If any plane is placed in a path curve transformation then it is being moved by that transformation. There
is generally one point in it which is momentarily stationary, that is, the plane is pivoting about that point.
It is known as the pivot point of the plane. If we place a surface in the transformation then every one of its tangent
planes has such a pivot point, and they form another surface known as the pivot transform of the first one.
They are described by Lawrence Edwards (Reference 7). The author has written a
brief summary assuming college level mathematics. The above animation shows how
the transform of a vortex varies as its lambda value is varied from -0.9 to -0.1, other parameters
being held constant.
The initial picture shows part of the vortex, the lower invariant plane of the bud transformation as a horizontal line,
and two centres of projection and an auxiliary line determined by the bud lambda and epsilon.
The final profile is shown by black dots where corresponding blue tangents and red lines meet. The blue lines represent tangent
planes orthogonal to the picture, and the red ones horizontal planes.
The vortex axis starts at 19o to the vertical at an azimuth of 180o, swinging round to 163o azimuth
and 62o to the vertical (for the largest image).
These images were obtained by calculating the angle of the tangent plane at each visible point, and setting the brightness according
to its orientation to the direction of illumination. This required a sophisticated bisection algorithm which could not always find
the required root of the equation, which is why there are blemishes.
Of course other surfaces can be transformed, and we see for example how bell forms can be obtained from quadric surfaces:
PIVOT TRANSFORMS AS GYNOECIUM FORMS
On the Path Curves page the application of those curves is briefly described. Below are two examples of actual results:
This shows a Kerria Japonica bud with the theoretical curve superimposed in red, which can be seen to be an excellent fit.
It is accomplished by selecting the axis by eye together with several points on the profile, and then finding the best mean
to fit them. The mathematical curve approaches the top and bottom points (marked by crosses) asymptotically, and we cannot expect
an actual physical form to accomplish that! So the top and bottom points are varied on the axis to minimise the deviation.
The top point is in this case above the physical bud, but more usually for other buds there is a physical prominence above the
mathematical top. The percentage deviation of the lambda value is 1.2%, a very good result as that is a more sensitive indicator
than the mean radius deviation. An added interest in this case was that only the right profile was analysed, yet the resulting fit
is also excellent for the left profile. Many buds, like the rose below, are asymmetrical and with a prominence at the top.
Clearly there is some kind of trade-off between the ideal form represented by the mathematical curves and the physical necessities of
actually producing it, together with the required structural integrity which requires a stalk, and a portion between the gynoecium and
the bud where the sepals were attached, and so on. The attempt to fit a gynoecium form is very sensitive to the relation between the
bud lambda and the actual gynoecium size, and will fail if the lambda is not determined accurately i.e. we do not just get a bad fit,
we get none at all as the mathematics fails with imaginary values where we require real ones.
In this case there is a large prominence at the top which evidently is not part of the bud, and any attempt to include it with a bad fit
fails to find any gynoecium form at all. It opens up the possibility for such buds of finding a criterion for judging what belongs to
the ideal form in physical reality, and reinforces the judgement made by eye, which is easy in this case.
Such results can only excite wonder at the processes occurring in Nature, and how much we have to learn about their holistic aspects
which can be investigated with this approach.