lx,-2),ParameterizeLine(ly,4)],t):
###############################################################################
#
# Pi is the plane containing li and the point pi
#
# These are the values of T for the four planes
#
p1:=lh(1/2):     # A horizontal plane for P1  x-z=0
p2:=lh(1/2):     # A horizontal plane for P2  x+z=0
p3:=lv(1/2):     # A vertical plane for P3  x-y=0
p4:=lv(1/2):     # A vertical plane for P4  y=0
#
###############################################################################
#
#  Equations for the planes
#
Eqs:=[PlaneEquation(l1,p1),PlaneEquation(l2,p2),
      PlaneEquation(l3,p3),PlaneEquation(l4,p4)]:

###############################################################################
#
#  Perturb the lines and tangents
#  
k1:=PerturbLine(l1,p1,-10,4/10000000000000):
c1:=traceHyperboloid(l4,l3,l2, Eqs[1]):
t1:=TangentLine(c1,1):
A:=1062856834547957347037860105/1000000000000000000000000000: 
B:=51692600238045/100000000000000:
m1:=[ ParameterizeLine(l1,A), ParameterizeLine([l1[1],p1],B)]:

###############################################################################
k2:=PerturbLine(l2,p2,-10,1/100000):
c2:=traceHyperboloid(l3,l1,l4, Eqs[2]):
t2:=TangentLine(c2,0):
m2:=perturbTan(t2,l2[1],894218989475/1000000000000,77965112/100000000):
###############################################################################
k3:=PerturbLine(l3,p3,10,-6/1000):
c3:=traceHyperboloid(l1,l4,l2, Eqs[3]):
t3:=TangentLine(c3,1):
m3:=perturbTan(t3,l3[2],185825/1000000,2073/10000):
###############################################################################
k4:=PerturbLine(l4,p4,10,32/100):
c4:=traceHyperboloid(l1,l2,l3, Eqs[4]):
t4:=TangentLine(c4,0):
m4:=perturbTan(t4,l4[2],11/1000,-3/40):
#############################################################################


#
#   Planei.maple plots the picture of the construction in the ith plane
#
#read("plane1.maple");
#read("plane2.maple");
#read("plane3.maple");
#read("plane4.maple");
read("plane4.www.maple");

#T:=[Triangle(l1,k1,m1), Triangle(l2,k2,m2), Triangle(l3,k3,m3), Triangle(l4,k4,m4)]:

#for i from 1 to 4 do
# for j from 1 to 3 do
#  for k from 1 to 3 do
#   T[i][j][k]:=  trunc(10^30*evalf(T[i][j][k],30))/10^30:
#  end do:
# end do:
#end do:
#fprintf(file,"Triangles:=%a:\n",T):

#numberOfTangents(T);