Predefined functions

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)