See, also, the downloadable
Word documents, (1) Practical Path Curve Calculations by N.C. Thomas (2) Practical Bud Observation by G. Calderwood (Web version here) (3) Practical Bud Measurement also by G. Calderwood (Web version here) (3) describes manual, rather than
computerassisted, methods, so should be of use to those without a PC! 
Fitting a Path
Curve: the Math of the Method


The General Approach The bud is fitted into an invariant triangle, XYZ, then measured at N heights, 7 in this case, as depicted on the left, on a Beech bud. Vertex X is called the Top Pole, and vertex Y is called the Bottom Pole. Then, to find λ, ( t is the stepping parameter)—

Thus, λ is just the ratio of the characteristic multipliers, q and k, of the two geometric progressions. 
We obtain a list of λ
s for the number of levels at which we have chosen to measure the bud, and
take their mean value to be the value of λ
for the whole bud. If the
bud were a perfect path curve, and if we were able to measure the bud with
perfect precision, all the values in the list would be the same. In
practice, there is variation, and we can use various statistical methods
to cope with it. We may of course choose to let variable a represent the measurements of radii, left or right, or of diameters. (The terms arise from the approximatelycircular, horizontal crosssection of a bud.) There are several specific methods* of analysis. Two are available in the Bud Workshop program  [* needs references] 
(a) The Projective Method

(b) The Regression
Method
This method was devised by Graham Calderwood. It regards the measures in the invariant triangle as coordinates, equivalent to socalled "ordinary", Cartesian coordinates. Indeed, it is easily shown that Cartesian coordinates are a special, very restricted case of these coordinates, so they are in fact extraordinary!. The "layout" of the coordinate grid, or graticule, is as shown on the left. The equation of a path curve reckoned with respect to these coordinates is identical to that of a straight line reckoned with respect to ordinary, Cartesian coordinates, namely  s = λt + c Where t is the independent variable (the stepping parameter, as above), and c and λ are constants. 
In other words, a path curve is a straight line in its own frame of reference  which is, of course, for buds, eggs and vortices, the semiimaginary invariant tetrahedron. (The invariant triangle in which we set the real bud is an "axial section" through this tetrahedron.)
So the business of finding the path curve best matching a real bud reduces to finding a straight line of best fit, and the wellknown, wellworn, statistical technique called linear regression is ideally suited to the task. Hence the name given to the method. 

On the left is the graph in "bud space", on seven diameters, for the beech bud shown above. The red line is the straightline fit by the projective method. The blue line is the fit by the regression method. The bud shows a reasonable fit over its profile, except at the topmost and bottommost heights. Such mismatches often indicate that

Signage 
It will be noticed that the numbers are flagged (signed) oppositely on the two horizontal limbs or sides of the triangle (those connecting to the Z vertex at infinity). Negatives are on the right on the top limb, and on the left on the bottom limb. This is done simply for convenience, to ensure that the λ of an eggshaped profile comes out positive. If such a λ is fractional (0.0 <= λ < 1.0), the egg/bud is "sharp end down". If λ is exactly unity (1.0), the form is equally rounded at both ends  it is an ellipse or a circle. Otherwise, it is "sharp end up". 
Correspondingly, by this convention,
a vortexprofile has a negative
λ , and if in this
case λ is a fraction,
the "throat" of the vortex lies in the bottom pole. If λ
is unity, it lies in neither pole (the path curve profile is a horizontal,
straight line terminating somewhere on the XY
axis). Otherwise, it lies in the top pole. If we regard the throat of a vortex as the "sharp" end of it (it surely looks sharp!), then we may say that the sharp ends of egg and vortex with the same λ share one pole, and the "blunt" ends share the other. Notice, also, that the curves pass between just two of the vertices of the triangle, and avoid the third. Finally, notice that forms that are identical, but inverted with respect to each other, have λs that are reciprocals of each other  so to invert the λ is to invert the form, and vice versa. 