Choreboy wrote:

With high frequency and amplitude, a sine wave could be very steep at 0
and 180 degrees. It could also turn sharply at 90 and 270, like the
corner of a square wave. You would need low frequency and amplitude for
a sine wave to approximate the flat peaks of a square wave.

That part is simple enough for me, but I don't understand harmonics. If
you overdrive an amplifier with a sine wave, the output will resemble a
square wave. I know the output can be broken down into the input
frequency and its odd multiples. I'll have to accept it on faith.

You might want to look into the basis of Fourier analysis. It all
falls out of a very simple mathematical property of the sine wave.

If you take any periodic waveform, and multiply its value at every
point in time with the value of any frequency of sine wave at the same
points in time, over all time and add up (integrate) all the products
and divide by the total time (an infinite amount of time), only sine
waves that fit an integral number of cycles within the period of the
waveform will produce nonzero results (infinite integral divided by
infinite time). In fact, it can be shown that you get the same
quotient for harmonics if you use any integral number of periods of
the waveform, including one period. Testing an infinite number of
waves is only necessary to show that non harmonics always produce a
zero contribution. For instance, if you test a sine wave that fits
1.000001 cycles into a cycle of the waveform, you don't reach the
first zero result till you include a million periods of the waveform
(and you get more zeros at every integer multiple of a million cycles,
with a smaller and smaller cycle of results between those millions as
the number of cycles increases because you are dividing by larger and
larger times).

Harmonics (sine waves that fit an integral number of cycles within the
waveform) will produce a finite result representing that frequencies
contribution to the waveform. (Actually you have to test both the
sine and cosine against the waveform to cover all possible phase
shifted versions of the sine. Any phase shifted sine can be broken
sown into sine and cosine components. Another nice property of sine
waves.) Since only harmonics contribute to the total wave shape, you
can skip all the other frequencies, and just evaluate the part each
harmonic contributes to making the total waveform.

That is Fourier analysis.

The rest is about making the math more efficient.