it's geometry, not mathematics but I am glad you are getting the feel for it. It is always good to have a deeper understanding of what is going on even if it seems to have no immediately revelance to what you are doing right now. The discussion has certainly helped me.
So here's where I am at with my
understanding of the relationship between tripod heads (including leveling bases), rotators, and nodal displacement slides. My understanding is based on a triad of thought experiment, modeling, and most crucially empirical testing by shooting actual photos.1)
When doing single row panoramas, best practice is to keep the nodal point or entrance pupil of the lens in the axis of rotation. This holds true where the plane of rotation is at any angle of tilt , including 0˚ ((AKA "level"). 2)
As long as the N.P. stays in that rotational axis it makes no difference if the rotator is below the tripod head or on top of it. 3)
A nodal displacement rail facilitates that alignment.4)
When shooting a multirow panoramic, ideally
a second rotator is needed to create a rotational axis perpendicular to the first rotator. This is necessary to keep the N.P. in the primary rotational axis when you change the pitch angle to shoot the additional rows.5)
How necessary it is to maintain the N.P. to the rotational axis alignment and get good high quality results is dependent on three factors:A)
Focal length of the lens as in most cases this governs the size of the subject to the size at which it is rendered on the sensor (see C.)B)
The difference in near-to-far subject distances in the subject field, relative to the distance of the camera to the subject field. C)
The degree of fine detail at the size the photograph is reproduced at (and there are a slews of perceptual factors inside that, but basically the smaller the reproduction and the greater the viewer's distance to the reproduction, the less ultra-fine detail matters.6)
Software: Contemporary stitching programs like PTGui Pro, Auto Pano Pro, and Photomerge in PsCS6 and PsCC are very good at correcting many common user errors during photography, and in some (but not all) cases can do a "good enough" job that specialized camera mounting gear may not be necessary. However it certainly will make life a lot easier. Where specialized rigs like the Really Right Stuff PG-02 package http://www.reallyrightstuff.com/s.nl/it.A/id.8872/.f
or the GigaPan EPIC http://www.bhphotovideo.com/c/product/689699-REG/Giga_Pan_EPIC_PRO_EPIC_Pro_Robotic_Camera.html
are essential is when the subject field is spatially complex and/or the reproduction size is large.
My practice when stitching is to have the stitching software output the panoramic composite as a "blended plus layers" PSD format document. Having the layers option allows me to go in and edit the contents of the photo especially important if some of the objects in the scene moved during the process of photographing the scene.