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.