# projection of a circle on sphere on C ( Stereographic projection)

Let $\lambda$ be a circle lying in $S$. Then there is a unique plane $P$ in $\mathbb R^3$ such that $P\cap S = \lambda$. Recall from analytic geometry that $$P = \{(x_1, x_2, x_3) : x_{1}\beta_{1} + x_{2}\beta_{2} + x_{3}\beta_{3} = l\}$$ where ($\beta_{1}, \beta_{2}, \beta_{3}$) is a vector orthogonal to $P$ and $l$ is some real number. It can be assumed that $\beta_{1}^{2} + \beta_{2}^{2} + \beta_{3}^{2} = 1$. Use this information to show that if $\lambda$ contains the point $N$ then its projection on $C$ is a straight line. Otherwise, projects onto a circle in $C$.

I am trying my hands on this question for fun, but I have no clue. Any help would be greatly appreciated.

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 Maybe I am missing something, but what does P\S equal in the second sentence? – Nicolás Kim Sep 5 '12 at 2:41 @NicolásKim corrected . thanks. – Brian Sep 5 '12 at 2:46 Please try to make the title of your question more informative. E.g., Why does \$a
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