NavList:

Brian:

I think part of the hang-up here is a focus on two-dimensional geometry drawn on flat surfaces. 

What you seem to be describing, predicting the sun's maximum altitude over the local horizon by taking measurements before or after noon, is called "ex-meridian sights" or "reduction to the meridian" in celestial navigation.  At particular latitudes and during a particular narrow band of time centered around noon, the particular piece of the celestial sphere you're focusing on is small enough that it can be treated as "flat enough" to get an accurate enough determination of the sun's maximum altitude that day just by assuming the sun's apparent motion is a parabola.  But if your latitude is too high or too low, or the sun is too far from noon, simple algebra isn't good enough.

The slice of sky where the "fancy parabola math" works best is shaped very much like the Canadian province of Saskatchewan.  Saskatchwan runs between 49° and 60° latitude and is 8° of longitude wide, which is very much like the sun's optimum altitudes and range of azimuths, respectively.  All four corners of Saskatchewan are 90° each, but the southern border of this rectangle more than 30% longer than the northern border, a trick that isn't possible for a rectangle that is flat.  As non-flat as Saskatchewan clearly is*, its shape is still flat enough for "fancy parabola math" in terms of x² to work, but outside the SK-shaped parcel of sky (too high/low/early/late) you will rapidly need x³, x⁴, and so forth, to the point where you're better off using trigonometry directly.

*(Local horizons notwithstanding)