Do you remember playing with a Spirograph when you were a kid? Did you ever go to one of those rock-music laser shows at the planetarium? (I think they went out of style in the mid 80's.) Those complicated-looking patterns can be described mathematically as parametric equations where x and y are sinusoidal functions of a parameter t.
Here's an example of the simplest spirograph, a circle:
x(t) = Axcos(2πt + φx)
y(t) = Aysin(2πt)
where
Amplitude: Ax = Ay = 40 pixels
Phase: φx = 0
Press play to vary the phase.
Now x(t) and y(t) have different frequencies:
x(t) = Axcos(2πfxt + φx)
y(t) = Aysin(2πfyt)
fx = 1/2 and fy = 1
Now 3x the period:
fx = 1/3
fy = 1
Now 3 against 4:
fx = 1/4
fy = 1/3
x(t) and y(t) are the sum of 2 sinusoids. Play varies A0 and A1
x(t) = A0cos(2πf0t) +
A1cos(2πf1t)
y(t) = A0sin(2πf0t) +
A1sin(2πf1t)
f0 = 1, f1 = 1/8
Same as above, but play varies the phase.
x(t) = A0cos(2πf0t+φx0) +
A1cos(2πf1t+φx1)
y(t) = A0sin(2πf0t) +
A1sin(2πf1t)
Now x(t) and y(t) are the product of 2 sinusoids. Play varies A0 and A1
x(t) = A0cos(2πt) * (A1cos(2πf1t)+C)
y(t) = A0sin(2πt) * (A1sin(2πf1t)+C)
f1 = 1/12, A1 + C = 1
Same as above but vary the phase.
I'll do more with these things later.