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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 256 25 "Equation f
or the evolute." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 586 "Given an algebraic curve C, and a point P on C, one take
s the circle that touches the curve C at the point P with the highest \+
possible multiplicity, so nearby P the circle and the curve will be as
 close as possible. Let R be the center of that curve. If we compute t
he point R=[R[1],R[2]] for all points P on the curve C, we get another
 curve called the evolute. The question is now, compute an algebraic r
elation F in Q[X,Y] between R[1] and R[2], i.e. compute a polynomial F
 such that subs(X=R[1],Y=R[2], F)=0. Then the evolute, the set of poin
ts R, is just the set of solutions of F." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 403 "The method of constructing R is t
he following. We take a point P=[x,y], and a point [x+delta_x, y+delta
_y] that is infinitely nearby (the delta_x and delta_y depend on a sma
ll epsilon, and we will take the limit epsilon -> 0, simply by substit
ing epsilon=0 at the end). Then we take the normal lines L_1 and L_2 t
hrough those two points, and compute the intersection using Maple's so
lve. Then we find R." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "f:=
x^2+2*y^2-1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG,(*$)%\"xG\"\"#
\"\"\"\"\"\"*$)%\"yGF)F*F)!\"\"F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
72 "The curve C is given by the solutions of f. We see that C is an el
lipse." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 311 "L_1:=[x,y] + t_1
 * [diff(f,x), diff(f,y)];\ndelta_x:=epsilon*diff(f,y) + error1 * epsi
lon^2;\ndelta_y:=-epsilon*diff(f,x) + error2 * epsilon^2;\nL_2:=subs(x
=x+delta_x,y=y+delta_y, t_1=t_2, L_1);\nzero:=expand(L_1-L_2);\nR:=map
(normal,subs(solve(\{op(zero)\},\{t_1,t_2\}),expand(L_1)));\nR:=map(no
rmal,subs(epsilon=0,R));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$L_1G,
&7$%\"xG%\"yG\"\"\"*&%$t_1GF)7$,$F'\"\"#,$F(\"\"%F)F)" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%(delta_xG,&*&%(epsilonG\"\"\"%\"yGF(\"\"%*&%'err
or1GF()F'\"\"#\"\"\"F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(delta_yG,
&*&%(epsilonG\"\"\"%\"xGF(!\"#*&%'error2GF()F'\"\"#\"\"\"F(" }}{PARA 
12 "" 1 "" {XPPMATH 20 "6#>%$L_2G,&7$,(%\"xG\"\"\"*&%(epsilonG\"\"\",(
%\"yGF)*&F+F)F(F)!\"#*&%'error2GF))F+\"\"#F,F)F)\"\"%*&%'error1GF)F3F,
F)F-F)*&%$t_2GF)7$,(F(F4F*\"\")F6F4,(F.F5F/!\")F1F5F)F)" }}{PARA 12 "
" 1 "" {XPPMATH 20 "6#>%%zeroG7$,6*&%\"xG\"\"\"%$t_2GF)!\"#*(%(epsilon
GF)%\"yGF)F*\"\"\"!\")*()F-\"\"#F/F(F/F*F/\"#;*(%'error2GF))F-\"\"$F/F
*F/F0*(%'error1GF)F2F/F*F/F+*&F(F/%$t_1GF)F3*&F-F/F.F/!\"%*&F2F/F(F/\"
\")*&F6F/F7F/F>*&F:F/F2F/!\"\",.*&F.F/F*F/F>*(F-F/F(F/F*F/F@*(F6F/F2F/
F*F/F>*&F.F/F<F/\"\"%*&F-F/F(F/F3*&F6F/F2F/FC" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%\"RG7$,$*&*&%\"xG\"\"\",4*(%(epsilonG\"\"\"F)F.%\"yGF
.!\")*(%'error2GF.)F-\"\"#F*F/F*\"\"%*&F3F*)F)F4F*\"#;*&)F2F4F*)F-F5F*
F5*$F7F*!\"#*(F)F*F2F*F-F*F.*()F-\"\"$F*F)F*F2F*!#;*(F3F*F)F*%'error1G
F.F=*(F2F*F@F*FDF*F.F.F*,.F<F4F>!\"\"*$)F/F4F*F5F,F0F1F5*(FDF*F-F*F/F*
F.!\"\"#FGF4,$*&*&F/F*,4F,FBF1\"\")FJF.F6F8F9F5F?FBFCF=FEF.FHF5F.F*FFF
KFG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"RG7$,$*&*$)%\"xG\"\"$\"\"\"
F,,&*$)F*\"\"#F,\"\"\"*$)%\"yGF0F,F0!\"\"#F1F0,$*&*$)F4F+F,F,F-F5!\"#
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Now we have the following equ
ations:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 
"f = 0" }}{PARA 0 "" 0 "" {TEXT -1 10 "X-R[1] = 0" }}{PARA 0 "" 0 "" 
{TEXT -1 10 "Y-R[2] = 0" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 200 "These are equations consisting of the variables x,y
,X,Y. The goal is to have just one equation F=0 involving X and Y, so \+
we want to eliminate x and y from these equations. Do this using the f
ollowing:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 
-1 0 "" }{TEXT 257 18 "General Principle:" }}{PARA 257 "" 0 "" {TEXT 
258 41 "If: you want to get rid of some variable." }}{PARA 258 "" 0 "
" {TEXT 259 24 "Then: use the resultant." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 260 155 "So remember: when
ever there's one variable too many, the question is not: \"should I ap
ply the resultant?\". The question is: \"what should I take as input?
\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "As
signment: Find the polynomial F in Q[X,Y] and e-mail it to hoeij@math.
fsu.edu" }}{PARA 0 "" 0 "" {TEXT -1 63 "Make your answer squarefree, b
ecause when F^2=0 then also: F=0." }}}}{MARK "4 6 0" 10 }{VIEWOPTS 1 
1 0 1 1 1803 }