%
% Data points for the final exam MAT 342, Fall 1996
%
p=[1 -2; 1 0; 5 0; 2 4; 3 4; 0 7; -3 4; -2 4; -4 1; -4 0; -1 0; -1 -2; 1 -2]'
q=[24 5;20 5;20 9;12 6;12 7; 6 4; 12 1; 12 2; 18 0; 20 0; 20 3; 24  3;24  5]'
%
%
ffill(p,'g')
hold on
ffill(q,[.5 .3 0])
grid
%
%
%
%
disp('strike any key to continue')
pause
%
%
%
disp('Hint: Given a 2x2-matrix A and a 2x2 vector v, draw the')
disp('      image of the polygon p under the map x --> Ax+v ')
disp('      using the following commands -- just an example.')
%
A=[2 0;0 .6]
v=[0;-10]
%
disp('You need to make v into a matrix of the same size as p')
disp('strike any key to continue')
pause
disp('Overlay the new plot -- look at the graphics window!')
%
%
disp(' ')
echo on
vv=diag(v')*ones(size(p))
ffill(A*p+vv,'b')
echo off
disp(' ')
disp(' ')
%
disp('Trial and error solutions will not earn credit. You need')
disp('to demonstrate how to use linear algebra effectively to')
disp('find the matrix A and the vector v such that x --> Ax+v')
disp('maps the green tree into the given brown one.')
%
disp(' ')
disp('Strike any key to continue for the solution.')
pause
%
disp('Write out the matrix equation  Pnew = A * Qnew + v in coordinates')
disp(' ')
disp('      xnew=a11*xold+a12*yold+v1       one equation for each point')
disp('      ynew=a21*xold+a22*yold+v2       one equation for each point')
disp(' ')
disp('For clarity we only use the first three points, and write the system')
disp('of equations in matrix from:  AA * XX = BB, where XX is the vector')
disp('of unknowns XX=[a11 a12 v1 a21 a22 v2]'', BB is a vector with coords')
disp('of the new points BB=[xnew1 xnew2 xnew3 ynew1 ynew2 ynew3]'', and AA')
disp('a 6x6 matrix containing the coordinates of the old points:')
disp(' ')
disp('Strike any key to continue')
pause
%
BB=[q(1,1:3) q(2,1:3)]'
AA=[p(1,1:3)' p(2,1:3)' ones([3,1]) zeros([3,3]); ...
    zeros([3,3]) p(1,1:3)' p(2,1:3)' ones([3,1])]
%
disp('You may decide to type in the matrix by hand; it is not too large.')
disp(' ')
disp('Next solve the system of equations -- create A and v, and check:')
disp('Strike any key to continue')
pause
%
XX=AA\BB
A=[XX(1) XX(2); XX(4) XX(5)]
v=[XX(3);XX(6)]
pplot(A*p+diag(v')*ones(size(p)),'m')
%
disp('Check the graph for a visual check.')
disp('Strike any key when ready to continue with an algebraic check:')
%
disp('old points')
p
disp('given new points')
q
disp('calculated new points')
qq=A*p+diag(v')*ones(size(p))
disp('take the difference of the two matrices and display min and max entry')
[min(min(q-qq)),max(max(q-qq))]
disp(' ')
disp('There is a much less naive solution -- but it takes some xperience')
disp('to immediately come up with the solution. The key is to first eliminate')
disp('the translation -- the trick is to only consider differences of points!')
disp('E.g. A*p1+v=q1 and A*p2+v=q2 leads to A(p2-p1)=(q2-q1) -- the v''s are gone!')
disp('Note, since A*pdiff=qdiff, we need A=qdiff/pdiff -- division on the right!')
disp(' ')
disp('Strike any key when ready to continue')
disp(' ')
echo on
pdiff=p(:,2:12)-p(:,1:11)
qdiff=q(:,2:12)-q(:,1:11)
AA=qdiff(:,1:2)/pdiff(:,1:2)
AA=qdiff/pdiff
vv=q-AA*p
v=v(:,1)


disp('Merry Xmas and Happy Holidays')