# Curvature of the trajectories by the example of *Navicula*

On this page we will discuss how the curvature of a path changes when the movement reverses. It is not about the question of the absolute value of curvature, but the question of the direction of curvature, whether it is positive or negative. The curvature of the path of a diatom is positive when it bends to the left for someone who is moving with it. With respect to the center of curvature, the diatom then moves counterclockwise. However, if the curve bends to the right, the curvature is negative. Correspondingly, the diatom moves clockwise.

An important prerequisite of the following considerations is therefore that the trajectories exhibit clearly recognizable curved and smooth path segments. Diatoms like *Cymatopleura elliptica* do not meet this requirement (see video on the page about Creating cultures and care). In diatoms that cover only short distances or frequently collide with other diatoms, the curvature is often not observable. In addition, only paths where the raphe is in contact with the substrate should be considered. Movements such as at *Pinnularia* in girdle band view (see video on page Description of the trajectories) are excluded. Furthermore, only path sections with "uncomplicated" changes of the direction of the movement are considered. In particular, the diatoms should not rotate around the apical axis and should not straighten up. Many motile diatoms meet these requirements over a sufficiently long observation period.

In the following, diatoms will be considered whose valves are symmetrical with respect to the transapical plane. This includes many genera such as *Navicula* or *Rhopalodia*. Sigmoid forms such as *Gyrosigma* are not treated here.

The drawing on the left (click to enlarge) shows the movement of a diatom where the valve is symmetrical with respect to the transapical plane. The partition into two raphe systems as shown in the figure and the changing position of the point P (see Analysis I) plays no significant role here. In addition, it is unimportant whether P is located on the leading or trailing apex side. However it is essential that the location of P changes only between points whose centers of curvature lie on the same side of the diatom. The curvature of the raphe is deliberately exaggerated in comparison with the curvature of the trajectory.

As already explained, it is assumed that the curvature of the path follows the curvature of the raphe. Consequently, the diatom moves in the first section of the path counterclockwise (positive curvature). From the viewpoint of a co-moving observer, the center of curvature is located to the left of the trajectory. Eventually, a reversal of direction occurs. At the same time the diatom usually rotates horizontally by a certain angle. In the second segment of the path, the diatom moves clockwise according to the curvature of the raphe (negative curvature). The center of curvature is now to the right of the trajectory from the viewpoint of the co-moving observer. After the next change of direction, the movement is counter-clockwise again. The rule is that the curvature changes its sign on every reversal. Clockwise and counter-clockwise rotations alternate:

.... → + (counterclockwise) → - (clockwise) → + (counterclockwise) →....

If there were no horizontal pivoting movements at the reversal points, and if the radius of curvature remained constant, the diatoms simply behaved as a car, which changes the direction of travel when the steering wheel is fixed.

It turns out that these ideas often correspond to the observation. An example of accordance with the rule in case of *Navicula* is shown in the picture on the left (click to enlarge).

In four studied *Navicua* spp. (length range: 30-50 µm) and some species with similar regular movements the rule has proved its worth. These observations support the thesis that the curvature of the raphe in P determines the curvature of the path.