## Harmonograph

- XLogo

A harmonograph is an instrument for drawing complex curves. They can often be found in good toy shops.

There are several models available, but generally they consist of a pen resting lightly on a flat plate. Both the pen and the plate move independently and are connected to heavy pendulums. Setting both pendulums swinging results in a complex pattern being drawn.

The pattern becomes smaller as friction reduces the swing of the pendulums, (the pendulums are damped).

This program mimics the harmonograph by two equations, one for horizontal motion and one for vertical. They are similar to the Spirograph equations, except that they incorporate a damping factor, which slowly reduces the size of the curve.

X = ke-rt [ Sin (f1t + p1) + Sin (f2t + p2) ]
Y = ke-rt [ Sin (f3t + p3) + Sin (f4t + p4) ]

where

k = 90 just a constant to fill the drawing screen (c is the speed of light)
e = 2.718, a famous constant
r = 0.00015 another constant, the 'damping' factor
f1,f2,f3,f4 = random pendulum frequencies, 1,2,3...9
p1,p2,p3,p4 = random phase angles, 0...359
T = time steps, 0,1,2,3,...5000

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

To Spir
SetPos XYCurve 0 PenDown
For [T 0 5000] [SetPC Hue2 :T SetPos XYCurve :T]
End

To XYCurve :T
Make "KK :K * Power :e ( Minus 0.00015 * :T )
Make "X :KK*((Cos ((:f1*:T) + :p1 )) + (Cos ((:f2*:T) + :p2 )))
Make "Y :KK*((Cos ((:f3*:T) + :p3 )) + (Cos ((:f4*:T) + :p4 )))
Output (List :X :Y)
End

To Init
Make "K 90
Make "e 2.718
Make "f1 Integer 1+ Random 9
Make "f2 Integer 1+ Random 9
Make "f3 Integer 1+ Random 9
Make "f4 Integer 1+ Random 9
Make "p1 Integer Random 360
Make "p2 Integer Random 360
Make "p3 Integer Random 360
Make "p4 Integer Random 360
End

To Display
# write header title and footer values
SetPC White
SetPos [-190 184] Label "Harmonograph
SetPos [-190 -190] Label ( List "Freqs: :f1 :f2 :f3 :f4 )
SetPos [34 -190] Label ( List "Phase: :p1 :p2 :p3 :p4 )
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
Repeat 12 [New Init Display Spir Wait 200]
End

Harmonograph

Procedures blue
Variables pink