62 METEOROLOGICAL OPTICS 
The effect of the amount of cloudiness is demon- 
strated in Table II by the most recent observations 
[37]. In qualitative agreement with observations by Rei- 
mann [c. 42], these data show that even a few clouds in 
the sky materially increase its apparent flatness, whereas 
with sky covers of > 340, the observer refers chiefly 
TasieE II. AverAGcE Hatr-Arc ANGLE FoR VARIOUS STEPS 
OF CLOUDINESS 
Clouds tenths 0 <1 1-3 4-7 8-9 >9 
a (°) 34.0) 32.6] 31.5] 30.6] 30.2) 29.9 
to the cloud layer in forming his impression of the sky 
shape. Therefore, for cloudiness > 440, the increase 
in flatness becomes practically negligible. 
Miller [87, 38] seems to be the only observer to investi- 
gate the effect of cloud types (and ceiling height) on the 
apparent sky shape. Table III shows his results for 
various cloud types arranged in ‘order of increasing 
average cloudiness for these types. The groups Ac and 
As, and Sc make the sky appear flatter than would 
result from the implicit cloudiness effect. This has been 
attributed to the structure of the underside of Ac and 
Sc, which facilitates depth perception and makes the 
observer aware of the extension of these cloud layers 
beyond the terrestrial horizon. This apparent increase 
of the horizontal extent of the cloud layers causes a 
decrease in a. 
Tasie III]. RELATIONSHIP BETWEEN CLoup TYPE AND 
Hatr-Arc ANGLE 
Cloud type Average cloudiness, tenths Average a(°) 
Cu, Fe 4 30.0 
Ci, Cs 5 29.1 
Ac, As 7 27.8 
Se 8 28.4 
St 10 29.0 
From Table III it is also obvious that the cloud 
height does not influence the apparent flatness of the 
sky, mm the manner which might be expected from 
Fig. 2.—Geometric relation between cloud height and half-are 
angle. 
purely geometric considerations (Fig. 2). The Ac- and 
As-group, for example, is associated with a smaller a 
2. Ac and As occurred simultaneously in almost all cases. 
than is the St-group, whereas the reverse should be true 
according to Fig. 2 if the visual space were a simple 
transformation of the physical space. A comparison 
between Table III and the last three columns of Table 
II shows that the type of clouds has a greater effect than 
the amount of clouds. 
In contrast to Dember and Uibe [9], Miller [37] found 
that the visual range has only a minor influence on the 
apparent shape of the sky, whereas the effect of the 
distance to the terrestrial horizon is very strong, as 
shown in Table IV. The differences in a between various 
Taste IV. Averace Hrrrects or VisuAL Rance (V) AnD 
Horizon Distance (D) on toe Hatr-Arc ANGLE (a) 
V (km) D=0.4 km a(°) D=12km a(°) 
<6 33.3 30.7 
6-10 32.1 28.8 
>10 32.6 28.9 
Mean 32.6 29.2 
groups of visual range are considerably smaller than 
between different horizon distances. This result was 
confirmed by measurements of @ at various distances 
from a mountain range [88]. In addition to the actual 
distance of the horizon, the facilities for subconscious 
estimation of this distance, such as 1s offered by suitable 
foreground configuration, have also proven of strong 
influence on the perceived sky shape [23, 37, 38]. For 
this reason, the sky dome cannot generally be considered 
a rotational geometric figure. 
Related Phenomena. Closely related to the shape of 
the sky is the well-known enlargement of sun or moon 
when near the horizon [23, 42]. As can be seen from the 
shaded angles in Fig. 3, the same angular subtense in- 
tersects a greater portion of the flattened sky near the 
(5° 
= BSS STSTK‘E | 
Fie. 3.—Relationship between true and apparent (slanted 
numbers) elevation angles and angles of subtense (bracketed 
numbers). 
horizon than at higher elevations, and this portion be- 
comes greater as the sky becomes flatter. The same 
holds true for the angular distance between stars. Jef- 
freys [25] raised an objection to this projection theory 
on the grounds that the sun or moon should appear 
elliptical with a long vertical axis, because only this 
axis would be distorted by the shape of the sky. How- 
ever, the angular subtense is too small to permit de- 
tection of such a deformation; moreover, this effect is 
probably compensated by an opposite effect of the 
