Lissajous curves (Bowditch curves)
The Lissajous curves, also known as Bodwitch curves, have applications in Physics, Astronomy and other sciences. The Lissajous curves are the family of curves described by the following parametric equations:
(1)
(2)
Time t is a parameter. These are equations describing simultaneous motions of an object (or mathematical point) along the x and y axes.
x(t) and y(t) are positions of moving point in the coordinate system.
Ax and Ay are amplitudes of these motions along x and y axes.
Amplitude: the maximum displacement in periodic type of motion.
ωx and ωy are angular frequencies in motion along x and y axes.
Φx and Φy are the initial phases of motions along the x and y axes (in radians).
Initial phase: angle describing position of moving point at t = 0 (beginning of motion).
Initial phase and phase are terms very often used in Physics.
Let us explain in more detail Eqs.1 and 2. First we assume that
Ax = Ay = R
ωx = ωy =ω
Φx = 0
Φy = π/2
Substituting these values to Eqs.1 and 2 we get
x(t) = Rcosωt (3)
y(t) = Rcos(ωt + π/2) = Rsin(ωt) (4)
Eqs.3 and 4 describe circular motion (see point “Circular motion, sin, cos”).
The circle is therefore one of the Lissajous curves.
Now let’s substitute to Eqs. 1 and 2
ωx = ωy =1
Φx = Φy = 0
We will get equations
x(t) = Ax cos t (5)
y(t) = Ay cos t (6)
Eqs 5 and 6 define the Lissajous curve shown in Fig.1.
It is a line segment in the coordinate system connecting points (-Ax, -Ay) and (Ax, Ay).
It is easy to realize that there is indefinite number of Lissajous curves, as indefinite is the number of sets of values of Ax, Ay, ωx, ωy, Φx and Φy. Their applications in some sections of Physics will be presented later.
An ellipse is also one of the Lissajous curves and is obtained with parametric equations
x(t) = Ax cosωt (7)
y(t) = Ay cos(ωt + π/2) (8)
when Ax not equal Ay
Lissajous curves are equally important in Physics and Mathematics. You can find information more about them at MathWorld.