#### 2.2. Analysis

We inspected all available optical images in the HASH DB for each PN. We assume the best optical images are a reasonable representation of the true PN “projected” shape. Whenever there was a Hubble image available we used that, otherwise we used the highest quality optical image resources in terms of both sensitivity and resolution that would give us the highest level of confidence. This was usually the H

α/Sloan

r’ quotient image (from the SHS or IPHAS surveys; Parker et al. [

12], Drew et al. [

13]) or an image from PanSTARRS (Chambers [

14]). For the morphological classification we adopted the classification from the HASH PN database which essentially follows Corradi and Schwarz [

15]. The PN orientation axis was taken to be that of the polar or major axis of the PN. Each PN examined was assigned a subjective confidence flag (1 for very confident, 2 for quite confident, 3 for not very confident) for the measurement of their Equatorial Position Angle (EPA).

The angle measurement is from North towards East, see

Figure 1. Out of the 766 PNe where we could with some confidence measure the EPA, we classified 144 elliptical and 99 bipolar PNe with the confidence flag 1, 194 elliptical and 129 bipolar PNe with the confidence flag 2, and 157 elliptical and 43 bipolar PNe with the confidence flag 3. Note that for PNe with more than one main EPA (e.g., for some multipolar examples) we included all EPAs. 55 PNe had more than one EPA recorded, giving a total of 824 measured EPAs.

For this study projection effects were not taken into account as we do not have information to disentangle the true 3-D aspect from that seen along the line of sight. The fact that the data is two dimensional has two effects: First, it raises the possibility that some bipolar PNe were observed approximately along the lobes and consequently misclassified. Secondly, if we consider the angles of the PNe taken from the Galactic North Pole, through the lobe-to-lobe axis of PN towards the Galactic Plane the true angles will be more concentrated towards 90° (Rees and Zijlstra [

5]).

The EPA was then converted to the Galactic Position Angle (GPA, measured from the direction of the Galactic North towards the East) utilising the formula adopted by Corradi et al. [

7] using standard relations for spherical triangles for epoch J2000. While Corradi et al. [

7] measured the EPAs in the interval [0°,90°) assuming a priori a symmetric problem in the sense that a preferred orientation of the PNe elongations very close to the Galactic equator will be easier to detect, Weidmann and Diaz [

4] demonstrated that this would blur any other preferred orientation. We therefore adopted the more widely used interval [0°,180°) for the EPAs and GPAs. The distribution on the sky of the PNe used for this study, colour-coded with their measured GPAs, is shown in

Figure 2.

Because a measured value of 178 degrees is closer to 0 than it is to 175 degrees, circular statistics must be applied. For the circular statistics analysis we doubled the GPA to get a full circle. The Rayleigh “randomness” Test (Wilkie [

16]) was applied to the GPA distribution to test the circular data. A

p value of 1.0 means the distribution is uniform, while small numbers indicate not uniformity. A

p value of 0.1 implies that there is a 90% probability that the distribution is non-uniform.

For the visualization of the GPA distributions (

Figure 3 and

Figure 4) we employed general histograms as well as “uniformity”, “rose”, and “vector” plots (Fisher [

17]). In the uniformity plots deviations from the diagonal indicate a non-uniform distribution. For the rose plots the radius of each finger was determined so that each PN contributes the same area. Only the left side has actually been measured and then rotated by 180 degrees for better visibility. Note that the vector plots show the doubled GPAs so the angle of the resulting radius vector

$\overrightarrow{R}$ (red line) needs to be divided by two to get the mean direction of the GPA distribution. For a given set of

n unity vectors with angles

${\alpha}_{i},\phantom{\rule{3.33333pt}{0ex}}{0}^{\circ}\le {\alpha}_{i}<{180}^{\circ},\phantom{\rule{3.33333pt}{0ex}}i=[1,\cdots ,n]$, the resulting vector

$\overrightarrow{R}$ can be calculated as

Thereby the half angle of

$\overrightarrow{R}$,

can be defined as the mean angle of the angular distribution and

as the deviation from the mean angle (See Jammalamadaka and Sengupta [

18] for the [0°360°) problem). We have repeated the analysis for individual directions, for different upper limits for the confidence flags, as well as for elliptical and bipolar PNe separately.