Computer-generated Harmonograph Figure
A harmonograph creates its figures using the movements of damped pendulums. The movement of a damped pendulum is described by the equation
in which represents frequency, represent phase, represent amplitude, represents damping and represents time. If that pendulum can move about two axes (in a circular or elliptical shape), due to the principle of superposition, the motion of a rod connected to the bottom of the pendulum along one axes will be described by the equation
A typical harmonograph has two pendulums that move in such a fashion, and a pen that is moved by two perpendicular rods connected to these pendulums. Therefore the path of the harmonograph figure is described by the parametric equations
An appropriate computer program can translate those equations into a graph that emulates a harmonograph. Applying the first equation a second time to each equation can emulate a moving piece of paper (see the figure below).
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