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I'm sure that I'm missing something simple here. Please can you help?

By stereographically projecting from the 'north pole' $(0,0,1)$ of the unit sphere $x^2+y^2+z^2=1$ onto the the plane $z=0$ we get a chart $\alpha_1: S^2\backslash (0,0,1) \to \mathbb R^2$ given by

$$\alpha_1(x,y,z)=\left(\frac{x}{1-z},\frac{y}{1-z}\right)$$

By stereographically projecting from the 'south pole' $(0,0,-1)$ of the unit sphere $x^2+y^2+z^2=1$ onto the the plane $z=0$ we get a chart $\alpha_2: S^2\backslash (0,0,-1) \to \mathbb R^2$ given by

$$\alpha_2(x,y,z) = \left(\frac{x}{1+z},\frac{y}{1+z}\right)$$

I'm reading something about charts $S^2 \to \mathbb C$.

I thought that $S^2 \subset \mathbb C \times \mathbb R$ could be identified with $S^2 \subset \mathbb R^3$ by $\{(a+\mathrm ib,c) \in \mathbb C \times \mathbb R : a^2+b^2+c^2 = 1\}$.

If so then the charts $S^2 \to \mathbb C$ would be given by

$$\psi_1(a,b,c) = \left(\frac{a}{1-c}\right) + \mathrm i \left(\frac{b}{1-c}\right)$$ $$\psi_2(a,b,c) = \left(\frac{a}{1+c}\right) + \mathrm i \left(\frac{b}{1+c}\right)$$

However, the text I'm reading says that the charts have

$$\psi_2(a,b,c) = \left(\frac{a}{1+c}\right) {\bf -} \, \mathrm i \left(\frac{b}{1+c}\right)$$

Any ideas what I'm missing? Thanks in advance.

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    There is nothing about maps $S^{2}\to\mathbb C$ in that article, and no map $S^2\to\mathbb C$ would be called a chart. – 2017-02-25
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    @ThomasAndrews I think you understand the meaning of my question. If you can answer it, then please leave a reply; I would be very grateful. – 2017-02-27
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    You might think I understand, but I don't. You haven't explained what you are talking about, and nobody calls a map $S^2\to$ anything a "chart." The article you linked to certainly does not. What are you asking? – 2017-02-27
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    @ThomasAndrews If we delete a point, as the body of my question and the document I linked to suggest, then $S^2 \backslash (0,0,1) \to \mathbb C$ is a chart. I missed $\backslash (0,0,1)$ from the question title, that's all. I make that clear in my definitions of $\alpha_1$ and $\alpha_2$. They are both well-defined charts from parts of $S^2$ to $\mathbb C$. See https://en.wikipedia.org/wiki/Manifold#Charts – 2017-02-27
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    Quote from the body: "I'm reading something about charts $S^2\to \mathbb C$." Literally, right where you link to the article. – 2017-02-27
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    Before you saw $S^2 \to \mathbb C$ you read: By stereographically projecting from the 'north pole' $(0,0,1)$ of the unit sphere $x^2+y^2+z^2=1$ onto the the plane $z=0$ we get a chart $\alpha_1: S^2\backslash (0,0,1) \to \mathbb R^2$ given by $$\alpha_1(x,y,z)=\left(\frac{x}{1-z},\frac{y}{1-z}\right)$$ You understand my intention. – 2017-02-27
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    Aw, now you are getting defensive. Perhaps you might consider that you haven't gotten help on this question because it was a mess. Help people help you. I made a specific comment at the beginning, and then you seemed to ignore it. – 2017-02-27
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    @ThomasAndrews If you say so... Sorry you couldn't help. – 2017-02-27
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    Sigh, you haven't even tried to fix your question. – 2017-02-28

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