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{SECT 0 {EXCHG {PARA 3 "" 0 "" {TEXT -1 81 "Procedures to draw the Pro
cedure Line and Approval Voting Region for 3 Candidates" }}{PARA 19 "
" 0 "" {TEXT -1 46 "Tommy Ratliff, Wheaton College, March 27, 2002" }}
{PARA 19 "" 0 "" {TEXT -1 27 "tratliff@wheatoncollege.edu" }}}{SECT 1 
{PARA 3 "" 0 "" {TEXT -1 10 "Directions" }}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 101 "You will need to execute the  commands in the next secti
on in order to have the functions available. " }}{PARA 0 "" 0 "" 
{TEXT -1 62 "It may take a little while for Maple to process the funct
ions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "
The most important syntax to know is:" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 111 "     plotprofile(a,b,c,d,e,f) will p
lot the procedure line and the approval voting region for the profile \+
where" }}{PARA 0 "" 0 "" {TEXT -1 451 "        a voters have preferenc
e A>B>C\n        b voters have preference A>C>B\n        c voters have
 preference C>A>B\n        d voters have preference C>B>A\n        e v
oters have preference B>C>A\n        f voters have preference B>A>C\n
\n   plur(a,b,c,d,e,f) calculates the coordinates in the representatio
n triangle using the plurality method\n   aplur(a,b,c,d,e,f) calculate
s the coordinates in the representation triangle using the antiplurali
ty method" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{SECT 1 {PARA 3 "" 0 "
" {TEXT -1 24 "Definition of Functions " }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "restart:with(plots):wit
h(geometry):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "plur:= (a,b,c,d,e,f
) -> [ (a+b)/(a+b+c+d+e+f), (e+f)/(a+b+c+d+e+f), (c+d)/(a+b+c+d+e+f)]:
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "aplur:= (a,b,c,d,e,f) -> [ (a+
b+c+f)/(2*(a+b+c+d+e+f)), (e+f+a+d)/(2*(a+b+c+d+e+f)), (c+d+b+e)/(2*(a
+b+c+d+e+f))]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "prdown := (a,b,c)
 -> [ -a/sqrt(2) + b/sqrt(2), -a/sqrt(6)-b/sqrt(6)+sqrt(2/3)*c]:" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "prup := (x,y) -> [-1/2*sqrt(2)*x-1
/6*sqrt(6)*y+1/3, 1/2*sqrt(2)*x-1/6*sqrt(6)*y+1/3, 1/3+1/3*sqrt(6)*y]:
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 158 "tri:=spacecurve(\{[ [1,0,0],[0,1,0],[0,0,1],[1,0,0]], [ [1,0,
0],[0,1/2,1/2]], [ [0,1,0],[1/2,0,1/2]], [ [0,0,1],[1/2,1/2,0]]\},colo
r=black,scaling=constrained):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "p
rocline:=(a,b,c,d,e,f) -> spacecurve(\{[ plur(a,b,c,d,e,f), aplur(a,b,
c,d,e,f)]\}, color=blue,thickness=2, axes=none,style=line):" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 150 "procpoints:=(a,b,c,d,e,f) -> spacecurve(
\{[ plur(a,b,c,d,e,f), aplur(a,b,c,d,e,f)]\}, color=green, axes=none,s
tyle=point, symbolsize=15, symbol=CIRCLE):" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 115 "normalize := pnt -> ( pnt[1]/(pnt[1]+pnt[2]+pnt[3]),
 pnt[2]/(pnt[1]+pnt[2]+pnt[3]), pnt[3]/(pnt[1]+pnt[2]+pnt[3])):" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 860 "apvertices := proc(a,b,c,d,e,f) lo
cal A,B,C,D,E,F,G,H, prof, i, po, apo,CH,L;\n    prof:=(a,b,c,d,e,f);
\n    po:= [a+b, e+f, c+d];\n    apo:= [a+b+c+f, e+f+a+d, c+d+b+e];\n
\n    point(A, prdown(normalize([po[1],po[2],po[3]])  ));\n    point(B
, prdown(normalize([po[1],po[2],apo[3]]) ));\n    point(C, prdown(norm
alize([po[1],apo[2],po[3]]) ));\n    point(D, prdown(normalize([po[1],
apo[2],apo[3]]) ));\n    point(E, prdown(normalize([apo[1],po[2],po[3]
]) ));\n    point(F, prdown(normalize([apo[1],po[2],apo[3]]) ));\n    \+
point(G, prdown(normalize([apo[1],apo[2],po[3]]) ));\n    point(H, prd
own(normalize([apo[1],apo[2],apo[3]]) ));\n    \n    CH:=convexhull(\{
A,B,C,D,E,F,G,H\});\n    L:=[]: \n    for i from 1 to nops(CH) do \n  \+
       L:=[op(L), prup(op(coordinates(CH[i])))]:\n         od: \n    L
:=[op(L),L[1]]:\n     \n    return(spacecurve(L,color=red,axes=none));
\nend proc:\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 
"" {MPLTEXT 1 0 119 "plotprofile := (a,b,c,d,e,f) -> display(tri, proc
line(a,b,c,d,e,f), procpoints(a,b,c,d,e,f), apvertices(a,b,c,d,e,f) ):
" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "plotprofile(10,0,6,6,3
,6);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "plur
(10,0,6,6,3,6);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "aplur(10
,0,6,6,3,6);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK 
"6 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 
1 }