iterations : keyword option of the form iterations=val, where val is a posint specifying the number of times the function defined by exprs will be applied at each (xvariable,yvariable) value. The default value is 1000.

outputimage : keyword option of the form outputimage=img where img is a color image as used by the ImageTools package, with dimensions of height and width. Escape will fill img with the generated image defined below.

height : keyword option of the form height=val, where val is a positive integer specifying the height of the image. If outputimage is defined the default value is the height of outputimage. Otherwise, the default value is 500.

width : keyword option of the form width=val, where val is a positive integer specifying the width of the image. If outputimage is defined the default value is the width of outputimage. Otherwise, the default value is 500.

xvariable : keyword option of the form xvariable=param, where param is a name specifying the parameter in exprs and/or init that will be varied along the horizontal axis. The default value is $'x'$

yvariable : keyword option of the form yvariable=param, where param is a name specifying the parameter in exprs and/or init that will be varied along the vertical axis. The default value is $'y'$

loopvariable : keyword option of the form loopvariable=var, where var is a name specifying the variable representing the current iteration index. The default value is $'i'$

redvariable : keyword option of the form redvariable=var, where var is a name specifying the red color variable at the current position in the image. The default value is $R$

greenvariable : keyword option of the form greenvariable=var, where var is a name specifying the green color variable at the current position in the image. The default value is $G$

bluevariable : keyword option of the form bluevariable=var, where var is a name specifying the blue color variable at the current position in the image. The default value is $B$

redexpression : keyword option of the form redexpression=expr, where expr is an algebraic expression defining the coloring procedure of the red color value at the current position the image. If redexpression=$\mathrm{null}$ then no procedure is applied to the red value at the current point. The default value is loopvariable

greenexpression : keyword option of the form greenexpression=expr, where expr is an algebraic expression defining the coloring procedure of the green color value at the current position the image. If greenexpression=$\mathrm{null}$ then no procedure is applied to the green value at the current point. The default value is $\mathrm{null}$

blueexpression : keyword option of the form blueexpression=expr, where expr is an algebraic expression defining the coloring procedure of the blue color value at the current position the image. If blueexpression=$\mathrm{null}$ then no procedure is applied to the blue value at the current point.The default value is $\mathrm{null}$

The Escape module computes the escape values of an iterative function (xvariable against yvariable) being applied iterations times at equally spaced (xvariable,yvariable) values, and constructs an image of the results.

The image is an Array with dimensions height * width * 3 and with float[8] datatype, which corresponds to a color image in the context of the ImageTools package.

Within the domain [xmin, xmax] x [ymin, ymax], (xvariable, yvariable) values will be selected at equally spaced intervals.

For each (xvariable, yvariable) value, starting at the point $X_0$=init in ${ℝ}^N$, where N is the number of variables in vars, the sequence $X_n$=$F\left(X_{n-1}\right)$, 0 $\le n\le$iterations, is generated.

The function $F:{ℝ}^n\to {ℝ}^n$, is defined by the following algorithm:

Start at the point $X_n=\left(X_{n,1}\,\mathrm{..}\,X_{n,N}\right)$, $j=1$

Apply the function $F_j$ to the current point: $y=F_j\left(X_n\right)$

Replace the jth component of $X_n$ with $y$: $X_n=\left(X_{n,1}\,\mathrm{..}\,X_{n,j-1}\,y\,\mathrm{..}\,X_{n,N}\right)$

Increment j by 1, $j=j+1$ and repeat the process with the new $X_n$. Stop when $j=N+1$, where $N$ is the number of variables in vars.

Each $F_j$ is defined by evaluating ${\mathrm{exprlist}}_j$ at the current values of the variables in vars.

If given an exprlistlist, then one of the functions in exprlistlist will be randomly selected to be applied to $X_n$, for each 0 $\le n\le$iterations-1. The probability that the ith function defined in exprlistlist is applied to $X_n$ is equal to $\frac{{\mathrm{probabilities}}_i}{\sum _{i=1}^N{\mathrm{probabilities}}_i}$.

Whenever this process yields a point $X_n$, the escapecond is examined. If escapecond is true, then the expressions $\mathrm{redexpression}$, $\mathrm{greenexpression}$, $\mathrm{blueexpression}$ are evaluated, substituting the values $X_{n,1},\mathrm{...},X_{n,N}$ for the variables ${\mathrm{vars}}_1,\mathrm{...},{\mathrm{vars}}_N$, and the current red, green, and blue values of the pixel, corresponding to the point (xvariable, yvariable), for the variables $\mathrm{redvariable}$, $\mathrm{greenvariable}$, and $\mathrm{bluevariable}$, respectively.

The red, green, and blue values of the pixel are then updated to the newly evaluated values, and the algorithm is repeated on the next (xvariable, yvariable).

If a coloring expression is $\mathrm{null}$ then the color, corresponding to the expression, of the pixel will be unchanged

Note that redexpression, greenexpression, blueexpression are applied to an image as defined by ImageTools:-Image, thus the minimum and maximum color values are 0 and 1 respectively.

Escape returns an image described above, as an Array. Further coloring procedures, defined in ImageTools, can be applied to the Array.

The defaults for options redexpression, greenexpression, and blueexpression generate an image that is suitable for use with ImageTools:-ColouringProcedures:-HueToRGB.

$\mathrm{with}\left(\mathrm{IterativeMaps}\right)\:$$\mathrm{with}\left(\mathrm{ImageTools}\right)\:$

$z≔{\left(\mathrm{zr}+\mathrm{zi}\mathrm{I}\right)}^4$

$z≔{\left(\mathrm{zr}+\mathrm{I}\mathrm{zi}\right)}^4$

$\mathrm{fzi}≔\mathrm{evalc}\left(\mathrm{\Im }\left(z\right)\right)$

$\mathrm{fzi}≔-4{\mathrm{zi}}^3\mathrm{zr}+4\mathrm{zi}{\mathrm{zr}}^3$

$\mathrm{fzr}≔\mathrm{evalc}\left(\mathrm{\Re }\left(\mathrm{subs}\left(\mathrm{zi}=\mathrm{zit}\,z\right)\right)\right)$

$\mathrm{fzr}≔{\mathrm{zit}}^4-6{\mathrm{zit}}^2{\mathrm{zr}}^2+{\mathrm{zr}}^4$

$\mathrm{mandelbroid}≔\mathrm{Escape}\left(\left[\mathrm{zi}\&comma;\mathrm{zr}\&comma;\mathrm{zit}\&comma;\mathrm{zrsqr}\&comma;\mathrm{zisqr}\right]\&comma;\left[\mathrm{fzi}+y\&comma;\mathrm{fzr}+x\&comma;\mathrm{zi}\&comma;{\mathrm{zr}}^2\&comma;{\mathrm{zi}}^2\right]\&comma;\left[y\&comma;x\&comma;y\&comma;x^2\&comma;y^2\right]\&comma;25<\mathrm{zrsqr}+\mathrm{zisqr}\&comma;-1.4\&comma;1.0\&comma;-1.2\&comma;1.2\&comma;\mathrm{iterations}=3000\&comma;\mathrm{redexpression}=i\right)\&colon;$

$\mathrm{ArrayTools}:-\mathrm{Dimensions}\left(\mathrm{mandelbroid}\right)$

$\left[1..500\&comma;1..500\&comma;1..3\right]$

$\mathrm{ColouringProcedures}:-\mathrm{HueToRGB}\left(\mathrm{mandelbroid}\right)\&colon;$$\mathrm{Embed}\left(\mathrm{mandelbroid}\right)$

$\mathrm{Hopalong}≔\mathrm{Escape}\left(\left[\mathrm{x1}\&comma;\mathrm{x2}\&comma;\mathrm{tx1}\right]\&comma;\left[\mathrm{x2}-\mathrm{signum}\left(\mathrm{x1}\right)\mathrm{sqrt}\left(\mathrm{abs}\left(1.1\mathrm{x1}-3\right)\right)\&comma;3.7-\mathrm{tx1}\&comma;\mathrm{x1}\right]\&comma;\left[x\&comma;y\&comma;x\right]\&comma;150<{\mathrm{x1}}^2+{\mathrm{x2}}^2\&comma;-20\&comma;20\&comma;-20\&comma;20\&comma;\mathrm{iterations}=5000\&comma;\mathrm{redexpression}=i\&comma;\mathrm{greenexpression}=\mathrm{sqrt}\left({\mathrm{x1}}^2+{\mathrm{x2}}^2\right)\right)\&colon;$

$\mathrm{ArrayTools}:-\mathrm{Dimensions}\left(\mathrm{Hopalong}\right)$

$\left[1..500\&comma;1..500\&comma;1..3\right]$

$\mathrm{Hopalong}≔\mathrm{Create}\left(\mathrm{Fractals}:-\mathrm{EscapeTime}:-\mathrm{Colorize}\left(500\&comma;\mathrm{Hopalong}\&comma;\mathrm{Array}\left(\left[2\&comma;3\&comma;7\right]\&comma;\mathrm{datatype}=\mathrm{integer}\left[4\right]\right)\&comma;\mathrm{Array}\left(\left[0.25\&comma;0.75\&comma;0.25\right]\&comma;\mathrm{datatype}=\mathrm{float}\left[8\right]\right)\&comma;\mathrm{rgb}=0\&comma;\mathrm{mode}=4\&comma;\mathrm{layer}=2\right)\&comma;\mathrm{fit}=\mathrm{true}\right)\&colon;$$\mathrm{Embed}\left(\mathrm{Hopalong}\right)$

The IterativeMaps:-Escape command was introduced in Maple 2015.

For more information on Maple 2015 changes, see Updates in Maple 2015.