Perhaps part of the problem is that the actual design
process is NOT about calculating the spacing and element
lengths. Rather, one CHOOSES a set of spacings and
element lengths/diameters, then models them to see whether
they give the desired pattern, gain, f/b ratio, impedance,
etc. Then you tweak one or two values and try again.
Eventually you get a feeling for which adjustments make
the biggest difference in the design, and (hopefully)
find an antenna that meets your requirements.
If you want to understand the modelling process, I'd
suggest starting with W4RNL's website - he has a whole
section on modeling, including pointers to many sources
of programs (and source code in several languages).
Another good resource is W2PV's book, "Yagi Antenna
Design". Dr. Lawson was one of the first to apply
computers to yagi modelling, and his book gives a lot
of background of how he did it. (His models don't give
exactly the same results as the current NEC-based models,
but it should give you a good idea what is involved.)
This also includes methods for calculating tapered
Basically, he divided each element up into a number of
segments. He assumed a 1 amp current in the feedpoint
segment, and, using the mutual coupling between every
pair of segments (which depends on spacing, distance,
and orientation) he solved a big hairy matrix to find
the current flowing in each segment. Given that, he
could then calculate the total field at any direction
from the antenna (far field) as the vector sum of the
currents in all the segments, with the relative distances
to each (and hence phase shifts) calculated relative
to the desired point in space.
On a more simple note, I have a spreadsheet that will
scale antennas for different frequencies and element
diameters, which I wrote based on W2PV's equations.
Basically, to scale an antenna the element spacings are
simply scaled by the frequency ratio. For the element
lengths, I use his formulas to calculate the actual
reactance of each element at the original frequency,
then work backwards and find what length of the new
diameter element would give the same reactance at the
new frequency. Rather than designing my own antennas
(since I haven't found a good antenna modelling program
yet that runs on my Macintosh) I generally find a W4RNL
design or one from another source that appears to meet
my needs and scale it for the desired frequency/element
diameter. This spreadsheet isn't perfect: it doesn't
necessarily take into account capacitive coupling between
elements, for example, or metal boom effects, but I've
had pretty good luck with it so far.
The W9CF applet that I pointed to earlier includes a
brief discussion of the computational algorithm it uses.
Beyond this, you can search for information on the
Method of Moments calculations - that is the computation
core at the root of most modern modelling programs.