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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 101 "x[c] is an x value on the constant axis.
 x[p] is on the perspective axis.\ncp converts x[c] into x[p]." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "cp := unapply(x*(1-u)/(x-u),
x):\ncp(x[c]); cp(-1);\nlimit(cp(x[c]),x[c]=infinity);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#*(&%\"xG6#%\"cG\"\"\",&F(F(%\"uG!\"\"F(,&F$F(F*F+F
+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,&\"\"\"F&%\"uG!\"\"F&,&F(F&F
'F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"\"\"F$%\"uG!\"\"" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "the inverse function is named pc. \+
Let's convert x[p] to an x[c] value." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 50 "pc := unapply(solve(cp(x[c])=x,x[c]),x):\npc(x[p]);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(&%\"xG6#%\"pG\"\"\"%\"uGF(,(!\"\"F
(F)F(F$F(F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "cu1 is  the scale \+
factor that takes point 1 to point x when dilated with center u." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "cu1 := unapply((x-u)/(1-u),x
):\ncu1(x[c]); cu1(0); cu1(1); cu1(u);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#*&,&&%\"xG6#%\"cG\"\"\"%\"uG!\"\"F),&F)F)F*F+F+" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#,$*&%\"uG\"\"\",&F&F&F%!\"\"F(F(" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "Which scale factor do we need to a
rrive at x[p] (after measuring x[p] on the constant axis)?" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "simplify(cu1(pc(x[p])));" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#*&%\"uG\"\"\",(!\"\"F%F$F%&%\"xG6#%\"pGF%F'
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "cu0 is the the scale factur t
hat takes point 0 to point x when dilated with center u." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "cu0 := unapply((u-x)/u,x):\ncu0(x[c
]); cu0(0); cu0(1); cu0(u);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&&%
\"xG6#%\"cG!\"\"%\"uG\"\"\"F+F*F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&!\"\"\"\"\"%\"uGF&F&F'F%
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 35 "The same question as above for cu1." }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 24 "simplify(cu0(pc(x[p])));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#*&,&!\"\"\"\"\"%\"uGF&F&,(F%F&F'F&&%\"xG6#%\"pGF&F%" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 154 "The reverse test. Take that scal
e factor and do the dilation (apply the factor to the inverse of cu0, \+
which yields x)\nand we should get back our pc(x[p])." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "unapply(solve(cu0(x)=f,x),f) (cu0(p
c(x[p])));\nsimplify(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,&*&,
&%\"uG\"\"\"*(&%\"xG6#%\"pGF)F(F),(!\"\"F)F(F)F+F)F0F0F)F(F0F)F0F)F)F(
F)F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(&%\"xG6#%\"pG\"\"\"%\"uGF(,(
!\"\"F(F)F(F$F(F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
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