## HopAlong

- Orbit Fractal

This plots random Hopalong functions. Each is mapped from the equation:

Xn+1 = Yn - Sign Xn * Sqrt Abs (B*Xn-C)
Yn+1 = A - Xn

where the new values of X and Y are determined from the previous X and Y values along with 3 constants A, B and C.

The Optimise procedure is not stricly necessary. It determines the maximum and minimum X and Y values and from this calculates the scale factor and offset values which allow the plot to be displayed so that it fills the drawing area.

To New
# set default screen, pen and turtle values
ResetAll SetScreenSize [400 400] HideTurtle
SetSC Black SetPC Green SetPS 1 PenUp
End

To Init
Make "A 0.1*((Random 199)-99)
Make "B 0.1*((Random 199)-99)
Make "C (Random 19)-9
Make "Cycle 6000
End

To Display :A :B :C :Cycle
SetPC White
SetPos [-190 184] Label "Hopalong
SetXY Minus 194 Minus 196 Label (List :A :B :C)
SetX 194-LabelLength :Cycle Label :Cycle
End

To Fit2Screen :A :B :C :Cycle
Make "X 0 Make "Y 0
Make "Xmax 0 Make "Xmin 0
Make "Ymax 0 Make "Ymin 0
Repeat :Cycle [HopAlong :A :B :C
If :X > :Xmax [Make "Xmax :X]
If :X < :Xmin [Make "Xmin :X]
If :Y > :Ymax [Make "Ymax :Y]
If :Y < :Ymin [Make "Ymin :Y] ]
Make "Xsize :Xmax + Abs :Xmin
Make "Scale 380/:XSize
Make "Xcent Minus (:Xmax + :Xmin) / 2
Make "Ycent Minus (:Ymax + :Ymin) / 2
End

To Hop :A :B :C :Cycle
Make "X 0 Make "Y 0
Repeat 180 [
SetPC Hue2 RepCount
Repeat Integer :Cycle/180 [HopAlong :A :B :C
Dot List :Scale*(:X+:XCent) :Scale*(:Y+:YCent) ] ]
End

To HopAlong :A :B :C
Make "Xnew :Y-((Sign :X)*Sqrt (Abs (:B*:X-:C)))
# Make "Xnew :Y-((Sign :X)*Abs((Sine :X)*(Cos :B)+:C-:X*Sine (:A+:B+:C)))     # 3-ply fractal
Make "Y :A-:X
Make "X :Xnew
End

To Sign :Num
# return sign (-1 or 1) of number
If :Num < 0 [Output Minus 1] [Output 1]
End

To Hue2 :Theta
# Output RGB hue list from angle :Theta
Make "Red Abs 255*Sin :Theta
Make "Green Abs 255*Sin (:Theta+120)
Make "Blue Abs 255*Sin (:Theta+240)
Output (List :Red :Green :Blue)
End

To Go
New Repeat 100 [
Init Fit2Screen :A :B :C :Cycle
Wash
Hop :A :B :C :Cycle
Display :A :B :C :Cycle Wait 300]
End

Comment in the second function to draw a series of 'Three Ply' orbit fractals. Best if A, B and C are all set to Random 100.

To draw just one curve use the following procedure:

To GoX :A :B :C :Cycle
New
Fit2Screen :A :B :C :Cycle
Hop :A :B :C :Cycle
Display :A :B :C :Cycle
End

Hopalong

Procedures blue
Variables pink