# Hoverflies

### Fixation point algorithm

For a quantitative evaluation it has been assumed that the male fixation point is situated at the axis of symmetry of the female. The fixation point is defined relative to the female body, as given in Fig. 1.

**Figure 1** : The coordinates of the fixation point.

The length of the female is the unit of length.

**Figure 2** : Geometry of rotating female.

*a.r*from the fixation point and the female scutellum is therefore at a distance

*(1-a).r*from the fixation point. The measurements of x-position are in a coördinate system defined by the position of the camera. The measurements for each frame are numbered by k. The x-position of the Male head in frame k is called

*xMale*. the x-positon of the female head in the same coördinates is

_{k}*xHead*The male head may have a small constant offset from the fixation point,

_{k }.*xMale*

_{k}-xFocus_{k}=b.*xHead _{k}-xFocus_{k}=a.r.cos(β_{k}).cos(α_{k}) (1)*

*xScutellum*

_{k}-xFocus_{k}= -(1-a).r.cos(βk).cos(αk) (2)By multiplication of (1) with

*(1-a)*and (2) with

*a*and addition of the results

*(1-a).(xHead*

Using xMale

(1-a).(xHead

a.[(xScutellum

_{k}-xFocus_{k})+a.(xScutellum_{k}-xFocus_{k})=0 for each k.Using xMale

_{k}-xFocus_{k}=b,(1-a).(xHead

_{k}-xMale_{k}+b)+a.(xScutellum_{k}-xMale_{k}+b)=0 ora.[(xScutellum

_{k}-xMale_{k})- (xHead_{k}-xMale_{k}

*)]+b= -(xHead*

_{k}-xMale_{k}).*Tracker*the coördinates are transformed to coördinates with the centre at the male head. So in these coördinates for example

*xScutellum*

_{k }is the distance in the x-direction between female scutellum and the male head in the k

^{th}frame. So at the end :

a.[xScutellum_{k} -xHead_{k}]+b= -xHead_{k} (3)

*a*has been solved by Least-squares with uncertainties in both coordinates and equal weights (Cantrell 2008). The method is equally applicable to other points of the female, in example the base of the antenna and the tip of the abdomen.

The parameter a has been solved by Least-squares with uncertainties in both coordinates and equal weights (Cantrell 2008). The method is equally applicable to other points of the female, in example base of antenna and the tip of the abdomen. In some tracks the direction of the female is constant in the first part and the last part of the track. The bold lines in Fig. 1 are drawn using the mean of the first part and the mean of the last part of the signal. In these cases an approximate value for the parameter a may be calculated by comparison of the first and the last part of the track.

**Figure 3 **: Least-Squares solution for track 090704_1504A_B10

**Cantrell C.A. **(2008) : Review of methods for linear least-squares fitting of data and application to atmospheric chemistry problems. *Atmod. Chem. Phys* **8** : 5477-5487

### Wobbling

When the female is not moving too much the hovering male is performing a rolling oscillation about the body axis combined with a horizontal oscillation perpendicular to the body axis. This movement will be called wobbling. The frequency of this oscillation is between 5 Hz and 9 Hz. Sometimes two males are wobbling above one female. See animation. The animation is 6X retarded.

### Materials and methods

The hovering flies have been filmed near Sint-Michielsgestel, the Netherlands, from April until August 2009. The camera is a Casio EX-F1. The camera has been used to film in HD with 30 frames per second and in a high speed mode with 300 frames per second.

**The fixation point**

*Start*to

*Finish*, the male remains positioned above the thorax of the female.

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