The whole combination of gene trees and species trees depends on deciding which is on the right and which on the left.

On
the Panbiogeographic track as Bezier Curve.

The Bezier curve is an interpolation curve methodology that
seems *prima facie* to capture significant aspects of the Panbiogeo

graphic track. Patterson raised an issue with Croizat’s
track by questioning whether the notion itself could be discriminated from
being instead of the eminent graphs they were subsequently argued to be (Page 1987) are rather simple statistical artifacts of drawing
connecting lines amongst large numbers of widely placed points. Patternson’s idea is that simple by cherry
picking points amongst a large set the appearance of order in the data can
manufactured by human error rather than by distillation of significant average
normalized behaviors.

The Bezier curve has the basic Croizat property to connect
two different points (having the starting and ending points) but it also has
the aspect of weighting the curve inside the convex hull of all control points
near to each point on average””.

The trick will be to pick the order of the points relative
to the vicariance. The construction of the Bezier curve with Bernstein
polynomials permits a definition of vicariance along the simple graph
representation of a track and nodes (line connecting different collection
localities). The vicariance is obtained at the midpoints.

The use of this curve results in the division of collection
localities as those that are “control points” and those that are distribution
points due to mobilism.

If vicariance in
immobilism is presented by the relation of the polynomials to the track (any
fracture of the track is equivalent despite the ordering of the points). This relation is developed as one that
connects the track width, node shape, mass density and baseline volume to the
remaining non-control points and can further be used to link the baselined track
graph to a phylogenetic trees proposed across the biogeography.

By altering the order (and hence affecting stability) one
can have different edges responsible for different polynomial order
vicariance. Thus different kinds of
vicariance can be explored for a set of points and compared to other sister groups.
Alternatively the use of specific track widths, node shapes, mass densities and
baseline models could suggest which amongst various possible are sisters. The scalablity and translation and rotation of
the curves enable one to search for sisters with complex embedded vicariance
distributed across lineages.