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## Testing of two plane sets for isometry

An isometry of the Euclidean plane is a distance-preserving transformation of the plane.  There are four types: translations,  rotations,  reflections,  and  glide reflections. See http://en.wikipedia.org/wiki/Euclidean_plane_isometry#Classification_of_Euclidean_plane_isometries

Procedure  AreIsometric  checks two plane sets  Set1  and  Set2  and  if there is an isometry of the plane mapping the first set to the second one, then the procedure returns true and false otherwise. Global variable  T  saves the type of isometry and all its parameters. For example, it returns  the rotation center and rotation angle, etc.

Each of the sets  Set1  and  Set2   is the set (or list) consisting of the following objects in any combinations:

1) The points that are defined as a list of coordinates  [х, y] .

2) Segments, which are defined as a set (or list)  of two points  {[x1, y1], [x2, y2]}  or  [[x1, y1], [x2, y2]]  .

3) Curves, which should  be defined as a list of points   [[x1, y1], [x2, y2], ..., [xn, yn]].

Of course, if  n = 2, then the curve is identical to the segment.

Code of the procedure:

AreIsometric:=proc(Set1::{set,list}, Set2::{set,list})

local n1, n2, n3, n4,s1, S, s, l1, l2, S11, f, x0, y0, phi, Sol, x, y, M1, M2, A1, A2, A3, A4, B1, B2, B3, B4, line1, line2, line3, line4, u, v, Sign, g, M, Line1, Line2, Line3, A, B, C, h, AB, CD, Eq, Eq1, T1, T2, i, S1, S2, T11;

global T;

uses combinat;

S1:={};  S2:={};  T1:={}; T2:={};

for i in Set1 do

if i[1]::realcons  then S1:={op(S1),i} else

S1:={op(i), op(S1)};  T1:={op(T1), seq({i[k],i[k+1]}, k=1..nops(i)-1)} fi;

od;

for i in Set2 do

if i[1]::realcons  then S2:={op(S2),i} else

S2:={op(i), op(S2)};  T2:={op(T2), seq({i[k],i[k+1]}, k=1..nops(i)-1)} fi;

od;

n1:=nops(S1);  n2:=nops(S2);  n3:=nops(T1); n4:=nops(T2);

if is(S1=S2) and is(T1=T2) then T:=identity;  return true fi;

if n1<>n2 or n3<>n4 then return false fi;

if n1=1 then T:=[translation, <S2[1,1]-S1[1,1], S2[1,2]-S1[1,2]>];  return true fi;

f:=(x,y,phi)->[(x-x0)*cos(phi)-(y-y0)*sin(phi)+x0, (x-x0)*sin(phi)+(y-y0)*cos(phi)+y0];  g:=(x,y)->[(B^2*x-A^2*x-2*A*B*y-2*A*C)/(A^2+B^2), (A^2*y-B^2*y-2*A*B*x-2*B*C)/(A^2+B^2)];

_Envsignum0 := 1;

s1:=[S1[1], S1[2]];  S:=select(s->is((s1[2,1]-s1[1,1])^2+(s1[2,2]-s1[1,2])^2=(s[2,1]-s[1,1])^2+(s[2,2]-s[1,2])^2),permute(S2, 2));

for s in S do

# Checking for translation

l1:=s[1]-s1[1]; l2:=s[2]-s1[2];

if is(l1=l2) then S11:=map(x->x+l1, S1);

if n3<>0 then T11:={seq(map(x->x+l1, T1[i]), i=1..nops(T1))}; fi;

if n3=0 then  if is(S11=S2) then T:=[translation, convert(l1, Vector)]; return true fi;  else

if is(S11=S2) and is(T11=T2) then T:=[translation, convert(l1, Vector)]; return true fi; fi;

fi;

# Checking for rotation

x0:='x0'; y0:='y0'; phi:='phi'; u:='u'; v:='v'; Sign:='Sign';

if  is(s1[1]-s[1]<>s1[2]-s[2]) then

M1:=[(s1[1,1]+s[1,1])/2, (s1[1,2]+s[1,2])/2]; M2:=[(s1[2,1]+s[2,1])/2, (s1[2,2]+s[2,2])/2]; A1:=s1[1,1]-s[1,1]; B1:=s1[1,2]-s[1,2]; A2:=s1[2,1]-s[2,1]; B2:=s1[2,2]-s[2,2];    line1:=A1*(x-M1[1])+B1*(y-M1[2])=0; line2:=A2*(x-M2[1])+B2*(y-M2[2])=0;

if is(A1*B2-A2*B1<>0) then Sol:=solve({line1, line2}); x0:=simplify(rhs(Sol[1]));   y0:=simplify(rhs(Sol[2])); u:=[s1[1,1]-x0,s1[1,2]-y0]; v:=[s[1,1]-x0,s[1,2]-y0];    else

if is(s[2]-s1[1]=s[1]-s1[2])  then   x0:=(s1[1,1]+s[1,1])/2;  y0:=(s1[1,2]+s[1,2])/2;

if is([x0,y0]<>s1[1]) then  u:=[s1[1,1]-x0,s1[1,2]-y0]; v:=[s[1,1]-x0,s[1,2]-y0]; else

u:=[s1[2,1]-x0,s1[2,2]-y0]; v:=[s[2,1]-x0,s[2,2]-y0]; fi;

else  A3:=s1[2,1]-s1[1,1];  B3:=s1[2,2]-s1[1,2]; A4:=s[2,1]-s[1,1];  B4:=s[2,2]-s[1,2];  line3:=B3*(x-s1[1,1])-A3*(y-s1[1,2])=0;  line4:=B4*(x-s[1,1])-A4*(y-s[1,2])=0;Sol:=solve({line3, line4}); x0:=simplify(rhs(Sol[1])); y0:=simplify(rhs(Sol[2]));

if is(s1[1]=s[1]) then    u:=s1[2]-[x0,y0]; v:=s[2]-[x0,y0]; else

u:=s1[1]-[x0,y0]; v:=s[1]-[x0,y0];  fi;  fi;  fi;

Sign:=signum(u[1]*v[2]-u[2]*v[1]);   phi:=Sign*arccos(expand(rationalize(simplify((u[1]*v[1]+u[2]*v[2])/sqrt(u[1]^2+u[2]^2)/sqrt(v[1]^2+v[2]^2)))));      S11:=expand(rationalize(simplify(map(x->f(op(x), phi), S1))));

if n3<>0 then T11:={seq(expand(rationalize(simplify(map(x->f(op(x), phi), T1[i])))), i=1..nops(T1))}; fi;

if n3=0 then  if is(S11=expand(rationalize(simplify(S2))))  then T:=[rotation, [x0,y0], phi]; return true fi;  else

if is(S11=expand(rationalize(simplify(S2)))) and  is(T11=expand(rationalize(simplify(T2)))) then

T:=[rotation, [x0,y0], phi]; return true fi;  fi;

fi;

od;

# Checking for reflection or glide reflection

for s in S do

AB:=s1[2]-s1[1]; CD:=s[2]-s[1];

if is(AB[1]*CD[2]-AB[2]*CD[1]=0) then  M:=(s1[2]+s[1])/2;

if  is(AB[1]*CD[1]+ AB[2]*CD[2]>0) then  A:=AB[2]; B:=-AB[1];    Line1:=A*(x-M[1])+B*(y-M[2])=0;  else

A:=AB[1]; B:=AB[2];  Line2:=A*(x-M[1])+B*(y-M[2])=0; fi;

else     u:=[AB[1]+CD[1], AB[2]+CD[2]];  A:=u[2]; B:=-u[1];     M:=[(s1[1,1]+s[1,1])/2, (s1[1,2]+s[1,2])/2]; Line3:=A*(x-M[1])+B*(y-M[2])=0;   fi;    C:=-A*M[1]-B*M[2];  h:= simplify(expand(rationalize(s[1]-g(op(s1[1])))));    S11:=expand(rationalize(simplify(map(x->g(op(x))+h, S1))));

if n3<>0 then T11:={seq(expand(rationalize(simplify(map(x->g(op(x))+h, T1[i])))), i=1..nops(T1))}; fi;

if n3=0 then   if is(S11=expand(rationalize(S2))) then

Eq:=A*x+B*y+C=0; Eq1:=`if`(is(coeff(lhs(Eq), y)<>0), y=solve(Eq, y),  x=solve(Eq, x));

if h=[0,0] then  T:=[reflection, Eq1] else T:=[glide_reflection,Eq1,convert(h, Vector)] fi; return true fi; else

if is(S11=expand(rationalize(S2))) and is(T11=expand(rationalize(T2))) then

Eq:=A*x+B*y+C=0; Eq1:=`if`(is(coeff(lhs(Eq), y)<>0), y=solve(Eq, y),  x=solve(Eq, x));

if h=[0,0] then T:=[reflection, Eq1] else

T:=[glide_reflection,Eq1,convert(h, Vector)] fi; return true fi;  fi;

od;

T:='T';   false;

end proc:

Three simple examples:

AreIsometric({[4, 0], [7, 4], [14, 0]}, {[4, 14], [9, 14], [10, 6]});  T;

AreIsometric({[2, 0], [2, 2], [5, 0]}, {[3, 3], [3, 6], [5, 3]});  T;

S1 := {[[5, 5], [5, 20], [10, 15], [15, 20], [15, 5]]}:
S2 := {[[21, 11], [30, 23], [31, 16], [38, 17], [29, 5]]}:
S3 := {[[50, 23], [41, 11], [51, 16], [49, 5], [58, 17]]}:
AreIsometric(S1, S2); T; AreIsometric(S1, S3);

plots[display](plottools[curve](op(S1), thickness = 2, color = green), plottools[curve](op(S2), thickness = 2, color = green), plottools[curve](op(S3), thickness = 2, color = red), scaling = constrained, view = [-1 .. 60, -1 .. 25]);  # Green sets are isometric,  green and red sets aren't

Example with animation:

S1:={[0,0],[-1,2],[2,4],[4,2]}:  S2:={[8,3],[6,6],[8,8],[10,4]}:

AreIsometric(S1, S2);  T;

with(plots): with(plottools):

#  For clarity, instead of points polygons depicted with vertices at these points

N:=50:

A:=seq(rotate(polygon([[0,0],[-1,2],[2,4],[4,2]], color=blue), (k*Pi)/(2*N), [3,7]), k=0..N):  B:=polygon([[8,3],[6,6],[8,8],[10,4]], color=green):  E:=line([3,7], [6,6], color=black, linestyle=2):

C:=seq(rotate(line([3,7], [2,4], color=black, linestyle=2), (k*Pi)/(2*N), [3,7]), k=0..N):  L:=curve([[3,7],[2,4], [-1,2], [0,0],[4,2], [2,4]], color=black, linestyle=2):  T:=textplot([3, 7.2, "Center of rotation"]):

Frames:=seq(display(A[k], B, E,T,L, C[k]), k=1..N+1):

display(seq(Frames[k],k=1..N+1), insequence=true, scaling=constrained);

Finding unique solutions to the problem of Queens (m chess queens on an n×n chessboard not attacking one another). Used the procedures  Queens  and  QueenPic  . See  http://www.mapleprimes.com/posts/200049-Queens-Problem-And-Its-Visualization-With-Maple

Queens(8, 8);  M := [ListTools[Categorize](AreIsometric, S)]:

nops(M);

seq(op(M[k])[1], k = 1 .. %);   # 12 unique solutions from total 92 solutions

QueensPic([%], 4);  #  Visualization of obtained solutions

Finding unique solutions to the problem "Polygons of matches"  (all polygons with specified perimeter and area). See http://www.mapleprimes.com/posts/142884-Polygons-Of-The-Matches

N := 12: S := 6: Polygons(N, S);

M := [ListTools[Categorize](AreIsometric, T)]:

n := nops(M);

seq([op(M[k][1])], k = 1 .. n);  #  7 unique solutions from total 35 solutions with perimeter 12 and area 6

Visual([%], 4);  #  Visualization of obtained solutions

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