{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal " -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "du:= (x,y,z)->r*f*x^ 2+s*g*y^2+t*h*z^2-(s*h+t*g)*y*z-(t*f+r*h)*z*x-(r*g+s*f)*x*y;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#duGR6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&ar rowGF*,.*(%\"rG\"\"\"%\"fGF1)9$\"\"#F1F1*(%\"sGF1%\"gGF1)9%F5F1F1*(%\" tGF1%\"hGF1)9&F5F1F1*(,&*&F7F1F=F1F1*&F " 0 "" {MPLTEXT 1 0 57 "factor(subs(u=-r/(s/v+t/w),du(g*w+h*v,f*w+h*u,g*u+f*v)));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&**%\"wG\"\"\"%\"vGF',(*(%\"rGF'%\" gGF'%\"hGF'F'*(%\"sGF'F-F'%\"fGF'F'*(%\"tGF'F,F'F0F'F'F',**(F0F'F/F'F& F'F'*(F0F'F2F'F(F'F'*(F,F'F+F'F&F'!\"\"*(F-F'F+F'F(F'F7F'F',&*&F/F'F&F 'F'*&F2F'F(F'F'F7!\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 93 "This is \+ the condition that the dual of the polar of U on C(R) wrt C(F) lies on the dual conic" }}{PARA 0 "" 0 "" {TEXT -1 51 "i.e. that the polar is tangent to the pivotal conic" }}{PARA 0 "" 0 "" {TEXT -1 131 "The las t factor on the top is zero if U is on C(F) - so we get the tangent to C(F) a tangent of the pivotal conic - confirmed below" }}{PARA 0 "" 0 "" {TEXT -1 21 "The middle factor is " }{TEXT 256 6 "always" }{TEXT -1 22 " zero if we have a cK0" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "factor(subs(\{u=1/(g*t-s*h),v=1/(h*r-t*f),w=1/(f*s-g*r)\},du(g*w +h*v,f*w+h*u,g*u+f*v)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "Now we find the tangent at the point to the dual du:" }} {PARA 0 "" 0 "" {TEXT -1 56 "The coefficients (i.e. the dual) is a poi nt on the pivot" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 62 "dx:=(subs(\{x=g*w+h*v,y=f*w+h*u,z=g*u+f*v\},di ff(du(x,y,z),x)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dxG,(*(%\"rG \"\"\"%\"fGF(,&*&%\"gGF(%\"wGF(F(*&%\"hGF(%\"vGF(F(F(\"\"#*&,&*&%\"tGF (F)F(F(*&F'F(F/F(F(F(,&*&F,F(%\"uGF(F(*&F)F(F0F(F(F(!\"\"*&,&*&F'F(F,F (F(*&%\"sGF(F)F(F(F(,&*&F)F(F-F(F(*&F/F(F9F(F(F(F;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "dy:=(subs(\{x=g*w+h*v,y=f*w+h*u,z=g*u+f*v\} ,diff(du(x,y,z),y)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dyG,(*(%\" sG\"\"\"%\"gGF(,&*&%\"fGF(%\"wGF(F(*&%\"hGF(%\"uGF(F(F(\"\"#*&,&*&F'F( F/F(F(*&%\"tGF(F)F(F(F(,&*&F)F(F0F(F(*&F,F(%\"vGF(F(F(!\"\"*&,&*&%\"rG F(F)F(F(*&F'F(F,F(F(F(,&*&F)F(F-F(F(*&F/F(F:F(F(F(F;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "dz:=(subs(\{x=g*w+h*v,y=f*w+h*u,z=g*u+f*v \},diff(du(x,y,z),z)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dzG,(*(% \"tG\"\"\"%\"hGF(,&*&%\"gGF(%\"uGF(F(*&%\"fGF(%\"vGF(F(F(\"\"#*&,&*&% \"sGF(F)F(F(*&F'F(F,F(F(F(,&*&F/F(%\"wGF(F(*&F)F(F-F(F(F(!\"\"*&,&*&F' F(F/F(F(*&%\"rGF(F)F(F(F(,&*&F,F(F9F(F(*&F)F(F0F(F(F(F;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "the conic c(R):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "cr:=(x,y,z)->r*y*z+s*z*x+t*x*y;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#crGR6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF* ,(*(%\"rG\"\"\"9%F19&F1F1*(%\"sGF1F3F19$F1F1*(%\"tGF1F6F1F2F1F1F*F*F* " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "when U is the fourth intersection we get a point of \+ C(R) - provided we have a cK0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "ff:=factor(subs(\{u=1/(g*t-s*h),v=1/(h*r-t*f),w=1/(f*s-g*r)\},cr (dx,dy,dz)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ffG,(*(%\"fG\"\"\" %\"sGF(%\"hGF(F(*(F'F(%\"tGF(%\"gGF(F(*(F*F(%\"rGF(F-F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "It also works for triangle vertices:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "fg:=factor(subs(\{u=1,v=0,w=0\},cr(dx,dy,dz)));" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#fgG*&),&*&%\"tG\"\"\"%\"gGF*F**&% \"sGF*%\"hGF*!\"\"\"\"#F*,(*(%\"fGF*F-F*F.F*F**(F3F*F)F*F+F*F**(F.F*% \"rGF*F+F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "What is the point involved ?" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "fx:=factor(subs(\{u=1/(g*t-s *h),v=1/(h*r-t*f),w=1/(f*s-g*r)\},dx));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#fxG,$*&,(*(%\"fG\"\"\"%\"sGF*%\"hGF*F**(F)F*%\"tGF*%\"gGF*F** *\"\"#F*F,F*%\"rGF*F/F*F*F*,&*&F.F*F/F*F**&F+F*F,F*!\"\"F6F6" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 83 "really r/(s/g-t/h) it is the inte rsection of C(R) with the tangent to C(F) at U - " }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "24 0 0" 83 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }