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**manjohn12** **Proposition**: Let $\displaystyle S $ be a circle on $\displaystyle \displaystyle \Sigma $ and let $\displaystyle T $ be its projection on $\displaystyle \mathbb{C} $. Then (a) if $\displaystyle S $ contains $\displaystyle (0,0,1) $, $\displaystyle T $ is a line; (b) if $\displaystyle S $ does not contain $\displaystyle (0,0,1) $, $\displaystyle T $ is a circle.

If $\displaystyle S $ contains $\displaystyle (0,0,1) $ then the circle in not entirely contained on the sphere. Is this an intuitive reason why the projection is a line? Whereas a circle that does not contain $\displaystyle (0,0,1) $ is a circle completely contained on the sphere. Thus its projection is a circle.