In the celestial sphere below, what is the $\omega$ and $\tau$ physically? I mean, In the normal world, how do we see them? for example, we can see the altitude of the sun (a) by examining the shadow of a pillar and comparing it with its height, How do we see $\omega$ and $\tau$ in real world?

S is the sun and W is the west.

Also I want to know how do they effect on the shadow of wall which is built in direction of "$SCP$ to $NCP$".

enter image description here

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    $\begingroup$ What is the source of this image? To remind you, homework (or similar) problems must be acknowledged as such. $\endgroup$
    – James K
    Apr 13, 2017 at 21:00
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    $\begingroup$ @JamesK It is from the answer of National Olympiad of an Asian country. about shadow of a wall which is built in direction of SCP to NCP in arbitary point on the north of the equator. $\endgroup$
    – titansarus
    Apr 14, 2017 at 5:45

1 Answer 1


There seems to be very little to go on here.

"Zenith" and "Horizon" are easy to understand, it suggests that this is a projection of the sky (and ground) In that context "W" would appear to be "West", and you give that "S" is "sun". The horizon is curved. If you project the entire hemisphere the horizon would go straight across this isn't a complete hemispheric projection.

If we are looking West, then North would be to the right, and South to the left. The point P seems to be identified with NCP. The first guess would be that N of NCP means North, but this contradicts the reasoning that North is on the right. You mention SCP in the text, but it isn't in the diagram. The sun appears to be West, or just North of West, which is not an unusual but not impossible position.

The lines SW, WP and PS are curved. This may be to indicate that they are great circles on the celestial sphere. If so then $\omega$ is the spherical angle SWP and $\tau$ is the great circle distance SW, perhaps given as an angle with vertex at the centre of the sphere. Given three points on a sphere (in alt-az coordinates), it is a small exercise in spherical geometry to find the spherical distance and angles between them.

I'm not sure why you are being coy about the source of this. If you could link to the question from which this is taken the meanings of $\delta$ and H could be clearer. I guess the solution is an exercise in spherical angles, and projections from a circle to the plane


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