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Does the latitude of your position on earth change the appearance of the cresent like it changes the position of other objects visible in the sky? Does it rotate it?

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Yes it does. If you are at a high latitude the crescent Moon will set looking like a reversed C orientation but in the tropics it will set more like a U orientation. Though the time of the year will also have some effect due to the angle of the ecliptic to the horizon.

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