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.

Lawrence Edwards discovered these transforms when investigating the shapes of plant seed pods. He found that if a suitably positioned watery vortex is transformed it gives a very good fit. The following animation shows how such a transform may be constructed in two dimensions:

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.

He then investigated how an airy vortex is transformed and found forms displaying invagination, reminiscent of embryonic forms. He calculated the horizontal profile of the transform of a particular vortex, and as the vortex axis was rotated the form changed as shown below.

The vortex axis starts at 19^o to the vertical at an azimuth of 180^o, swinging round to 163^o azimuth and 62^o to the vertical (for the largest image).

The full three-dimensional forms containing these profiles (which were 30 percent up from the bottom) are shown below:

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.

The following image shows some other such forms where the vortex axis always contains the upper invariant point of the bud transformation (hence the symmetry).

Of course other surfaces can be transformed, and we see for example how bell forms can be obtained from quadric surfaces:

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 lambda 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.

Lawrence Edwards discovered the Pivot Transform when seeking a way to describe the gynoecium or seed pod. His idea was to use the projective transformation that produces the bud form to transform another surface. The path curves arise as the invariant curves of a linear transformation, and that very transformation is then used to transform another surface. He found that transforming a water vortex gave the form of the gynoecium (in contrast to the transformation of the airy vortex shown above). The picture below shows a rose bud and its seed pod. As it is asymmetrical the left profile of the bud was analysed, and the resulting fit is shown in red on the bud. Then the transformation corresponding to that was used to find a vortex that transformed into the gynoecium, the result being superimposed on the left side of the seed pod. The closeness of the fit is striking. What is more striking is that this process applies to many buds i.e. in every case it is a watery vortex that is transformed by the bud transformation to give the gynoecium. The vortex is coaxial with the bud, and its invariant plane lies between those of the bud transformation.

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.

The next picture shows the fit for another rose bud , illustrating that the gynoecium really does depend upon the bud shape and is not just a standard one, as the shape is more elliptic than the above one:

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.

Although the right hand profile is less precise, bearing the above remarks in mind it is nevertheless possible to find a good path curve for it, and a surprisingly good gynoecium fit:

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.

The prints shown above were obtained by placing the bud directly in an enlarger to obtain the profile, and the lines drawn on them were for hand calculation of the parameters. However the red curves were obtained by computer methods.

The mathematics of the pivot transform is described in Reference 7 and also in the document Pivot Transforms.