{VERSION 6 0 "IBM INTEL NT" "6.0" } {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 Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE " Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Dark Red Emphasis" -1 259 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 260 "Times" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Purple Emphasis" -1 261 "Times " 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Grey Emphasis" -1 262 "Times" 1 12 96 52 84 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Blue emphasi s" -1 263 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 } {CSTYLE "" -1 268 "" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 270 "" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 272 "" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 } {CSTYLE "" -1 273 "" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 274 "" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 275 "" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 276 "" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 277 "" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 } {CSTYLE "" -1 278 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 281 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 282 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 283 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 284 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 285 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 286 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 287 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 288 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 289 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 262 290 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 291 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 292 "" 0 1 0 0 0 0 0 0 1 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 "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Time s" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 } {PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "T imes" 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 "Normal" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 33 "A procedure for drawing fractals \+ " }}{PARA 0 "" 0 "" {TEXT -1 38 "by Peter Stone, Nanaimo, B.C., Canada " }}{PARA 0 "" 0 "" {TEXT -1 17 "Version: 6.3.2007" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "load " }{TEXT 0 11 "fractalplot" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 262 9 "fractal. m" }{TEXT -1 38 " contains the code for the procedures " }{TEXT 0 11 " fractalplot" }{TEXT -1 5 " and " }{TEXT 0 5 "blend" }{TEXT -1 25 " use d in this worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 122 "It can be read \+ into a Maple session by a command similar to the one that follows, whe re the file path gives its location. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "read \"K:\\\\MapleMath/procdrs/fractal.m\";" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 51 "load ext ra colours and utility routines for colours" }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 262 9 "colours.m" }{TEXT -1 87 " cont ains the code for the procedures giving the extra colours used in this worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 122 "It can be read into a Ma ple session by a command similar to the one that follows, where the fi le path gives its location. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "read \"K:\\\\MapleMath/procdrs/colours.m\";" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 42 "A procedure for drawing fr actal pictures: " }{TEXT 0 11 "fractalplot" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 11 "fr actalplot" }{TEXT -1 80 " uses Maple's 3-dimensionals graphics data st ructure to display fractal images. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 18 "fractalplot : usage" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 269 17 "Calling Sequence:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT 278 2 " " }{TEXT -1 46 " fractalplot( colorpro c, horiz_rng, vert_rng )" }}{PARA 0 "" 0 "" {TEXT -1 2 "or" }}{PARA 0 "" 0 "" {TEXT -1 44 " fractalplot( colorproc, centre, width ) " }} {PARA 0 "" 0 "" {TEXT -1 2 "or" }}{PARA 0 "" 0 "" {TEXT -1 39 " fra ctalplot( horiz_rng, vert_rng ) " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " } }{PARA 0 "" 0 "" {TEXT -1 33 " fractalplot( centre, width ) " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 11 "Paramet ers:" }}{PARA 0 "" 0 "" {TEXT -1 4 " " }}{PARA 0 "" 0 "" {TEXT 23 17 " colorproc - " }{TEXT -1 72 " a procedure with argument a \+ pair of floating point numbers x and y " }}{PARA 0 "" 0 "" {TEXT -1 129 " corresponding to a complex number x + y*I such that the returned value is either a float " }} {PARA 0 "" 0 "" {TEXT -1 133 " be tween 0 and 1 to give the HUE value for the colour to be associated wi th x + y*I or an array " }}{PARA 0 "" 0 "" {TEXT -1 124 " \+ consisting of 3 hardware floats between 0 and 1 to give the RGB (red, green and blue) " }}{PARA 0 "" 0 "" {TEXT -1 94 " components of the colour to \+ be associated with x + y*I. " }}{PARA 0 "" 0 "" {TEXT -1 30 " \+ " }{TEXT 261 11 " Note" }{TEXT -1 97 ": The procedure must be constructed in such a way that it can be evaluated \+ with hardware floating" }}{PARA 0 "" 0 "" {TEXT -1 61 " \+ point arithmetic using " }{TEXT 0 6 "evalhf" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 116 " If the 1st argument is omitted, certain standard \" built-in\" fractals can be drawn. See the \"type\" option below." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 " " } {TEXT 23 15 " horiz_rng - " }{TEXT 276 98 "a range of real constants specifing the horizontal range for the displayed region of the fracta l. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 " \+ " }{TEXT 23 15 " vert_rng - " }{TEXT 277 98 "a range of real const ants specifing the verticalal range for the displayed region of the fr actal. " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 23 17 " centre - " }{TEXT 270 84 "a c omplex number which is the central point of the displayed region of th e fractal. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }{TEXT 23 14 " width - " }{TEXT 272 76 "a real numbe r which gives the width of the displayed region of the fractal. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 " \+ " }}{PARA 258 "" 0 "" {TEXT -1 12 "Description:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 11 "fractalplot" }{TEXT 273 62 " he draws a region of a fractal colour ed by a given procedure." }{TEXT 0 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 271 8 "Options:" }{TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 74 "grid=[n,m]\nThis option allows the grid used fo r the plot to be specified. " }}{PARA 0 "" 0 "" {TEXT -1 108 "The defa ult is a 25 by 25 grid which only produces a rather crude and coarse r epresentation of the fractal. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 28 "colortype/colourtype=HUE/RGB" }}{PARA 0 " " 0 "" {TEXT -1 105 "For standard \"built-in\" fractals, the colour ca n be specified either by a single HUE value or by neans of" }}{PARA 0 "" 0 "" {TEXT -1 55 "the RGB (red, green and blue) components of the c olour." }}{PARA 0 "" 0 "" {TEXT -1 139 "The option \"colortype=RGB\" g ives more flexibility in the colouring of the fractal, but the resulti ng 3d plot stucture consumes more memory." }}{PARA 0 "" 0 "" {TEXT -1 44 "The lower case versions hue/rgb may be used." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "type=Mandelbrot .. Logist ic .. ChebyshevT3 .. Newton3 .. etc." }}{PARA 0 "" 0 "" {TEXT -1 47 "T hese are the (current) built-in fractal types." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 291 15 "ESCAPE FRACTALS" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 262 10 "Mandelbrot" }{TEXT -1 23 " initialisation: " }{XPPEDIT 18 0 "z = k;" "6#/%\" zG%\"kG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 38 " \+ formula: " }{TEXT 287 1 "z" }{TEXT -1 1 "'" }{XPPEDIT 18 0 "`` = z^2+k;" "6#/%!G,&*$%\"zG\"\"#\"\"\"%\"kGF)" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 262 11 "Mandelbrot3" }{TEXT -1 21 " initialisation: " }{XPPEDIT 18 0 "z = k;" "6#/%\"zG%\"kG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 38 " \+ formula: " }{TEXT 286 1 "z" }{TEXT -1 1 "'" }{XPPEDIT 18 0 "`` = z^3+k;" "6#/%!G,&*$%\"zG\"\"$\"\"\"%\"kGF)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 262 11 "Mandelbrot4" }{TEXT -1 21 " \+ initialisation: " }{XPPEDIT 18 0 "z = k;" "6#/%\"zG%\"kG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 38 " formul a: " }{TEXT 285 1 "z" }{TEXT -1 1 "'" }{XPPEDIT 18 0 "`` = z^4+k;" " 6#/%!G,&*$%\"zG\"\"%\"\"\"%\"kGF)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 262 8 "Logistic" }{TEXT -1 30 " init ialisation: " }{XPPEDIT 18 0 "z = 0" "6#/%\"zG\"\"!" }{TEXT -1 1 "." }{XPPEDIT 18 0 "5+0*i;" "6#,&\"\"&\"\"\"*&\"\"!F%%\"iGF%F%" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 38 " formul a: " }{TEXT 284 1 "z" }{TEXT -1 1 "'" }{XPPEDIT 18 0 "`` = k*z*(1-z) ;" "6#/%!G*(%\"kG\"\"\"%\"zGF',&F'F'F(!\"\"F'" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 262 11 "ChebyshevT3" }{TEXT -1 20 " initialisation: " }{XPPEDIT 18 0 "z = 0" "6#/%\"zG\"\"!" } {TEXT -1 1 "." }{XPPEDIT 18 0 "5+0*i;" "6#,&\"\"&\"\"\"*&\"\"!F%%\"iGF %F%" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 38 " \+ formula: " }{TEXT 283 1 "z" }{TEXT -1 1 "'" }{XPPEDIT 18 0 "``=k*(4*z^3-3*z)" "6#/%!G*&%\"kG\"\"\",&*&\"\"%F'*$%\"zG\"\"$F'F'*&F- F'F,F'!\"\"F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " } {TEXT 262 11 "ChebyshevT4" }{TEXT -1 20 " initialisation: " } {XPPEDIT 18 0 "z = 0" "6#/%\"zG\"\"!" }{TEXT -1 1 "." }{XPPEDIT 18 0 " 70710678118654752440 + 0*i" "6#,&\"5SCva'=\"y1rq\"\"\"*&\"\"!F%%\"iGF% F%" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 38 " \+ formula: " }{TEXT 282 1 "z" }{TEXT -1 1 "'" }{XPPEDIT 18 0 " ``=k*(8*z^4-8*z^2+1)" "6#/%!G*&%\"kG\"\"\",(*&\"\")F'*$%\"zG\"\"%F'F'* &F*F'*$F,\"\"#F'!\"\"F'F'F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 262 11 "ChebyshevT5" }{TEXT -1 20 " initialisation: \+ " }{XPPEDIT 18 0 "z = 0" "6#/%\"zG\"\"!" }{TEXT -1 1 "." }{XPPEDIT 18 0 "30901699437494742410 + 0*i" "6#,&\"55Cu%\\P%*p,4$\"\"\"*&\"\"!F% %\"iGF%F%" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 38 " \+ formula: " }{TEXT 281 1 "z" }{TEXT -1 1 "'" } {XPPEDIT 18 0 "``=k*(16*z^5-20*z^3+5*z)" "6#/%!G*&%\"kG\"\"\",(*&\"#;F '*$%\"zG\"\"&F'F'*&\"#?F'*$F,\"\"$F'!\"\"*&F-F'F,F'F'F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 262 11 "ChebyshevT6" } {TEXT -1 20 " initialisation: " }{XPPEDIT 18 0 "z = 0" "6#/%\"zG\" \"!" }{TEXT -1 4 ".5 " }}{PARA 0 "" 0 "" {TEXT -1 40 " \+ formula: z'" }{XPPEDIT 18 0 "``= k*(32*z^6-48*z^4+18*z^2 -1))" "6#/%!G*&%\"kG\"\"\",**&\"#KF'*$%\"zG\"\"'F'F'*&\"#[F'*$F,\"\"%F '!\"\"*&\"#=F'*$F,\"\"#F'F'F'F2F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 262 11 "ChebyshevT7" }{TEXT -1 19 " initialisat ion: " }{XPPEDIT 18 0 "z = 0" "6#/%\"zG\"\"!" }{TEXT -1 1 "." } {XPPEDIT 18 0 "62348980185873353053 + 0*i" "6#,&\"5`INte=!)*[B'\"\"\"* &\"\"!F%%\"iGF%F%" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 38 " \+ formula: " }{TEXT 280 1 "z" }{TEXT -1 1 "'" }{XPPEDIT 18 0 "``= k*(64*z^7-112*z^5+56*z^3-7*z)" "6#/%!G*&%\"kG\"\" \",**&\"#kF'*$%\"zG\"\"(F'F'*&\"$7\"F'*$F,\"\"&F'!\"\"*&\"#cF'*$F,\"\" $F'F'*&F-F'F,F'F2F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 262 11 "ChebyshevT8" }{TEXT -1 20 " initialisation: " } {XPPEDIT 18 0 "z = 0" "6#/%\"zG\"\"!" }{TEXT -1 1 "." }{XPPEDIT 18 0 " 70710678118654752440 + 0*i" "6#,&\"5SCva'=\"y1rq\"\"\"*&\"\"!F%%\"iGF% F%" }{TEXT -1 40 " \n formula: " }{TEXT 279 1 "z" }{TEXT -1 1 "'" }{XPPEDIT 18 0 "`` = k*(128*z^8-256*z^6+160* z^4-32*z^2+1);" "6#/%!G*&%\"kG\"\"\",,*&\"$G\"F'*$%\"zG\"\")F'F'*&\"$c #F'*$F,\"\"'F'!\"\"*&\"$g\"F'*$F,\"\"%F'F'*&\"#KF'*$F,\"\"#F'F2F'F'F' " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 262 4 "Sine " }{TEXT -1 36 " initialisation: " }{XPPEDIT 18 0 " z = k;" "6#/%\"zG%\"kG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 38 " formula: " }{TEXT 288 1 "z" }{TEXT -1 1 "'" }{XPPEDIT 18 0 "`` = k*sin(z);" "6#/%!G*&%\"kG\"\"\"-%$sinG6#%\"zG F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 292 15 "NEWTON FRACTALS" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "For " }{XPPEDIT 18 0 "m=3*` . . . `*8" "6#/%\"mG*(\"\"$\" \"\"%(~.~.~.~GF'\"\")F'" }{TEXT -1 3 ": " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 262 7 "Newton(" }{TEXT 290 1 "m" }{TEXT 262 1 ")" }{TEXT -1 25 " initialisation: " }{XPPEDIT 18 0 "z = k;" "6#/%\"zG%\" kG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 38 " \+ formula: " }{TEXT 289 1 "z" }{TEXT -1 1 "'" }{XPPEDIT 18 0 " `` = z-phi(z)/(phi*`'`(z));" "6#/%!G,&%\"zG\"\"\"*&-%$phiG6#F&F'*&F*F' -%\"'G6#F&F'!\"\"F0" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "phi(z) = z ^m-1;" "6#/-%$phiG6#%\"zG,&)F'%\"mG\"\"\"F+!\"\"" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "colortabl e/colourtable=C" }}{PARA 0 "" 0 "" {TEXT -1 119 "For escape fractals s uch as the Mandelbrot fractal it is possible to construct a custom col our table via the procedure " }{TEXT 0 5 "blend" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "numcolors /numcolours=n" }}{PARA 0 "" 0 "" {TEXT -1 328 "For escape fractals suc h as the Mandelbrot fractal the default colouring scheme is based on a HUE value associated with the number of iterations required to \"esca pe\". The number of colours to be used before cycling back to the star ting colour can be specified by this option.\nThis option subsitutes t he value n for the variable " }{TEXT 262 10 "_numcolors" }{TEXT -1 122 " within the colouring procedure, or, when \"colortype=RGB, it det ermines the number of colours in the default colour table." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "offset=n" }}{PARA 0 "" 0 "" {TEXT -1 92 "For escape fractals such as the Mandelbrot frac tal this number \"rotates\" the colour values. " }}{PARA 0 "" 0 "" {TEXT -1 52 "This option subsitutes the value n for the variable " } {TEXT 262 7 "_offset" }{TEXT -1 32 " within the colouring procedure." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "maxiter ations/maxit=n" }}{PARA 0 "" 0 "" {TEXT -1 117 "This option controls t he maximum number of iterations performed within the colouring procedu re for built-in fractals." }}{PARA 0 "" 0 "" {TEXT -1 52 "This option \+ subsitutes the value n for the variable " }{TEXT 262 8 "_maxiter" } {TEXT -1 32 " within the colouring procedure." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 16 "How to activate:" }{TEXT 263 1 "\n" }{TEXT -1 154 "To ma ke the procedure active open the subsection, place the cursor anywhere after the prompt [ > and press [Enter].\nYou can then close up the s ubsection." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 27 "fractalplot: implem entation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54470 "fractalplot := proc()\n local aa,bb,tt,i,j,a,b,c ,d,f,points,m,n,t,cn,cr,ci,wd,\n A,B,zero_hf,color_proc,ab1,ab2,c d1,cd2,xlist,ylist,err,\n output_type,typ,clrtyp,xx,yy,rngs,C,num clrs,offst,mxt,clrproc,\n startopts,ctab,Options;\n\n if nargs< 2 then\n error \"at least 2 arguments are required; the basic syn tax is: 'fractalplot(centre,width)','fractalplot(horiz_range,vert_rang e)','fractalplot(colorproc,centre,width)' or 'fractalplot(colorproc,ho riz_range,vert_range)'.\"\n end if;\n\n if type(args[1],procedure) then\n if nops([op(1,eval(args[1]))])<>2 then\n error \" the 1st argument, %1, is invalid .. it should be a procedure with two \+ arguments\",args[1];\n end if;\n typ := 0;\n clrproc := args[1];\n if nargs<3 then\n error \"when the 1st argume nt is a procedure at least 3 arguments are required; the basic syntax \+ is: 'fractalplot(colorproc,centre,width)' or 'fractalplot(colorproc,ho riz_range,vert_range)'.\"\n end if;\n startopts := 4;\n \+ aa := args[2];\n bb := args[3];\n elif not type(args[1],\{cons tant,constant..constant\}) then\n error \"the 1st argument, %1, i s invalid .. it should be a procedure with two arguments, or, if a 'bu ilt-in' fractal is to be drawn via the option \\\"type\\\", it should \+ be a complex constant or a range of real constants\",args[1];\n else \n typ := 2;\n clrproc := `fractalplot/newton4_hue`;\n \+ startopts := 3;\n aa := args[1];\n bb := args[2];\n end if ; \n\n if type(aa,complex(constant)) and type(bb,constant) the n\n cn := evalf(aa,15);\n if not type(cn,complex(numeric)) t hen\n error \"%-1 argument %2 should evaluate to a complex num eric\",startopts-2,aa;\n end if; \n wd := evalf(bb,15);\n \+ if not type(wd,numeric) then\n error \"%-1 argument %2 sho uld evaluate to a numeric\",startopts-1,bb;\n end if;\n cr : = Re(cn);\n ci := Im(cn);\n wd := evalf(wd*0.5,15);\n a := cr-wd;\n b := cr+wd;\n c := ci-wd;\n d := ci+wd;\n \+ elif type(aa,constant..constant) and type(bb,constant..constant) the n\n a := op(1,aa);\n b := op(2,aa);\n c := op(1,bb);\n \+ d := op(2,bb);\n f := proc(x)\n local y;\n opt ion `Copyright`;\n y := traperror(evalf(x,15));\n \+ if type(y,numeric) then return y end if;\n error \"real c onstants expected in ranges: %1\",x\n end proc;\n a := f(a );\n b := f(b);\n c := f(c);\n d := f(d);\n if b=a or d=c then error \"ranges must be non-empty\" end if;\n if b=starto pts then\n Options := [args[startopts..nargs]];\n if not typ e(Options,list(equation)) then\n error \"each optional argumen t must be an equation\"\n end if;\n if hasoption(Options,'gr id','tt','Options') then \n if type(tt,[integer,integer]) then \n m := tt[1]-1;\n n := tt[2]-1;\n if m<1 or n<1 then error \"grid dimensions must be > 1\" end if\n \+ else error \"bad grid option: %1\", tt\n end if;\n end \+ if;\n if hasoption(Options,'numcolors','numclrs','Options') or\n \+ hasoption(Options,'numcolours','numclrs','Options') then\n \+ if not type(numclrs,posint) then\n error \"\\\"numcol ors\\\" or \\\"numcolours\\\" must be a positive integer\"\n e nd if;\n end if;\n if hasoption(Options,'offset','offst','Op tions') then\n if not type(offst,nonnegint) then\n \+ error \"\\\"offset\\\" must be a non-negative integer\"\n end \+ if;\n end if;\n if hasoption(Options,'colortype','tt','Optio ns') or\n hasoption(Options,'colourtype','tt','Options') then \n if member(tt,\{'hue','HUE'\}) then\n clrtyp := 0 ;\n elif member(tt,\{'rgb','RGB'\}) then\n clrtyp : = 1;\n if typ=1 then clrproc := `fractalplot/newton4_rgb` e nd if;\n else\n error \"\\\"color_type\\\",\\\"colo ur_type\\\",\\\"colortype\\\" or \\\"colourtype\\\"\nmust be one of 'h ue','HUE','rgb' or 'RGB'.\";\n end if;\n end if;\n i f hasoption(Options,'type','tt','Options') then\n if tt='Mande lbrot' then\n typ := 2;\n mxt := 100;\n \+ if clrtyp=0 then\n clrproc := `fractalplot/mandelbrot _hue`\n else\n clrproc := `fractalplot/mandel brot_rgb`\n end if;\n elif tt='Mandelbrot3' then\n \+ typ := 2;\n mxt := 100;\n if clrtyp=0 then\n clrproc := `fractalplot/mandelbrot3_hue`\n \+ else\n clrproc := `fractalplot/mandelbrot3_rgb`\n \+ end if;\n elif tt='Mandelbrot4' then\n t yp := 2;\n mxt := 100;\n if clrtyp=0 then\n \+ clrproc := `fractalplot/mandelbrot4_hue`\n else\n clrproc := `fractalplot/mandelbrot4_rgb`\n e nd if;\n elif tt='Logistic' then\n typ := 2;\n \+ mxt := 100;\n if clrtyp=0 then\n clrpr oc := `fractalplot/logistic_hue`\n else\n clr proc := `fractalplot/logistic_rgb`\n end if;\n elif tt='ChebyshevT3' then\n typ := 2;\n mxt := 100; \n if clrtyp=0 then\n clrproc := `fractalplot /chebyshevT3_hue`\n else\n clrproc := `fracta lplot/chebyshevT3_rgb`\n end if;\n elif tt='Chebysh evT4' then\n typ := 2;\n mxt := 100;\n \+ if clrtyp=0 then\n clrproc := `fractalplot/chebyshevT4 _hue`\n else\n clrproc := `fractalplot/chebys hevT4_rgb`\n end if;\n elif tt='ChebyshevT5' then\n typ := 2;\n mxt := 100;\n if clrtyp= 0 then\n clrproc := `fractalplot/chebyshevT5_hue`\n \+ else\n clrproc := `fractalplot/chebyshevT5_rgb`\n end if;\n elif tt='ChebyshevT6' then\n \+ typ := 2;\n mxt := 100;\n if clrtyp=0 then\n \+ clrproc := `fractalplot/chebyshevT6_hue`\n else \n clrproc := `fractalplot/chebyshevT6_rgb`\n \+ end if;\n elif tt='ChebyshevT7' then\n typ := 2;\n mxt := 100;\n if clrtyp=0 then\n \+ clrproc := `fractalplot/chebyshevT7_hue`\n else\n \+ clrproc := `fractalplot/chebyshevT7_rgb`\n end if;\n \+ elif tt='ChebyshevT8' then\n typ := 2;\n \+ mxt := 100;\n if clrtyp=0 then\n clrproc := ` fractalplot/chebyshevT8_hue`\n else\n clrproc := `fractalplot/chebyshevT8_rgb`\n end if;\n elif \+ tt='Sine' then\n typ := 2;\n mxt := 100;\n \+ if clrtyp=0 then\n clrproc := `fractalplot/sine_hu e`\n else\n clrproc := `fractalplot/sine_rgb` \n end if;\n elif tt='Newton3' then\n ty p := 1;\n if clrtyp=0 then\n clrproc := `frac talplot/newton3_hue`\n else\n clrproc := `fra ctalplot/newton3_rgb`\n end if;\n elif tt='Newton4' then\n typ := 1;\n if clrtyp=0 then\n \+ clrproc := `fractalplot/newton4_hue`\n else\n \+ clrproc := `fractalplot/newton4_rgb`\n end if;\n \+ elif tt='Newton5' then\n typ := 1;\n if clrt yp=0 then\n clrproc := `fractalplot/newton5_hue`\n \+ else\n clrproc := `fractalplot/newton5_rgb`\n \+ end if;\n elif tt='Newton6' then\n typ := 1; \n if clrtyp=0 then\n clrproc := `fractalplot /newton6_hue`\n else\n clrproc := `fractalplo t/newton6_rgb`\n end if;\n elif tt='Newton7' then\n typ := 1;\n if clrtyp=0 then\n cl rproc := `fractalplot/newton7_hue`\n else\n c lrproc := `fractalplot/newton7_rgb`\n end if;\n eli f tt='Newton8' then\n typ := 1;\n if clrtyp=0 th en\n clrproc := `fractalplot/newton8_hue`\n e lse\n clrproc := `fractalplot/newton8_rgb`\n \+ end if;\n else\n error \"\\\"type\\\" must be one o f 'Mandelbrot','Mandelbrot3','Mandelbrot4','Logistic','ChebyshevT3','C hebyshevT4','ChebyshevT5','ChebyshevT6','ChebyshevT7','ChebyshevT8','S ine','Newton3','Newton4','Newton5','Newton6','Newton7' or 'Newton8'.\" \n end if;\n end if;\n if hasoption(Options,'maxit', 'mxt','Options') or \n hasoption(Options,'maxiterations','mxt ','Options') then\n if not type(mxt,posint) then\n \+ error \"\\\"maxiterations\\\" or \\\"maxit\\\" must be a positive inte ger\"\n end if;\n end if;\n if hasoption(Options,'co lortable','C','Options') or\n hasoption(Options,'colourtable' ,'C','Options') then \n if type(C,hfarray) then\n \+ if evalb(\{op(4,C)\}<>\{datatype=float[8],storage=rectangular,o rder=C_order\}) then\n error \"wrong type of hfarray in \+ colour table\"\n end if;\n rngs := [op(2,C)];\n \+ if nops(rngs)<>2 or rngs[2]<>1..3 or op(1,rngs[1])<>1 then \n error \"wrong dimensions in colour table\"\n \+ end if;\n numclrs := op(2,rngs[1]);\n else\n \+ error \"the colour table must be an array of hardware floats\" \n end if;\n end if;\n if nops(Options)>0 then\n \+ error \"%1 is not a valid option for %2\",op(1,Options),procname; \n end if;\n end if;\n\n if typ=2 and clrtyp=1 and not assign ed(C) then\n ctab := proc(n)\n local C,i,j,h,s,v,f,p,q,t;\n \+ option `Copyright`;\n C := hfarray(1..n,1..3);\n \+ for j to n do\n\011 h := irem(j,n)/n;\n\011 s : = .9;\n \011 v := 1.;\n h := h*6;\n i : = floor(h);\n\011 \011\011 f := h-i;\n\011 \011 p := v* (1-s);\n\011 \011 q := v*(1-s*f);\n\011 \011 t := v*(1 -s*(1-f));\n \011\011 if i=0 then\n C[j,1] := v; C[j,2] := t; C[j,3] := p;\n\011\011\011 elif i=1 then\n \+ C[j,1] := q; C[j,2] := v; C[j,3] := p;\n elif i=2 \+ then\n C[j,1] := p; C[j,2] := v; C[j,3] := t;\n \+ elif i=3 then\n C[j,1] := p; C[j,2] := q; C[j,3] := v ;\n elif i=4 then\n C[j,1] := t; C[j,2] := p; C[j,3] := v;\n\011\011\011 else # i=5 then\n C[ j,1] := v; C[j,2] := p; C[j,3] := q;\n end if;\n en d do;\n C\n end proc;\n C := evalhf(ctab(numclrs)); \n end if; \n\n # detect and test form of procedure output\n if typ=0 then\n ab1 := .381966011*a+.618033989*b;\n cd1 := .38 1966011*c+.618033989*d;\n ab2 := .7101449275*a+.2898550725*b;\n \+ cd2 := .7101449275*c+.2898550725*d;\n xlist := [ab1,ab1,ab2,a b2];\n ylist := [cd1,cd2,cd1,cd2];\n output_type := [0,0,0,0 ];\n for i to 4 do\n xx := xlist[i];\n yy := ylis t[i];\n err := traperror(evalhf(clrproc(xx,yy)));\n if err<>lasterror then\n if type(err,float) then\n \+ if err<0 or err>1 then\n error \"numerical values of the 1st argument procedure should lie between 0 and 1\"\n \+ end if;\n elif type(err,hfarray) then \n \+ err := traperror(convert(err,list)); \n if err=lasterr or or not type(err,list(numeric)) or not nops(err)=3 then\n \+ error \"each value of the 1st argument procedure should be an ' hfarray' with three numerical entries\"\n end if;\n \+ for j to 3 do\n if err[j]<0 or err[j]>1 the n\n error \"numerical entries of each 'hfarray' va lue of the 1st argument procedure should lie between 0 and 1\"\n \+ end if;\n end do;\n output_typ e[i] := 1;\n else\n error \"1st argument proc edure should have values which are of type 'float' or 'hfarray'\"\n \+ end if;\n else\n error \"could not evaluat e 1st argument procedure\"\n end if; \n end do;\n if \{op(output_type)\}=\{0\} then\n clrtyp := 0;\n \+ elif \{op(output_type)\}=\{1\} then\n clrtyp := 1;\n else \n error \"1st argument procedure gives inconsistent output: a ll values must be of type 'float' or all values must be of type 'hf array'\"\n end if;\n end if;\n\n zero_hf := proc(m,n)\n \+ local i,j,A;\n A := hfarray(1..m+1,1..n+1);\n for i to m+1 d o\n for j to n+1 do\n A[i,j] := 0.; \n end \+ do;\n end do;\n A\n end proc;\n A := evalhf(zero_hf(m,n) );\n\n if clrtyp=0 then\n color_proc := proc(f,a,b,c,d,m,n)\n \+ local i,j,A,t,x,y,xinc,yinc;\n A := hfarray(1..m+1,1..n +1);\n xinc := (b-a)/m;\n yinc := (d-c)/n;\n x := a;\n for i to m+1 do\n y := c;\n for j to n+1 do\n A[i,j] := f(x,y);\n y := y+ yinc\n end do;\n x := x+xinc\n end do;\n A\n end proc;\n clrproc := subs(\{_maxiter=mxt,_num colors=numclrs,\n _offset=offst\},eval(clrproc));\n B := traperror(evalhf(color_proc(clrproc,a,b,c,d,m,n)));\n if last error=B then \n error \"could not construct image using the pr ocedure provided\"\n end if;\n points := 'GRID'(a..b,c..d,A, \n COLOR('HUE',op(map(op,[seq([seq(B[i,j],j=1..n+1)],i=1..m+1)] ))));\n else # clrtyp=1\n if typ<2 then # no colour table\n \+ color_proc := proc(f,a,b,c,d,m,n)\n local i,j,A,t,x,y, xinc,yinc;\n A := hfarray(1..m+1,1..n+1,1..3);\n \+ xinc := (b-a)/m;\n yinc := (d-c)/n;\n x := a;\n for i to m+1 do\n y := c;\n fo r j to n+1 do\n t := f(x,y);\n A[i,j ,1] := t[1];\n A[i,j,2] := t[2];\n A [i,j,3] := t[3];\n y := y+yinc\n end do ;\n x := x+xinc\n end do;\n A\n \+ end proc;\n clrproc := subs(\{_maxiter=mxt,_numcolors=nu mclrs,\n _offset=offst\},eval(clrproc));\n B := tr aperror(evalhf(color_proc(clrproc,a,b,c,d,m,n)));\n else # typ=2 \+ (colour table used)\n color_proc := proc(f,C,a,b,c,d,m,n)\n \+ local i,j,A,t,x,y,xinc,yinc;\n A := hfarray(1..m+1 ,1..n+1,1..3);\n xinc := (b-a)/m;\n yinc := (d-c )/n;\n x := a;\n for i to m+1 do\n \+ y := c;\n for j to n+1 do\n t := f(x,y ,C);\n A[i,j,1] := t[1];\n A[i,j,2] \+ := t[2];\n A[i,j,3] := t[3];\n y := \+ y+yinc\n end do;\n x := x+xinc\n \+ end do;\n A\n end proc;\n clrproc := subs (\{_maxiter=mxt,_numcolors=numclrs,\n _offset=offst\},eval (clrproc));\n B := traperror(evalhf(color_proc(clrproc,C,a,b,c ,d,m,n)));\n end if;\n if lasterror=B then \n error \+ \"could not construct image using the procedure provided\"\n end \+ if;\n points := 'GRID'(a..b,c..d,A,COLOR('RGB',eval(B)));\n end if; \n PLOT3D(points,STYLE(PATCHNOGRID),ORIENTATION(-90.,0.),\n \+ VIEW(-wd..wd),SCALING(CONSTRAINED));\nend proc: # of fractalpl ot\n\n`fractalplot/newton3_hue` := proc(x,y)\n local xx,yy,xs,ys,w,s um,i,p,q,r,comp,notclosetoroot,\n a,b,c,d,e,rootThree,yTimesRootT hree,twoThirds,oneThird;\n \n xx := x;\n yy := y;\n xs := xx^ 2;\n ys := yy^2;\n notclosetoroot := true;\n rootThree := 1.7320 5080756887729;\n oneThird := 0.333333333333333333;\n twoThirds := \+ 0.666666666666666667;\n\n for i from 1 to _maxiter while notclosetor oot do\n\011\011\011 w := xx*yy;\n\011\011\011\011\011 c := xs-ys; \011# Real part of z^2\n\011\011\011\011\011 d := w+w;\011\011 # Imag \+ part of z^2\011\011\011\n\n\011\011\011\011\011 if abs(c)>abs(d) then \n\011\011\011\011\011\011 e := d/c;\n\011\011\011\011\011\011 a : = 1/(c+d*e); # real part of 1/z^3\011\n\011\011\011\011\011\011 b := e*a;\011 \011 # minus imag part of 1/z^3\n\011\011\011\011\011\011 else\n\011\011\011\011\011\011 e := c/d;\n\011\011\011\011\011\011 \+ b := 1/(c*e+d);\011# minus imag part of 1/z^3\n\011\011\011\011\011 \011 a := e*b;\011 # real part of 1/z^3\n\011\011\011\011\011 e nd if;\n\n # real part of 2/3*z+1/(3*z^3)\n\011\011\011\011\011 x x := twoThirds*xx+oneThird*a; \n # imag part of 2/3*z+1/(3*z^3)\n \011\011\011\011\011 yy := twoThirds*yy-oneThird*b;\n\011\011\011\011 \011\n\011\011\011\011\011 # How close are we to one of the roots?\n \011\011\011\011\011 xs := xx^2;\n ys := yy^2;\n sum := xs+y s;\n\n\011\011\011\011\011 comp := sum+.99;\n\011\011\011\011\011 yTim esRootThree := yy*rootThree;\n\n\011\011\011\011\011 # Are we close to the root -1/2-sqrt(3)/2*I?\n\011\011\011\011\011 p := evalb(-xx-yTime sRootThree>comp);\n\n\011\011\011\011\011 # Are we close to the troot \+ -1/2+sqrt(3)/2*I?\n\011\011\011\011 \011q := evalb(yTimesRootThree-xx> comp);\n\n\011\011\011\011\011 # Are we close to the root 1?\n\011\011 \011\011 r := evalb(2*xx>comp);\n\n # Set flag to end loop if su fficiently close to a root.\n notclosetoroot := not(p or q or r); \n end do;\n\n # Determine the value to return.\n if p then .666 6666667 elif q then .1666666667 \n elif r then 0. else .5416666667 e nd if;\nend proc: # end of `fractalplot/newton3_hue`\n\n`fractalplot/n ewton3_rgb` := proc(x,y)\n local xx,yy,xs,ys,w,sum,i,p,q,r,comp,notc losetoroot,\n a,b,c,d,e,rootThree,yTimesRootThree,twoThirds,oneTh ird,h,ans;\n \n xx := x;\n yy := y;\n xs := xx^2;\n ys := y y^2;\n notclosetoroot := true;\n rootThree := 1.73205080756887729; \n oneThird := 0.333333333333333333;\n twoThirds := 0.666666666666 666667;\n\n for i from 1 to _maxiter while notclosetoroot do\n\011 \011\011 w := xx*yy;\n\011\011\011\011\011 c := xs-ys;\011# Real par t of z^2\n\011\011\011\011\011 d := w+w;\011\011 # Imag part of z^2 \011\011\011\n\n\011\011\011\011\011 if abs(c)>abs(d) then\n\011\011 \011\011\011\011 e := d/c;\n\011\011\011\011\011\011 a := 1/(c+d*e ); # real part of 1/z^3\011\n\011\011\011\011\011\011 b := e*a;\011 \+ \011 # minus imag part of 1/z^3\n\011\011\011\011\011\011else\n\011 \011\011\011\011\011 e := c/d;\n\011\011\011\011\011\011 b := 1/(c *e+d);\011# minus imag part of 1/z^3\n\011\011\011\011\011\011 a := \+ e*b;\011 # real part of 1/z^3\n\011\011\011\011\011 end if;\n\n \+ # real part of 2/3*z+1/(3*z^3)\n\011\011\011\011\011 xx := twoThir ds*xx+oneThird*a; \n # imag part of 2/3*z+1/(3*z^3)\n\011\011\011 \011\011 yy := twoThirds*yy-oneThird*b;\n\011\011\011\011\011\n\011 \011\011\011\011 # How close are we to one of the roots?\n\011\011\011 \011\011 xs := xx^2;\n ys := yy^2;\n sum := xs+ys;\n\n\011 \011\011\011\011 comp := sum+.99;\n\011\011\011\011\011 yTimesRootThre e := yy*rootThree;\n\n\011\011\011\011\011 # Are we close to the root \+ -1/2-sqrt(3)/2*I?\n\011\011\011\011\011 p := evalb(-xx-yTimesRootThree >comp);\n\n\011\011\011\011\011 # Are we close to the troot -1/2+sqrt( 3)/2*I?\n\011\011\011\011 \011q := evalb(yTimesRootThree-xx>comp);\n\n \011\011\011\011\011 # Are we close to the root 1?\n\011\011\011\011 \+ r := evalb(2*xx>comp);\n\n # Set flag to end loop if sufficiently close to a root.\n notclosetoroot := not(p or q or r);\n end d o;\n\n h := irem(i,25)^2;\n h := 1+0.000004*(h-625)*h;\n\n # Det ermine the value to return.\n ans := hfarray(1..3);\n if p then\n \+ ans[1]:=0; ans[2]:=0; ans[3]:=h; \n elif q then\n ans[1]:= h; ans[2]:=h; ans[3]:=0; \n elif r then\n ans[1]:=h; ans[2]:=0; ans[3]:=0; \n else\n ans[1]:=.5; ans[2]:=.5; ans[3]:=.5\n en d if;\n ans;\nend proc: # end of `fractalplot/newton3_rgb`\n\n`fract alplot/newton4_hue` := proc(x,y)\n local xx,yy,xs,ys,sum,i,p,q,r,s,c omp,notclosetoroot,\n a,b,c,d,e;\n \n xx := x;\n yy := y; \n xs := xx^2;\n ys := yy^2;\n notclosetoroot := true;\n\n for i from 1 to _maxiter while notclosetoroot do\n c := (xs-ys*3)*xx ; # real part of z^3\n d := (xs*3-ys)*yy; # imag part of z^3\n\n \011\011\011\011\011 if abs(c)>abs(d) then\n\011\011\011\011\011\011 \+ e := d/c;\n\011\011\011\011\011\011 a := 1/(c+d*e); # real part of \+ 1/z^3\011\n\011\011\011\011\011\011 b := e*a;\011 \011# minus imag p art of 1/z^3\n\011\011\011\011\011\011else\n\011\011\011\011\011\011 \+ e := c/d;\n\011\011\011\011\011\011 b := 1/(c*e+d);\011# minus imag part of 1/z^3\n\011\011\011\011\011\011 a := e*b;\011 # real part \+ of 1/z^3\n\011\011\011\011\011 end if;\n\n\011\011\011\011\011 xx := 0 .75*xx+0.25*a; # real part of 3/4*z+1/(4*z^3)\n\011\011\011\011\011 yy := 0.75*yy-0.25*b; # imag part of 3/4*z+1/(4*z^3)\n\011\011\011\011 \011\n\011\011\011\011\011 # How close are we to one of the roots?\n \011\011\011\011\011 xs := xx^2;\n ys := yy^2;\n sum := xs+y s;\n\n comp := 0.5*sum+.495;\n\n # Are we close to the root \+ -1?\n p := evalb(xx+comp<0);\n\n # Are we close to the root \+ -i?\n q := evalb(yy+comp<0);\n\n # Are we close to the root \+ i?\n r := evalb(yy>comp);\n\n # Are we close to the root 1? \n s := evalb(xx>comp);\n\n # Set flag to end loop if suffic iently close to a root.\n notclosetoroot := not(p or q or r or s) ;\n end do;\n\n # Determine the value to return.\n if p then .66 66666667 elif q then .1666666667 \n elif r then 0 elif s then .33333 33333\n else .5416666667 end if;\nend proc: # end of `fractalplot/ne wton4_hue`\n\n`fractalplot/newton4_rgb` := proc(x,y)\n local xx,yy,x s,ys,sum,i,p,q,r,s,comp,notclosetoroot,\n a,b,c,d,e,h,ans;\n \n \+ xx := x;\n yy := y;\n xs := xx^2;\n ys := yy^2;\n notclosetor oot := true;\n\n for i from 1 to _maxiter while notclosetoroot do\n \+ c := (xs-ys*3)*xx; # real part of z^3\n d := (xs*3-ys)*yy; # imag part of z^3\n\n\011\011\011\011\011 if abs(c)>abs(d) then\n\011 \011\011\011\011\011 e := d/c;\n\011\011\011\011\011\011 a := 1/(c +d*e); # real part of 1/z^3\011\n\011\011\011\011\011\011 b := e*a; \011 \011# minus imag part of 1/z^3\n\011\011\011\011\011\011else\n \011\011\011\011\011\011 e := c/d;\n\011\011\011\011\011\011 b := \+ 1/(c*e+d);\011# minus imag part of 1/z^3\n\011\011\011\011\011\011 a := e*b;\011 # real part of 1/z^3\n\011\011\011\011\011 end if;\n\n \011\011\011\011\011 xx := 0.75*xx+0.25*a; # real part of 3/4*z+1/(4*z ^3)\n\011\011\011\011\011 yy := 0.75*yy-0.25*b; # imag part of 3/4*z+1 /(4*z^3)\n\011\011\011\011\011\n\011\011\011\011\011 # How close are w e to one of the roots?\n\011\011\011\011\011 xs := xx^2;\n ys := \+ yy^2;\n sum := xs+ys;\n\n comp := 0.5*sum+.495;\n\n # A re we close to the root -1?\n p := evalb(xx+comp<0);\n\n # A re we close to the root -i?\n q := evalb(yy+comp<0);\n\n # A re we close to the root i?\n r := evalb(yy>comp);\n\n # Are \+ we close to the root 1?\n s := evalb(xx>comp);\n\n # Set fla g to end loop if sufficiently close to a root.\n notclosetoroot : = not(p or q or r or s);\n end do;\n\n h := irem(i,25)^2;\n h := 1+0.000004*(h-625)*h;\n\n # Determine the value to return.\n ans \+ := hfarray(1..3);\n if p then\n ans[1]:=0; ans[2]:=0; ans[3]:=h ; \n elif q then\n ans[1]:=h; ans[2]:=h; ans[3]:=0; \n elif r then\n ans[1]:=h; ans[2]:=0; ans[3]:=0; \n elif s then\n \+ ans[1]:=0; ans[2]:=h; ans[3]:=0; \n else\n ans[1]:=.5; ans[2]: =.5; ans[3]:=.5\n end if;\n ans;\nend proc: # end of `fractalplot/ newton4_rgb`\n\n`fractalplot/newton5_hue` := proc(x,y)\n local xx,yy ,xs,ys,w,ws,ds,sum,i,p,q,r,s,t,comp,notclosetoroot,\n a,b,c,d,e,x c1,ys1,xc2,ys2;\n \n xx := x;\n yy := y;\n xs := xx^2;\n ys := yy^2;\n notclosetoroot := true;\n\n for i from 1 to _maxiter w hile notclosetoroot do\n\011\011\011 w := xx*yy;\n\011\011\011\011 \011 ws := w*w;\n\011\011\011\011\011 ds := xs-ys;\n\011\011\011\011 \011 c := ds*ds-ws*4;\011 # Real part of z^4\n\011\011\011\011\011 d : = ds*w*4;\011\011\011 # Imag part of z^4\n\011\n\011\011\011\011\011 if abs(c)>abs(d) then\n\011\011\011\011\011\011 e := d/c;\n\011\011 \011\011\011\011 a := 1/(c+d*e); # real part of 1/z^4\011\n\011\011 \011\011\011\011 b := e*a;\011 \011 # minus imag part of 1/z^4\n \011\011\011\011\011\011else\n\011\011\011\011\011\011 e := c/d;\n \011\011\011\011\011\011 b := 1/(c*e+d);\011# minus imag part of 1/z ^4\n\011\011\011\011\011\011 a := e*b;\011 # real part of 1/z^4 \n\011\011\011\011\011 end if;\n\n\011\011\011\011\011 xx := 0.8*xx + \+ 0.2*a;\011 # Real part of 4/5*z + 1/(5*z^4)\n\011\011\011\011\011 yy : = 0.8*yy - 0.2*b; # Imag part of 4/5*z + 1/(5*z^4)\n\011\011\011\011 \011\n\011\011\011\011\011 # How close are we to one of the roots?\n \011\011\011\011\011 xs := xx^2;\n ys := yy^2;\n sum := xs+y s;\n\011\011\011\011\011 comp := 0.5*sum+.495;\n\011\011\011\011\011\n xc1 := xx*0.80901699437494742410;\011# xx times cos 36 degrees\n \011\011\011\011\011 ys1 := yy*0.58778525229247312917;\011# yy times s in 36 degrees\n\011\011\011\011\011 xc2 := xx*0.30901699437494742410; \+ # xx times cos 72 degrees\n\011\011\011\011\011 ys2 := yy*0.9510565162 9515357212;\011# yy times sin 72 degrees\n\011\011\011\011\011\n\011 \011\011\011\011 # Are we close to the root -cos36 - isin36?\n\011\011 \011\011\011 p := evalb(-ys1-xc1-comp>0);\n\011\011\011\011\011\n\011 \011\011\011\011 # Are we close to the root -cos36 + isin36?\n\011\011 \011\011\011 q := evalb(ys1-xc1-comp>0);\n\011\011\011\011\011\n\011 \011\011\011\011 # Are we close to the root cos72 - isin72?\n\011\011 \011\011\011 r := evalb(xc2-ys2-comp>0);\n\011\011\011\011\011\n\011 \011\011\011\011 # Are we close to the root cos72 + isin72?\n\011\011 \011\011\011 s := evalb(xc2+ys2-comp>0);\n\011\011\011\011\011\n\011 \011\011\011\011 # Are we close to the root 1?\n\011\011\011\011\011 t := evalb(xx-comp>0);\n\n # Set flag to end loop if sufficiently \+ close to a root.\n notclosetoroot := not(p or q or r or s or t); \n end do;\n\n # Determine the value to return.\n if p then .833 3333333 elif q then .6666666667 \n elif r then .1666666667 elif s th en 0 \n elif t then .3333333333 else .5416666667 end if;\nend proc: \+ # end of `fractalplot/newton5_hue`\n\n`fractalplot/newton5_rgb` := pro c(x,y)\n local xx,yy,xs,ys,w,ws,ds,sum,i,p,q,r,s,t,comp,notclosetoro ot,\n a,b,c,d,e,xc1,ys1,xc2,ys2,h,ans;\n \n xx := x;\n yy \+ := y;\n xs := xx^2;\n ys := yy^2;\n notclosetoroot := true;\n\n \+ for i from 1 to _maxiter while notclosetoroot do\n\011\011\011 w : = xx*yy;\n\011\011\011\011\011 ws := w*w;\n\011\011\011\011\011 ds := \+ xs-ys;\n\011\011\011\011\011 c := ds*ds-ws*4;\011 # Real part of z^4\n \011\011\011\011\011 d := ds*w*4;\011\011\011 # Imag part of z^4\n \011\n\011\011\011\011\011 if abs(c)>abs(d) then\n\011\011\011\011\011 \011 e := d/c;\n\011\011\011\011\011\011 a := 1/(c+d*e); # real pa rt of 1/z^4\011\n\011\011\011\011\011\011 b := e*a;\011 \011 # mi nus imag part of 1/z^4\n\011\011\011\011\011\011else\n\011\011\011\011 \011\011 e := c/d;\n\011\011\011\011\011\011 b := 1/(c*e+d);\011# \+ minus imag part of 1/z^4\n\011\011\011\011\011\011 a := e*b;\011 \+ # real part of 1/z^4\n\011\011\011\011\011 end if;\n\n\011\011\011 \011\011 xx := 0.8*xx + 0.2*a;\011 # Real part of 4/5*z + 1/(5*z^4)\n \011\011\011\011\011 yy := 0.8*yy - 0.2*b; # Imag part of 4/5*z + 1/( 5*z^4)\n\011\011\011\011\011\n\011\011\011\011\011 # How close are we \+ to one of the roots?\n\011\011\011\011\011 xs := xx^2;\n ys := yy ^2;\n sum := xs+ys;\n\011\011\011\011\011 comp := 0.5*sum+.495;\n \011\011\011\011\011\n xc1 := xx*0.80901699437494742410;\011# xx \+ times cos 36 degrees\n\011\011\011\011\011 ys1 := yy*0.587785252292473 12917;\011# yy times sin 36 degrees\n\011\011\011\011\011 xc2 := xx*0. 30901699437494742410; # xx times cos 72 degrees\n\011\011\011\011\011 \+ ys2 := yy*0.95105651629515357212;\011# yy times sin 72 degrees\n\011 \011\011\011\011\n\011\011\011\011\011 # Are we close to the root -cos 36 - isin36?\n\011\011\011\011\011 p := evalb(-ys1-xc1-comp>0);\n\011 \011\011\011\011\n\011\011\011\011\011 # Are we close to the root -cos 36 + isin36?\n\011\011\011\011\011 q := evalb(ys1-xc1-comp>0);\n\011 \011\011\011\011\n\011\011\011\011\011 # Are we close to the root cos7 2 - isin72?\n\011\011\011\011\011 r := evalb(xc2-ys2-comp>0);\n\011 \011\011\011\011\n\011\011\011\011\011 # Are we close to the root cos7 2 + isin72?\n\011\011\011\011\011 s := evalb(xc2+ys2-comp>0);\n\011 \011\011\011\011\n\011\011\011\011\011 # Are we close to the root 1?\n \011\011\011\011\011 t := evalb(xx-comp>0);\n\n # Set flag to end loop if sufficiently close to a root.\n notclosetoroot := not(p \+ or q or r or s or t);\n end do;\n\n h := irem(i,25)^2;\n h := 1+ 0.000004*(h-625)*h;\n\n # Determine the value to return.\n ans := \+ hfarray(1..3);\n if p then\n ans[1]:=h; ans[2]:=0; ans[3]:=h; \+ \n elif q then\n ans[1]:=0; ans[2]:=0; ans[3]:=h; \n elif r t hen\n ans[1]:=h; ans[2]:=h; ans[3]:=0; \n elif s then\n an s[1]:=h; ans[2]:=0; ans[3]:=0; \n elif t then\n ans[1]:=0; ans [2]:=h; ans[3]:=0; \n else\n ans[1]:=.5; ans[2]:=.5; ans[3]:=.5 \n end if;\n ans;\nend proc: # end of `fractalplot/newton5_rgb`\n \n`fractalplot/newton6_hue` := proc(x,y)\n local xx,yy,xs,ys,xf,yf,a ,b,c,d,e,sum,i,p,q,r,s,t,u,\n comp,notclosetoroot,dx,rootThree,yTime sRootThree,\n oneSixth,fiveSixths;\011\011\011\011\n \n xx := x ; \n yy := y;\n xs := xx^2;\n ys := yy^2;\n\011\011\011\011\n \+ notclosetoroot := true;\n rootThree := 1.73205080756887729;\n one Sixth := .166666666666666667;\n fiveSixths := .833333333333333333;\n \011\011\011\n for i from 1 to _maxiter while notclosetoroot do\n \011\011\011\011\011 # Apply Newton's method for calculating roots of \+ z^6-1=0.\n\011\011\011\011\011 xf := xs^2;\n\011\011\011\011\011 yf := ys^2;\n\011\011\011\011\011 c := ((ys-2*xs)*ys*5+xf)*xx;\011# real pa rt of z^5\n\011\011\011\011\011 d := ((xs-2*ys)*xs*5+yf)*yy; # imag pa rt of z^5\n\011\011\011\011\011\n\011\011\011\011\011 if abs(c)>abs(d) then\n\011\011\011\011\011\011 e := d/c;\n\011\011\011\011\011\011 \+ a := 1/(c+d*e); # real part of 1/z^5\011\n\011\011\011\011\011\011 \+ b := e*a;\011 \011# minus imag part of 1/z^5\n\011\011\011\011\011 \011else\n\011\011\011\011\011\011 e := c/d;\n\011\011\011\011\011 \011 b := 1/(c*e+d);\011# minus imag part of 1/z^5\n\011\011\011\011 \011\011 a := e*b;\011 # real part of 1/z^5\n\011\011\011\011\011 e nd if;\011\n # real part of 5/6*z+1/(6*z^5)\011\011\011\011\011 \011\011\011\011\011\011\n\011\011\011\011\011 xx := fiveSixths*xx+one Sixth*a;\n # imag part of 5/6*z+1/(6*z^5)\n\011\011\011\011\011 y y := fiveSixths*yy-oneSixth*b;\n\011\011\011\011\011\n\011\011\011\011 \011 # How close are we to one of the roots?\n\011\011\011\011\011 xs \+ := xx^2;\n ys := yy^2;\n\011\011\011\011\011 sum := xs+ys;\n\n \011\011\011\011\011 comp := sum+.99;\n\011\011\011\011\011 dx := 2*xx ;\n\011\011\011\011\011 yTimesRootThree := yy*rootThree;\n\n\011\011 \011\011\011 # Are we close to the root -1?\n\011\011\011\011 \011p := evalb(-dx>comp);\n\011\011\011\011\011\n\011\011\011\011\011 # Are we close to the root -1/2-sqrt(3)/2*I?\n\011\011\011\011\011 q := evalb( -xx-yTimesRootThree>comp);\n\011\011\011\011\011\n\011\011\011\011\011 # Are we close to the root 1/2-sqrt(3)/2*I?\n\011\011\011\011\011 r : = evalb(xx-yTimesRootThree>comp);\n\n\011\011\011\011\011 # Are we clo se to the root 1?\n\011\011\011\011 s := evalb(dx>comp);\n\011\011 \011\011\011\n\011\011\011\011\011 # Are we close to the root 1/2+sqrt (3)/2*I?\n\011\011\011\011\011 t := evalb(xx+yTimesRootThree>comp);\n \n\011\011\011\011\011 # Are we close to the troot -1/2+sqrt(3)/2*I?\n \011\011\011\011 \011u := evalb(yTimesRootThree-xx>comp);\n\011\011 \011\011\011\n\011\011\011\011\011 # Set flag to end loop if sufficien tly close to a root.\n\011\011\011\011\011 notclosetoroot := not(p or \+ q or r or s or t or u);\n end do;\n\n # Determime the value to ret urn.\n if p then .6666666667 elif q then .1666666667 \n elif r the n .8333333333 elif s then .3333333333\n elif t then .75 \n elif u \+ then 0 else .5416666667 end if;\nend proc: # end of `fractalplot/newto n6_hue`\n\n`fractalplot/newton6_rgb` := proc(x,y)\n local xx,yy,xs,y s,xf,yf,a,b,c,d,e,sum,i,p,q,r,s,t,u,\n comp,notclosetoroot,dx,rootTh ree,yTimesRootThree,h,ans,\n oneSixth,fiveSixths;\011\011\011\011\n \+ \n xx := x; \n yy := y;\n xs := xx^2;\n ys := yy^2;\n\011 \011\011\011\n notclosetoroot := true;\n rootThree := 1.7320508075 6887729;\n oneSixth := .166666666666666667;\n fiveSixths := .83333 3333333333333;\n\011\011\011\n for i from 1 to _maxiter while notclo setoroot do\n\011\011\011\011\011 # Apply Newton's method for calculat ing roots of z^6-1=0.\n\011\011\011\011\011 xf := xs^2;\n\011\011\011 \011\011 yf := ys^2;\n\011\011\011\011\011 c := ((ys-2*xs)*ys*5+xf)*xx ;\011# real part of z^5\n\011\011\011\011\011 d := ((xs-2*ys)*xs*5+yf) *yy; # imag part of z^5\n\011\011\011\011\011\n\011\011\011\011\011 if abs(c)>abs(d) then\n\011\011\011\011\011\011 e := d/c;\n\011\011 \011\011\011\011 a := 1/(c+d*e); # real part of 1/z^5\011\n\011\011 \011\011\011\011 b := e*a;\011 \011# minus imag part of 1/z^5\n\011 \011\011\011\011\011else\n\011\011\011\011\011\011 e := c/d;\n\011 \011\011\011\011\011 b := 1/(c*e+d);\011# minus imag part of 1/z^5\n \011\011\011\011\011\011 a := e*b;\011 # real part of 1/z^5\n\011 \011\011\011\011 end if;\011\011\011\011\011\011\011\011\011\011\011 \011\n # real part of 5/6*z+1/(6*z^5)\011\011\011\011\011\011\011 \011\011\011\011\n\011\011\011\011\011 xx := fiveSixths*xx+oneSixth*a; \n # imag part of 5/6*z+1/(6*z^5)\n\011\011\011\011\011 yy := fiv eSixths*yy-oneSixth*b;\n\011\011\011\011\011\n\011\011\011\011\011 # H ow close are we to one of the roots?\n\011\011\011\011\011 xs := xx^2; \n ys := yy^2;\n\011\011\011\011\011 sum := xs+ys;\n\n\011\011 \011\011\011 comp := sum+.99;\n\011\011\011\011\011 dx := 2*xx;\n\011 \011\011\011\011 yTimesRootThree := yy*rootThree;\n\n\011\011\011\011 \011 # Are we close to the root -1?\n\011\011\011\011 \011p := evalb(- dx>comp);\n\011\011\011\011\011\n\011\011\011\011\011 # Are we close t o the root -1/2-sqrt(3)/2*I?\n\011\011\011\011\011 q := evalb(-xx-yTim esRootThree>comp);\n\011\011\011\011\011\n\011\011\011\011\011 # Are w e close to the root 1/2-sqrt(3)/2*I?\n\011\011\011\011\011 r := evalb( xx-yTimesRootThree>comp);\n\n\011\011\011\011\011 # Are we close to th e root 1?\n\011\011\011\011 s := evalb(dx>comp);\n\011\011\011\011 \011\n\011\011\011\011\011 # Are we close to the root 1/2+sqrt(3)/2*I? \n\011\011\011\011\011 t := evalb(xx+yTimesRootThree>comp);\n\n\011 \011\011\011\011 # Are we close to the troot -1/2+sqrt(3)/2*I?\n\011 \011\011\011 \011u := evalb(yTimesRootThree-xx>comp);\n\011\011\011 \011\011\n\011\011\011\011\011 # Set flag to end loop if sufficiently \+ close to a root.\n\011\011\011\011\011 notclosetoroot := not(p or q or r or s or t or u);\n end do;\n\n h := irem(i,25)^2;\n h := 1+0. 000004*(h-625)*h;\n\n # Determine the value to return.\n ans := hf array(1..3);\n if p then\n ans[1]:=0; ans[2]:=0; ans[3]:=h; \n \+ elif q then\n ans[1]:=h; ans[2]:=h; ans[3]:=0; \n elif r then \n ans[1]:=h; ans[2]:=0; ans[3]:=h; \n elif s then\n ans[1 ]:=0; ans[2]:=h; ans[3]:=0; \n elif t then\n ans[1]:=h*0.6; an s[2]:=0; ans[3]:=h; \n elif u then\n ans[1]:=h; ans[2]:=0; ans[ 3]:=0; \n else\n ans[1]:=.5; ans[2]:=.5; ans[3]:=.5\n end if; \n ans;\nend proc: # end of `fractalplot/newton6_rgb`\n\n`fractalplo t/newton7_hue` := proc(x,y)\n local xx,yy,xs,ys,xf,yf,ds,sum,i,p,q,r ,s,t,u,v,comp,notclosetoroot,\n a,b,c,d,e,a1x,b1y,a2x,b2y,a3x,b3y ,oneSeventh,sixSevenths;\n \n xx := x;\n yy := y;\n xs := xx^ 2;\n ys := yy^2;\n notclosetoroot := true;\n oneSeventh := .1428 57142857142857;\n sixSevenths := .857142857142857143;\n\n for i fr om 1 to _maxiter while notclosetoroot do\n\011\011\011\011\011 xf := x s*xs;\n\011\011\011\011\011 yf := ys*ys;\n\011\011\011\011\011 ds := x s-ys;\n\011\011\011\011\011 c := ds*(xf+yf-14*xs*ys);\011\011 # Rea l part of z^6\n\011\011\011\011\011 d := (6*xf-20*xs*ys+6*yf)*xx*yy; \011# Imag part of z^6\n\011\n\011\011\011\011\011 if abs(c)>abs(d) th en\n\011\011\011\011\011\011 e := d/c;\n\011\011\011\011\011\011 a := 1/(c+d*e); # real part of 1/z^4\011\n\011\011\011\011\011\011 b \+ := e*a;\011 \011 # minus imag part of 1/z^4\n\011\011\011\011\011 \011else\n\011\011\011\011\011\011 e := c/d;\n\011\011\011\011\011 \011 b := 1/(c*e+d);\011# minus imag part of 1/z^4\n\011\011\011\011 \011\011 a := e*b;\011 # real part of 1/z^4\n\011\011\011\011 \011 end if;\n\n # Real part of 6/7*z + 1/(7*z^4)\n\011\011\011 \011\011 xx := sixSevenths*xx+oneSeventh*a;\n # Imag part of 6/7* z + 1/(7*z^4)\011\n\011\011\011\011\011 yy := sixSevenths*yy-oneSevent h*b;\n\011\011\011\011\011\n\011\011\011\011\011 # How close are we to one of the roots?\n\011\011\011\011\011 xs := xx^2;\n ys := yy^2 ;\n sum := xs+ys;\n\011\011\011\011\011 comp := 0.5*sum+.495;\n\n \011\011\011\011\011 a1x := -0.900968867902419126*xx;\n\011\011\011 \011\011 b1y := 0.4338837391175581205*yy;\n\011\011\011\011\011 a2x : = -0.222520933956314404*xx;\n\011\011\011\011\011 b2y := 0.97492791218 1823607*yy;\n\011\011\011\011\011 a3x := 0.623489801858733531*xx;\n \011\011\011\011\011 b3y := 0.781831482468029809*yy;\n\011\011\011\011 \011\011\011\011\011\011\n\011\011\011\011\011 # Are we close to the r oot a1-b1*i?\n\011\011\011\011\011 p := evalb(a1x-b1y>comp);\n\011\011 \011\011\011\n\011\011\011\011\011 # Are we close to the root a1+b1*i? \n\011\011\011\011\011 q := evalb(a1x+b1y>comp);\n\011\011\011\011\011 \n\011\011\011\011\011 # Are we close to the root a2-b2*i?\n\011\011 \011\011\011 r := evalb(a2x-b2y>comp);\n\011\011\011\011\011\n\011\011 \011\011\011 # Are we close to the root a2+b2*i?\n\011\011\011\011\011 s := evalb(a2x+b2y>comp);\011\011\011\011\011\n\n\011\011\011\011\011 # Are we close to the root a3-b3*i?\n\011\011\011\011\011 t := evalb( a3x-b3y>comp);\n\011\011\011\011\011\n\011\011\011\011\011 # Are we cl ose to the root a3+b3*i?\n\011\011\011\011\011 u := evalb(a3x+b3y>comp );\n\n\011\011\011\011\011 # Are we close to the root 1?\n\011\011\011 \011\011 v := evalb(xx>comp);\n\n # Set flag to end loop if suffi ciently close to a root.\n notclosetoroot := not(p or q or r or s or t or u or v);\n end do;\n\n # Determine the value to return.\n if p then .6666666667 elif q then .1666666667 \n elif r then 0. \+ elif s then .3333333333\n elif t then .75 elif u then .08333333333 \+ \n elif v then .8333333333 else .5416666667 end if;\nend proc: # end of `fractalplot/newton7_hue`\n\n`fractalplot/newton7_rgb` := proc(x,y )\n local xx,yy,xs,ys,xf,yf,ds,sum,i,p,q,r,s,t,u,v,comp,notclosetoro ot,\n a,b,c,d,e,a1x,b1y,a2x,b2y,a3x,b3y,oneSeventh,sixSevenths,h, ans;\n \n xx := x;\n yy := y;\n xs := xx^2;\n ys := yy^2;\n notclosetoroot := true;\n oneSeventh := .142857142857142857;\n \+ sixSevenths := .857142857142857143;\n\n for i from 1 to _maxiter whi le notclosetoroot do\n\011\011\011\011\011 xf := xs*xs;\n\011\011\011 \011\011 yf := ys*ys;\n\011\011\011\011\011 ds := xs-ys;\n\011\011\011 \011\011 c := ds*(xf+yf-14*xs*ys);\011\011 # Real part of z^6\n\011 \011\011\011\011 d := (6*xf-20*xs*ys+6*yf)*xx*yy;\011# Imag part of z^ 6\n\011\n\011\011\011\011\011 if abs(c)>abs(d) then\n\011\011\011\011 \011\011 e := d/c;\n\011\011\011\011\011\011 a := 1/(c+d*e); # rea l part of 1/z^4\011\n\011\011\011\011\011\011 b := e*a;\011 \011 \+ # minus imag part of 1/z^4\n\011\011\011\011\011\011else\n\011\011\011 \011\011\011 e := c/d;\n\011\011\011\011\011\011 b := 1/(c*e+d); \011# minus imag part of 1/z^4\n\011\011\011\011\011\011 a := e*b; \011 # real part of 1/z^4\n\011\011\011\011\011 end if;\n\n \+ # Real part of 6/7*z + 1/(7*z^4)\n\011\011\011\011\011 xx := sixSevent hs*xx+oneSeventh*a;\n # Imag part of 6/7*z + 1/(7*z^4)\011\n\011 \011\011\011\011 yy := sixSevenths*yy-oneSeventh*b;\n\011\011\011\011 \011\n\011\011\011\011\011 # How close are we to one of the roots?\n \011\011\011\011\011 xs := xx^2;\n ys := yy^2;\n sum := xs+y s;\n\011\011\011\011\011 comp := 0.5*sum+.495;\n\n\011\011\011\011\011 a1x := -0.900968867902419126*xx;\n\011\011\011\011\011 b1y := 0.4338 837391175581205*yy;\n\011\011\011\011\011 a2x := -0.222520933956314404 *xx;\n\011\011\011\011\011 b2y := 0.974927912181823607*yy;\n\011\011 \011\011\011 a3x := 0.623489801858733531*xx;\n\011\011\011\011\011 b3y := 0.781831482468029809*yy;\n\011\011\011\011\011\011\011\011\011\011 \n\011\011\011\011\011 # Are we close to the root a1-b1*i?\n\011\011 \011\011\011 p := evalb(a1x-b1y>comp);\n\011\011\011\011\011\n\011\011 \011\011\011 # Are we close to the root a1+b1*i?\n\011\011\011\011\011 q := evalb(a1x+b1y>comp);\n\011\011\011\011\011\n\011\011\011\011\011 # Are we close to the root a2-b2*i?\n\011\011\011\011\011 r := evalb( a2x-b2y>comp);\n\011\011\011\011\011\n\011\011\011\011\011 # Are we cl ose to the root a2+b2*i?\n\011\011\011\011\011 s := evalb(a2x+b2y>comp );\011\011\011\011\011\n\n\011\011\011\011\011 # Are we close to the r oot a3-b3*i?\n\011\011\011\011\011 t := evalb(a3x-b3y>comp);\n\011\011 \011\011\011\n\011\011\011\011\011 # Are we close to the root a3+b3*i? \n\011\011\011\011\011 u := evalb(a3x+b3y>comp);\n\n\011\011\011\011 \011 # Are we close to the root 1?\n\011\011\011\011\011 v := evalb(xx >comp);\n\n # Set flag to end loop if sufficiently close to a roo t.\n notclosetoroot := not(p or q or r or s or t or u or v);\n \+ end do;\n\n h := irem(i,25)^2;\n h := 1+0.000004*(h-625)*h;\n\n \+ # Determine the value to return.\n ans := hfarray(1..3);\n if p th en\n ans[1]:=0; ans[2]:=0; ans[3]:=h; \n elif q then\n ans [1]:=h; ans[2]:=h; ans[3]:=0; \n elif r then\n ans[1]:=h; ans[2 ]:=0; ans[3]:=0; \n elif s then\n ans[1]:=0; ans[2]:=h; ans[3]: =0; \n elif t then\n ans[1]:=h*0.6; ans[2]:=0; ans[3]:=h; \n \+ elif u then\n ans[1]:=h; ans[2]:=0.6*h; ans[3]:=0;\n elif v th en\n ans[1]:=h; ans[2]:=0; ans[3]:=h; \n else\n ans[1]:=.5 ; ans[2]:=.5; ans[3]:=.5\n end if;\n ans;\nend proc: # end of `fra ctalplot/newton7_rgb`\n\n`fractalplot/newton8_hue` := proc(x,y)\n lo cal xx,yy,xs,ys,xf,yf,ds,sum,i,p,q,r,s,t,u,v,w,comp,comp2,\n notc losetoroot,a,b,c,d,e,rootTwo;\n \n xx := x;\n yy := y;\n xs : = xx^2;\n ys := yy^2;\n notclosetoroot := true;\n rootTwo := 1.4 1421356237309504880; \n\n for i from 1 to _maxiter while notclosetor oot do\n\011\011\011\011\011 xf := xs*xs;\n\011\011\011\011\011 yf := \+ ys*ys;\n\011\011\011\011\011 c := ((xs-21*ys)*xf+(35*xs-7*ys)*yf)*xx; \011# Real part of z^7\n\011\011\011\011\011 d := ((xs-5*ys)*7*xf+(21* xs-ys)*yf)*yy;\011 # Imag part of z^7\n\011\n\011\011\011\011\011 if a bs(c)>abs(d) then\n\011\011\011\011\011\011 e := d/c;\n\011\011\011 \011\011\011 a := 1/(c+d*e); # real part of 1/z^4\011\n\011\011\011 \011\011\011 b := e*a;\011 \011 # minus imag part of 1/z^4\n\011 \011\011\011\011\011else\n\011\011\011\011\011\011 e := c/d;\n\011 \011\011\011\011\011 b := 1/(c*e+d);\011# minus imag part of 1/z^4\n \011\011\011\011\011\011 a := e*b;\011 # real part of 1/z^4\n \011\011\011\011\011 end if;\n\n # Real part of 7/8*z+1/(8*z^4)\n \011\011\011\011\011 xx := 0.875*xx+0.125*a;\n # Imag part of 7/8 *z+1/(8*z^4)\011\n\011\011\011\011\011 yy := 0.875*yy-0.125*b;\n\011 \011\011\011\011\n\011\011\011\011\011 # How close are we to one of th e roots?\n\011\011\011\011\011 xs := xx^2;\n ys := yy^2;\n s um := xs+ys;\n\011\011\011\011\011 comp := 0.5*sum+.495;\n comp2 \+ := comp*rootTwo;\n\n\011\011\011\011\011 # Are we close to the root -1 ?\n\011\011\011\011\011 p := evalb(-xx>comp);\n\011\011\011\011\011\n \011\011\011\011\011 # Are we close to the root -1/sqrt(2)-1/sqrt(2)*i ?\n\011\011\011\011\011 q := evalb(-xx-yy>comp2);\n\011\011\011\011 \011\n\011\011\011\011\011 # Are we close to the root -1/sqrt(2)+1/sqr t(2)*i?\n\011\011\011\011\011 r := evalb(-xx+yy>comp2);\n\011\011\011 \011\011\n\011\011\011\011\011 # Are we close to the root -i?\n\011 \011\011\011\011 s := evalb(-yy>comp);\n\011\011\011\011\011\n\011\011 \011\011\011 # Are we close to the root i?\n\011\011\011\011\011 t := \+ evalb(yy>comp);\n\011\011\011\011\011\n\011\011\011\011\011 # Are we c lose to the root 1/sqrt(2)-1/sqrt(2)*i?\n\011\011\011\011\011 u := eva lb(xx-yy>comp2);\n\011\011\011\011\011\n\011\011\011\011\011 # Are we \+ close to the root 1/sqrt(2)+1/sqrt(2)*i?\n\011\011\011\011\011 v := ev alb(xx+yy>comp2);\n\n\011\011\011\011\011 # Are we close to the root 1 ?\n\011\011\011\011\011 w := evalb(xx>comp);\n\011\011\011\011\011\011 \011\011\011\011\n # Set flag to end loop if sufficiently close t o a root.\n notclosetoroot := not(p or q or r or s or t or u or v or w);\n end do;\n\n # Determine the value to return.\n if p th en .6666666667 elif q then .1666666667 \n elif r then 0 elif s the n .3333333333\n elif t then .75 elif u then .08333333333\n elif v \+ then .8333333333 elif w then .5\n else .5416666667 end if;\nend proc : # end of `fractalplot/newton8_hue`\n\n`fractalplot/newton8_rgb` := p roc(x,y)\n local xx,yy,xs,ys,xf,yf,ds,sum,i,p,q,r,s,t,u,v,w,comp,com p2,\n notclosetoroot,a,b,c,d,e,rootTwo,h,ans;\n \n xx := x; \n yy := y;\n xs := xx^2;\n ys := yy^2;\n notclosetoroot := tr ue;\n rootTwo := 1.41421356237309504880; \n\n for i from 1 to _max iter while notclosetoroot do\n\011\011\011\011\011 xf := xs*xs;\n\011 \011\011\011\011 yf := ys*ys;\n\011\011\011\011\011 c := ((xs-21*ys)*x f+(35*xs-7*ys)*yf)*xx;\011# Real part of z^7\n\011\011\011\011\011 d : = ((xs-5*ys)*7*xf+(21*xs-ys)*yf)*yy;\011 # Imag part of z^7\n\011\n \011\011\011\011\011 if abs(c)>abs(d) then\n\011\011\011\011\011\011 \+ e := d/c;\n\011\011\011\011\011\011 a := 1/(c+d*e); # real part of \+ 1/z^4\011\n\011\011\011\011\011\011 b := e*a;\011 \011 # minus im ag part of 1/z^4\n\011\011\011\011\011\011else\n\011\011\011\011\011 \011 e := c/d;\n\011\011\011\011\011\011 b := 1/(c*e+d);\011# minu s imag part of 1/z^4\n\011\011\011\011\011\011 a := e*b;\011 # \+ real part of 1/z^4\n\011\011\011\011\011 end if;\n\n # Real part \+ of 7/8*z+1/(8*z^4)\n\011\011\011\011\011 xx := 0.875*xx+0.125*a;\n \+ # Imag part of 7/8*z+1/(8*z^4)\011\n\011\011\011\011\011 yy := 0.875 *yy-0.125*b;\n\011\011\011\011\011\n\011\011\011\011\011 # How close a re we to one of the roots?\n\011\011\011\011\011 xs := xx^2;\n ys := yy^2;\n sum := xs+ys;\n\011\011\011\011\011 comp := 0.5*sum+. 495;\n comp2 := comp*rootTwo;\n\n\011\011\011\011\011 # Are we cl ose to the root -1?\n\011\011\011\011\011 p := evalb(-xx>comp);\n\011 \011\011\011\011\n\011\011\011\011\011 # Are we close to the root -1/s qrt(2)-1/sqrt(2)*i?\n\011\011\011\011\011 q := evalb(-xx-yy>comp2);\n \011\011\011\011\011\n\011\011\011\011\011 # Are we close to the root \+ -1/sqrt(2)+1/sqrt(2)*i?\n\011\011\011\011\011 r := evalb(-xx+yy>comp2) ;\n\011\011\011\011\011\n\011\011\011\011\011 # Are we close to the ro ot -i?\n\011\011\011\011\011 s := evalb(-yy>comp);\n\011\011\011\011 \011\n\011\011\011\011\011 # Are we close to the root i?\n\011\011\011 \011\011 t := evalb(yy>comp);\n\011\011\011\011\011\n\011\011\011\011 \011 # Are we close to the root 1/sqrt(2)-1/sqrt(2)*i?\n\011\011\011 \011\011 u := evalb(xx-yy>comp2);\n\011\011\011\011\011\n\011\011\011 \011\011 # Are we close to the root 1/sqrt(2)+1/sqrt(2)*i?\n\011\011 \011\011\011 v := evalb(xx+yy>comp2);\n\n\011\011\011\011\011 # Are we close to the root 1?\n\011\011\011\011\011 w := evalb(xx>comp);\n\011 \011\011\011\011\011\011\011\011\011\n # Set flag to end loop if \+ sufficiently close to a root.\n notclosetoroot := not(p or q or r or s or t or u or v or w);\n end do;\n\n h := irem(i,25)^2;\n h := 1+0.000004*(h-625)*h;\n\n # Determine the value to return.\n a ns := hfarray(1..3);\n if p then\n ans[1]:=0; ans[2]:=0; ans[3] :=h; \n elif q then\n ans[1]:=h; ans[2]:=h; ans[3]:=0; \n eli f r then\n ans[1]:=h; ans[2]:=0; ans[3]:=0; \n elif s then\n \+ ans[1]:=0; ans[2]:=h; ans[3]:=0; \n elif t then\n ans[1]:=h *0.6; ans[2]:=0; ans[3]:=h; \n elif u then\n ans[1]:=h; ans[2]: =0.6*h; ans[3]:=0;\n elif v then\n ans[1]:=h; ans[2]:=0; ans[3] :=h;\n elif w then\n ans[1]:=0; ans[2]:=h; ans[3]:=h; \n else \n ans[1]:=.5; ans[2]:=.5; ans[3]:=.5\n end if;\n ans;\nend p roc: # end of `fractalplot/newton8_rgb`\n\n`fractalplot/mandelbrot_hue ` := proc(x,y)\n local xx,yy,xs,ys,prevx,k;\n xx := x;\n yy := y ;\n xs := xx^2;\n ys := yy^2;\n for k from 0 to _maxiter while x s+ys<16 do\n prevx := xx;\n xx := xs-ys+x;\n yy := 2*pr evx*yy+y;\n xs := xx^2;\n ys := yy^2;\n end do;\n if k<_ maxiter then irem(k+_offset,_numcolors)/_numcolors else .5625 end if; \nend proc: # of `fractalplot/mandelbrot_hue`\n\n`fractalplot/mandelbr ot_rgb` := proc(x,y,C)\n local xx,yy,xs,ys,prevx,k,A;\n xx := x;\n yy := y;\n xs := xx^2;\n ys := yy^2;\n for k from 0 to _maxit er while xs+ys<16 do\n prevx := xx;\n xx := xs-ys+x;\n \+ yy := 2*prevx*yy+y;\n xs := xx^2;\n ys := yy^2;\n end do; \n A := hfarray(1..3);\n if k<_maxiter then\n\011 k := irem(k+ _offset,_numcolors)+1;\n A[1] := C[k,1]; A[2] := C[k,2]; A[3] := \+ C[k,3];\n else\n A[1] := 0.; A[2] := 0.; A[3] := 0.;\n end if ;\n A;\nend proc: # end of `fractalplot/mandelbrot_rgb`\n\n`fractalp lot/mandelbrot3_hue` := proc(x,y)\n local xx,yy,xs,ys,k;\n xx := x ;\n yy := y;\n xs := xx^2;\n ys := yy^2;\n for k from 0 to _ma xiter while xs+ys<16 do\n xx := (xs-ys*3)*xx+x;\n yy := (xs* 3-ys)*yy+y;\n xs := xx^2;\n ys := yy^2;\n end do;\n if k <_maxiter then irem(k+_offset,_numcolors)/_numcolors else .5625 end if ;\nend proc: # of `fractalplot/mandelbrot3_hue`\n\n`fractalplot/mandel brot3_rgb` := proc(x,y,C)\n local xx,yy,xs,ys,k,A;\n xx := x;\n \+ yy := y;\n xs := xx^2;\n ys := yy^2;\n for k from 0 to _maxiter \+ while xs+ys<16 do\n xx := (xs-ys*3)*xx+x;\n yy := (xs*3-ys)* yy+y;\n xs := xx^2;\n ys := yy^2;\n end do;\n A := hfarr ay(1..3);\n if k<_maxiter then\n\011 k := irem(k+_offset,_numcol ors)+1;\n A[1] := C[k,1]; A[2] := C[k,2]; A[3] := C[k,3];\n els e\n A[1] := 0.; A[2] := 0.; A[3] := 0.;\n end if;\n A;\nend p roc: # end of `fractalplot/mandelbrot3_rgb`\n\n`fractalplot/mandelbrot 4_hue` := proc(x,y)\n local xx,yy,xs,ys,w,ws,ds,k;\n xx := x;\n \+ yy := y;\n xs := xx^2;\n ys := yy^2;\n for k from 0 to _maxiter \+ while xs+ys<16 do\n\011\011\011 w := xx*yy;\n\011\011\011\011\011 ws := w*w;\n\011\011\011\011\011 ds := xs-ys;\n\011\011\011\011\011 xx : = ds*ds-ws*4+x;\n\011\011\011\011\011 yy := ds*w*4+y;\n xs := xx^ 2;\n ys := yy^2;\n end do;\n if k<_maxiter then irem(k+_offse t,_numcolors)/_numcolors else .5625 end if;\nend proc: # of `fractalpl ot/mandelbrot4_hue`\n\n`fractalplot/mandelbrot4_rgb` := proc(x,y,C)\n \+ local xx,yy,xs,ys,w,ws,ds,k,A;\n xx := x;\n yy := y;\n xs := x x^2;\n ys := yy^2;\n for k from 0 to _maxiter while xs+ys<16 do\n \011\011\011 w := xx*yy;\n\011\011\011\011\011 ws := w*w;\n\011\011 \011\011\011 ds := xs-ys;\n\011\011\011\011\011 xx := ds*ds-ws*4+x;\n \011\011\011\011\011 yy := ds*w*4+y;\n xs := xx^2;\n ys := y y^2;\n end do;\n A := hfarray(1..3);\n if k<_maxiter then\n\011 \+ k := irem(k+_offset,_numcolors)+1;\n A[1] := C[k,1]; A[2] := \+ C[k,2]; A[3] := C[k,3];\n else\n A[1] := 0.; A[2] := 0.; A[3] : = 0.;\n end if;\n A;\nend proc: # end of `fractalplot/mandelbrot4_ rgb`\n\n`fractalplot/logistic_hue` := proc(x,y)\n local xx,yy,xs,ys, p,q,k;\n xx := 0.5;\n yy := 0;\n xs := xx^2;\n ys := yy^2;\n \+ for k from 0 to _maxiter while xs+ys<16 do\n p := xx+ys-xs;\n \+ q := yy-2*xx*yy;\n xx := p*x-q*y;\n yy := q*x+p*y;\n \+ xs := xx^2;\n ys := yy^2;\n end do;\n if k<_maxiter then ire m(k+_offset,_numcolors)/_numcolors else .5625 end if;\nend proc: # of \+ `fractalplot/logistic_hue`\n\n`fractalplot/logistic_rgb` := proc(x,y,C )\n local xx,yy,xs,ys,p,q,k,A;\n xx := .5;\n yy := 0;\n xs := \+ xx^2;\n ys := yy^2;\n for k from 0 to _maxiter while xs+ys<16 do\n p := xx+ys-xs;\n q := yy-2*xx*yy;\n xx := p*x-q*y;\n \+ yy := q*x+p*y;\n xs := xx^2;\n ys := yy^2;\n end do;\n A := hfarray(1..3);\n if k<_maxiter then\n\011 k := irem(k+_o ffset,_numcolors)+1;\n A[1] := C[k,1]; A[2] := C[k,2]; A[3] := C[ k,3];\n else\n A[1] := 0.; A[2] := 0.; A[3] := 0.;\n end if; \n A;\nend proc: # end of `fractalplot/logistic_rgb`\n\n`fractalplot /chebyshevT3_hue` := proc(x,y)\n local xx,yy,xs,ys,p,q,k;\n xx := \+ 0.5;\n yy := 0;\n xs := xx^2;\n ys := yy^2;\n for k from 0 to \+ _maxiter while xs+ys<16 do\n\011\011 p := (xs*4-ys*12-3)*xx;\n \+ q := (xs*12-ys*4-3)*yy;\n xx := p*x-q*y;\n yy := q*x+p*y;\n xs := xx^2;\n ys := yy^2;\n end do;\n if k<_maxiter the n irem(k+_offset,_numcolors)/_numcolors else .5625 end if;\nend proc: \+ # of `fractalplot/chebyshevT3_hue`\n\n`fractalplot/chebyshevT3_rgb` := proc(x,y,C)\n local xx,yy,xs,ys,p,q,k,A;\n xx := 0.5;\n yy := 0 ;\n xs := xx^2;\n ys := yy^2;\n for k from 0 to _maxiter while x s+ys<16 do\n\011\011 p := (xs*4-ys*12-3)*xx;\n q := (xs*12-ys* 4-3)*yy;\n xx := p*x-q*y;\n yy := q*x+p*y;\n xs := xx^2 ;\n ys := yy^2;\n end do;\n A := hfarray(1..3);\n if k<_max iter then\n\011 k := irem(k+_offset,_numcolors)+1;\n A[1] := \+ C[k,1]; A[2] := C[k,2]; A[3] := C[k,3];\n else\n A[1] := 0.; A[ 2] := 0.; A[3] := 0.;\n end if;\n A;\nend proc: # end of `fractalp lot/chebyshevT3_rgb`\n\n`fractalplot/chebyshevT4_hue` := proc(x,y)\n \+ local xx,yy,xs,ys,w,xf,yf,ws,ds,p,q,k;\n xx := 0.707106781186547524 40;\n yy := 0;\n xs := xx^2;\n ys := yy^2;\n for k from 0 to _ maxiter while xs+ys<16 do\n w := xx*yy;\n xf := xs*xs;\n \+ yf := ys*ys;\n ws := w*w;\n ds := xs-ys;\n\011\011 p := (xf-ws*6+yf-ds)*8+1;\n q := (ds*2-1)*16*w; \n xx := p*x-q*y ;\n yy := q*x+p*y;\n xs := xx^2;\n ys := yy^2;\n end \+ do;\n if k<_maxiter then irem(k+_offset,_numcolors)/_numcolors else \+ .5625 end if;\nend proc: # of `fractalplot/chebyshevT4_hue`\n\n`fracta lplot/chebyshevT4_rgb` := proc(x,y,C)\n local xx,yy,xs,ys,w,xf,yf,ws ,ds,p,q,k,A;\n xx := 0.70710678118654752440;\n yy := 0;\n xs := \+ xx^2;\n ys := yy^2;\n for k from 0 to _maxiter while xs+ys<16 do\n w := xx*yy;\n xf := xs*xs;\n yf := ys*ys;\n ws := w*w;\n ds := xs-ys;\n\011\011 p := (xf-ws*6+yf-ds)*8+1;\n \+ q := (ds*2-1)*16*w; \n xx := p*x-q*y;\n yy := q*x+p*y;\n \+ xs := xx^2;\n ys := yy^2;\n end do;\n A := hfarray(1..3); \n if k<_maxiter then\n\011 k := irem(k+_offset,_numcolors)+1;\n A[1] := C[k,1]; A[2] := C[k,2]; A[3] := C[k,3];\n else\n \+ A[1] := 0.; A[2] := 0.; A[3] := 0.;\n end if;\n A;\nend proc: # en d of `fractalplot/chebyshevT4_rgb`\n\n`fractalplot/chebyshevT5_hue` := proc(x,y)\n local xx,yy,xs,ys,xf,yf,ts,p,q,k;\n xx := 0.309016994 37494742410;\n yy := 0;\n xs := xx^2;\n ys := yy^2;\n for k fr om 0 to _maxiter while xs+ys<16 do\n\011\011 xf := xs*xs;\n yf := ys*ys;\n ts := 10*xs*ys;\n\011\011 p := ((xf-ts+5*yf)*16-( xs-ys*3)*20+5)*xx;\n\011\011 q := ((yf-ts+5*xf)*16-(xs*3-ys)*20+5)* yy; \n xx := p*x-q*y;\n yy := q*x+p*y;\n xs := xx^2;\n \+ ys := yy^2;\n end do;\n if k<_maxiter then irem(k+_offset,_nu mcolors)/_numcolors else .5625 end if;\nend proc: # of `fractalplot/ch ebyshevT5_hue`\n\n`fractalplot/chebyshevT5_rgb` := proc(x,y,C)\n loc al xx,yy,xs,ys,xf,yf,ts,p,q,k,A;\n xx := 0.30901699437494742410;\n \+ yy := 0;\n xs := xx^2;\n ys := yy^2;\n for k from 0 to _maxiter while xs+ys<16 do\n\011\011 xf := xs*xs;\n yf := ys*ys;\n \+ ts := 10*xs*ys;\n\011\011 p := ((xf-ts+5*yf)*16-(xs-ys*3)*20+5)*x x;\n\011\011 q := ((yf-ts+5*xf)*16-(xs*3-ys)*20+5)*yy; \n xx : = p*x-q*y;\n yy := q*x+p*y;\n xs := xx^2;\n ys := yy^2; \n end do;\n A := hfarray(1..3);\n if k<_maxiter then\n\011 \+ k := irem(k+_offset,_numcolors)+1;\n A[1] := C[k,1]; A[2] := C[k, 2]; A[3] := C[k,3];\n else\n A[1] := 0.; A[2] := 0.; A[3] := 0. ;\n end if;\n A;\nend proc: # end of `fractalplot/chebyshevT5_rgb` \n\n`fractalplot/chebyshevT6_hue` := proc(x,y)\n local xx,yy,xs,ys,w ,xf,yf,ws,ds,p,q,k;\n xx := 0.5;\n yy := 0;\n xs := xx^2;\n ys := yy^2;\n for k from 0 to _maxiter while xs+ys<16 do\n w := x x*yy;\n xf := xs*xs;\n yf := ys*ys;\n ws := w*w;\n \+ ds := xs-ys;\n\011\011 p := ((xs-ys*15)*xf+(xs*15-ys)*yf)*32-(xf-w s*6+yf)*48+ds*18-1;\n\011\011 q := ((xf*3-ws*10+yf*3)*16-ds*48+9)*w *4;\n xx := p*x-q*y;\n yy := q*x+p*y;\n xs := xx^2;\n \+ ys := yy^2;\n end do;\n if k<_maxiter then irem(k+_offset,_num colors)/_numcolors else .5625 end if;\nend proc: # of `fractalplot/che byshevT6_hue`\n\n`fractalplot/chebyshevT6_rgb` := proc(x,y,C)\n loca l xx,yy,xs,ys,w,xf,yf,ws,ds,p,q,k,A;\n xx := 0.5;\n yy := 0;\n x s := xx^2;\n ys := yy^2;\n for k from 0 to _maxiter while xs+ys<16 do\n w := xx*yy;\n xf := xs*xs;\n yf := ys*ys;\n \+ ws := w*w;\n ds := xs-ys;\n\011\011 p := ((xs-ys*15)*xf+(xs*15 -ys)*yf)*32-(xf-ws*6+yf)*48+ds*18-1;\n\011\011 q := ((xf*3-ws*10+yf *3)*16-ds*48+9)*w*4;\n xx := p*x-q*y;\n yy := q*x+p*y;\n \+ xs := xx^2;\n ys := yy^2;\n end do;\n A := hfarray(1..3);\n if k<_maxiter then\n\011 k := irem(k+_offset,_numcolors)+1;\n \+ A[1] := C[k,1]; A[2] := C[k,2]; A[3] := C[k,3];\n else\n A[ 1] := 0.; A[2] := 0.; A[3] := 0.;\n end if;\n A;\nend proc: # end \+ of `fractalplot/chebyshevT6_rgb`\n\n`fractalplot/chebyshevT7_hue` := p roc(x,y)\n local xx,yy,xs,ys,u,v,p,q,k;\n xx := 0.6234898018587335 3053;\n yy := 0;\n xs := xx^2;\n ys := yy^2;\n for k from 0 to _maxiter while xs+ys<16 do\n\011\011 u := 1344*xs;\n\011\011 v \+ := 2240*xs;\n\011\011 p := (((56+(64*xs-112)*xs)*xs-7)+(((1120-u)*x s-168)+(v-560-448*ys)*ys)*ys)*xx;\n\011\011 q := (((168+(448*xs-560 )*xs)*xs-7)+(((1120-v)*xs-56)+(u-112-64*ys)*ys)*ys)*yy;\n xx := p *x-q*y;\n yy := q*x+p*y;\n xs := xx^2;\n ys := yy^2;\n \+ end do;\n if k<_maxiter then irem(k+_offset,_numcolors)/_numcolors else .5625 end if;\nend proc: # of `fractalplot/chebyshevT7_hue`\n\n` fractalplot/chebyshevT7_rgb` := proc(x,y,C)\n local xx,yy,xs,ys,u,v, p,q,k,A;\n xx := 0.62348980185873353053;\n yy := 0;\n xs := xx^2 ;\n ys := yy^2;\n for k from 0 to _maxiter while xs+ys<16 do\n\011 \011 u := 1344*xs;\n\011\011 v := 2240*xs;\n\011\011 p := ((( 56+(64*xs-112)*xs)*xs-7)+(((1120-u)*xs-168)+(v-560-448*ys)*ys)*ys)*xx; \n\011\011 q := (((168+(448*xs-560)*xs)*xs-7)+(((1120-v)*xs-56)+(u- 112-64*ys)*ys)*ys)*yy;\n xx := p*x-q*y;\n yy := q*x+p*y;\n \+ xs := xx^2;\n ys := yy^2;\n end do;\n A := hfarray(1..3); \n if k<_maxiter then\n\011 k := irem(k+_offset,_numcolors)+1;\n A[1] := C[k,1]; A[2] := C[k,2]; A[3] := C[k,3];\n else\n \+ A[1] := 0.; A[2] := 0.; A[3] := 0.;\n end if;\n A;\nend proc: # en d of `fractalplot/chebyshevT7_rgb`\n\n`fractalplot/chebyshevT8_hue` := proc(x,y)\n local xx,yy,xs,ys,u,v,p,q,k;\n xx := 0.70710678118654 752440;\n yy := 0;\n xs := xx^2;\n ys := yy^2;\n for k from 0 \+ to _maxiter while xs+ys<16 do\n u := 3584*xs;\n v := 7168*xs ;\n\011\011 p := 1+((160+(128*xs-256)*xs)*xs-32)*xs+((32+((3840-u)* xs-960)*xs)+\n (160+(8960*xs-3840)*xs+((256-u)+128*ys)*ys)* ys)*ys;\n\011\011 q := ((640+(1024*xs-1536)*xs)*xs-64+((5120-v)*xs- 640+\n (v-1536-1024*ys)*ys)*ys)*xx*yy;\n xx := p*x-q*y ;\n yy := q*x+p*y;\n xs := xx^2;\n ys := yy^2;\n end \+ do;\n if k<_maxiter then irem(k+_offset,_numcolors)/_numcolors else \+ .5625 end if;\nend proc: # of `fractalplot/chebyshevT8_hue`\n\n`fracta lplot/chebyshevT8_rgb` := proc(x,y,C)\n local xx,yy,xs,ys,u,v,p,q,k, A;\n xx := 0.70710678118654752440;\n yy := 0;\n xs := xx^2;\n \+ ys := yy^2;\n for k from 0 to _maxiter while xs+ys<16 do\n u := 3584*xs;\n v := 7168*xs;\n\011\011 p := 1+((160+(128*xs-256)* xs)*xs-32)*xs+((32+((3840-u)*xs-960)*xs)+\n (160+(8960*xs-3 840)*xs+((256-u)+128*ys)*ys)*ys)*ys;\n\011\011 q := ((640+(1024*xs- 1536)*xs)*xs-64+((5120-v)*xs-640+\n (v-1536-1024*ys)*ys)*ys )*xx*yy;\n xx := p*x-q*y;\n yy := q*x+p*y;\n xs := xx^2 ;\n ys := yy^2;\n end do;\n A := hfarray(1..3);\n if k<_max iter then\n\011 k := irem(k+_offset,_numcolors)+1;\n A[1] := \+ C[k,1]; A[2] := C[k,2]; A[3] := C[k,3];\n else\n A[1] := 0.; A[ 2] := 0.; A[3] := 0.;\n end if;\n A;\nend proc: # end of `fractalp lot/chebyshevT8_rgb`\n\n`fractalplot/sine_hue` := proc(x,y)\n local \+ xx,yy,xs,ys,p,q,k;\n xx := x;\n yy := y;\n xs := xx^2;\n ys := yy^2;\n for k from 0 to _maxiter while xs+ys<16 do\n\011\011 p : = sin(xx)*cosh(yy); # Real part of sin(z)\n\011\011 q := cos(xx)*si nh(yy); # Imag part of sin(z)\n xx := p*x-q*y;\n yy := q*x+p *y;\n xs := xx^2;\n ys := yy^2;\n end do;\n if k<_maxite r then irem(k+_offset,_numcolors)/_numcolors else .5625 end if;\nend p roc: # of `fractalplot/sine_hue`\n\n`fractalplot/sine_rgb` := proc(x,y ,C)\n local xx,yy,xs,ys,p,q,k,A;\n xx := x;\n yy := y;\n xs := xx^2;\n ys := yy^2;\n for k from 0 to _maxiter while xs+ys<16 do \n\011\011 p := sin(xx)*cosh(yy); # Real part of sin(z)\n\011\011 \+ q := cos(xx)*sinh(yy); # Imag part of sin(z)\n xx := p*x-q*y;\n yy := q*x+p*y;\n xs := xx^2;\n ys := yy^2;\n end do; \n A := hfarray(1..3);\n if k<_maxiter then\n\011 k := irem(k+ _offset,_numcolors)+1;\n A[1] := C[k,1]; A[2] := C[k,2]; A[3] := \+ C[k,3];\n else\n A[1] := 0.; A[2] := 0.; A[3] := 0.;\n end if ;\n A;\nend proc: # end of `fractalplot/sine_rgb`" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "Examples \+ are given in a following section." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 71 "A procedure for constructing colour tables for use with \"fractalplot\": " }{TEXT 0 5 "blend" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " } {TEXT 0 5 "blend" }{TEXT -1 63 " constructs colour tables which can be used with the procedure " }{TEXT 0 11 "fractalplot" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 261 4 "Note" }{TEXT -1 2 ": " }{TEXT 0 5 "blen d" }{TEXT -1 14 " requires the " }{TEXT 261 23 "utility colour routine s" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 12 "blend: usage" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 264 18 "Calling S equence:\n" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 " blend( c olor_list, number_list ) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 11 "Parameters:" }}{PARA 0 "" 0 "" {TEXT -1 4 " \+ " }{TEXT 23 15 " color_list - " }{TEXT 265 70 "a list of colours to provide \"pivotal colours\" to be blended between. " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }{TEXT 23 15 "num ber_list - " }{TEXT 267 70 "a list of non-negative integers to specif y how many \"blended colours\" " }}{PARA 0 "" 0 "" {TEXT 275 76 " \+ to insert between the \"pivotal colours \". " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 " " }} {PARA 258 "" 0 "" {TEXT -1 12 "Description:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 " The procedure " }{TEXT 0 5 "b lend" }{TEXT 268 81 " extends a list of colours to a table of colours by inserting a specified number" }}{PARA 0 "" 0 "" {TEXT 274 59 " o f blended colours between the given \"pivotal colours\". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 266 8 "Options:" }{TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 72 "output=hfarray/color_list\nTh e desired output for use with the procedure " }{TEXT 0 11 "fractalplot " }{TEXT -1 33 " is an array of hardware floats. " }}{PARA 0 "" 0 "" {TEXT -1 74 "The blended colours may also be obtained in the form of a list of colours." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "draw=true/false/0/1/2" }}{PARA 0 "" 0 "" {TEXT -1 33 "Thi s option allows the procedure " }{TEXT 0 5 "blend" }{TEXT -1 52 " to d isplay the blended colours in a graphical form." }}{PARA 0 "" 0 "" {TEXT -1 66 "When \"draw=1\", the table is displayed as a 2 dimensiona l graphic. " }}{PARA 0 "" 0 "" {TEXT -1 140 "When \"draw=2\" and the o utput is an array of hardware floats, the blended colours can be displ ayed using a 3-dimensional graphics structure. " }}{PARA 0 "" 0 "" {TEXT -1 36 "\"draw=true\" is the same as \"draw=1\" " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 16 "How to activate:" }{TEXT 263 1 "\n" }{TEXT -1 154 "To ma ke the procedure active open the subsection, place the cursor anywhere after the prompt [ > and press [Enter].\nYou can then close up the s ubsection." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 "blend: implementati on" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3430 "blend := proc(colrs,nums)\n local n,m,clrs,i,j,bl endedclrs,steps,pp,\n unknownopts,s,t,tt,hfa,drw,A,B,C,points,tp; \n\n if nargs<2 then\n error \"at least 2 arguments are require d; the basic syntax is: 'blend(colours,numbers)',where 'colours' is a \+ list of colours and 'numbers' is a list of non-negative integers.\"\n \+ end if;\n tp := traperror(type(colrs,list(color)));\n if tp=last error then\n error \"please ensure that you have loaded the utili ty colour routines\"\n end if;\n if not tp then\n error \"the 1st argument must be a list of colours\"\n end if;\n n := nops(co lrs);\n m := nops(nums);\n if not type(nums,list(nonnegint)) or n- m<>1 then\n error \"the 2nd argument must be a list of %1 non-neg ative integer(s) to specify the number of blended colours to insert be tween the colours supplied\",n-1;\n end if;\n unknownopts := [];\n hfa := true;\n drw := 0;\n for t in [args[3..nargs]] do\n \+ if not type(t,equation) then\n error \"incorrect optional argu ment: %1\",t;\n end if;\n s := op(1,t);\n tt := op(2,t) ;\n if s='output' then\n if member(tt,\{'colour_list','co lor_list','colourlist','colorlist'\}) then\n hfa := false; \+ \n else\n if not member(tt,\{'hfarray','hf_array'\} ) then\n error \"\\\"output\\\" must be 'hfarray','hf_ar ray','colour_list','color_list','colourlist' or 'colorlist'.\"\n \+ end if; \n end if;\n elif s='draw' then\n i f member(tt,\{0,1,2\}) then drw := tt\n elif tt=true then drw \+ := 1\n else drw := 0 end if;\n else unknownopts := [op(un knownopts),s=tt];\n end if\n end do;\n if nops(unknownopts)<> 0 then \n error \"unknown or inapplicable option: %1\",op(1,unkno wnopts);\n end if;\n\n clrs := [];\n for i to n do\n clrs : = [op(clrs),convert(colrs[i],RGB)];\n end do;\n blendedclrs := []; \n for i to m do\n steps := [seq((op(k+1,clrs[i+1])-op(k+1,clrs [i]))/(nums[i]+1),k=1..3)];\n for j from 0 to nums[i] do\n \+ blendedclrs := [op(blendedclrs),\n COLOUR('RGB',seq(op(k+1 ,clrs[i])+j*steps[k],k=1..3))];\n end do;\n end do; \+ \n blendedclrs := [op(blendedclrs),clrs[n]];\n n := nops(blendedcl rs);\n pp := NULL;\n j := 1; \n for i from 1 to n do\n pp : = pp,plots[polygonplot]([[j-1,0],[j-1,1],[j,1],[j,0]],\n \+ style=patchnogrid,color=blendedclrs[i]);\n j := j+1;\n end do; \n \n if hfa then\n C := hfarray(1..n,1..3);\n for i to \+ n do\n C[i,1] := op(2,blendedclrs[i]);\n C[i,2] := op( 3,blendedclrs[i]);\n C[i,3] := op(4,blendedclrs[i]);\n en d do;\n if drw=1 then\n print(plots[display]([pp],axes=NO NE));\n elif drw=2 then\n A := hfarray(1..n,1..n);\n \+ for i to n do\n for j to n do\n A[i,j] := 0.; \n end do;\n end do;\n B := hfarray(1. .n,1..n,1..3);\n for i to n do\n for j to n do\n \+ B[i,j,1] := C[i,1];\n B[i,j,2] := C[i,2];\n B[i,j,3] := C[i,3];\n end do;\n end \+ do;\n points := 'GRID'(-1..1,-1..1,A,COLOR('RGB',eval(B)));\n \+ print(PLOT3D(points,STYLE(PATCHNOGRID),ORIENTATION(-90.,0.),\n VIEW(-1..1),SCALING(UNCONSTRAINED))); \n end if;\n \+ C; \n else \n if drw>0 then\n print(plots[display]( [pp],axes=NONE));\n end if;\n blendedclrs;\n end if;\nend \+ proc: " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "Examples are given in the following section." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 0 5 "blend" }{TEXT -1 11 ": examples " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" 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