Gerneral rule for Lissajou traces
b frequency of the oscillation in z direction cos(b*t)
c frequency in xy-plane
x contains term cos(ct), y contains sin(ct)
The character of the envelope is determined by how the term of z enters the formulas for x and y
a generally defines the scaling of axes
Default value of a, b, and c is 0.5. This creates most simple Lissajou traces that often do not indicate the envelope. They become evident when parameters b and c are varied such that their quotient is a non integer rational number.
Approximately: Small values of b create traces spiraling perpendicular to the z axis, small
values of c create traces spiraling in planes that include the z axis. Try with the Torus!
Speed of the object is determined by p via the time interval of calculation, and via ct and bt as arguments in the coordinate functions. Very high speed may be associated with low resolution; you can reduce it by slider p.
Linear−linear (arithmetic) spiral
x = a*t/20*cos(t)
y = b*t/20*sin(t)
z = -1+c*t/20
Exponential−linear spiral
x = a*t/20*cos(t)
y = b*t/20*sin(t)
z = -2+exp(c*t/20)
Exponential−exponential spiral
x = a*(exp(t/50)-1)*cos(t)");
y = b*(exp(t/50)-1)*sin(t)
z = -2+exp(c*t/20)
Flat Lissajou trace
x = a*cos(t)
y = a*sin(c*t)
z = 0
Non closed, flat Lissajou trace
x = a*cos(t)
y = b*sin((c+sqrt(2))*t)
z = 0
Lissajou on cylinder
x = a*cos(t)
y = c*sin(t)
z = sin(b*t)
Lissajou on cone
x =a*(1-cos(b*t))*cos(c*t)
y = a*(1-cos(b*t))*sin(c*t)
z = cos(b*t)
Lissajou on double cone
x = a*cos(b*t)*cos(c*t)");
y = a*cos(b*t)*sin(c*t)
z = cos(b*t)
Lissajou on hollow object
x = a*(1+(cos(b*t))^2)*cos(c*t)
y = a*(1+(cos(b*t))^2)*sin(c*t)");
z = cos(b*t)
Lissajou on spindle
x = a*(1-(cos(b*t))^2)*cos(c*t)
y = a*(1-(cos(b*t))^2)*sin(c*t)
z = cos(b*t)
Lissajou on sphere
x = a*cos(b*t)*cos(c*t)
y = a*cos(b*t)*sin(c*t)
z = a*sin(b*t)
Lissajou on torus (r = 0,25; R = a)
x = (a+0.25*cos(b*t))*cos(c*t)
y = (a+0.25*cos(b*t))*sin(c*t)
z = 0.25*sin(b*t)