# Find an orientation preserving isometry $f (z) = \frac{az+b}{cz+d}$ such that $f (i) = 17 + 3i$

This is probably a very simple questions but I am not clear on Möbius transformations and how to solve this problem. I'd appreciate if somebody can point me towards a method to do these sort of questions or a webpage that explains what I need to solve this problem.

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you could just use a linear map, $f(z)=3z+17$ for instance – yoyo Apr 20 '13 at 21:14

Consider $f(z)=3z+17$. It sends $i$ to $17+3i$ as desired and also preserves orientation since it is a polynomial over $z$ and hence a holomorphic function.

Edit: As "yoyo" already stated in the comments... I was being careless to see the comment.

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Thank you! I'd like to read more about polynomials over z preserving orientation and holomorphic functions because I don't understand those yet. Any specific sources you suggest ? – devcoder Apr 20 '13 at 21:54
I meant to say all polynomials are holomorphic functions which are orientation preserving maps. en.wikipedia.org/wiki/Holomorphic_function – Metin Y. Apr 20 '13 at 22:06

When you use the word "isometry" you have to indicate the exact domain that should be mapped and the used metric(s) on the domain and codomain.

If you have just ${\mathbb C}$ with its euclidean metric in mind the simplest Möbius transformation mapping the point $i$ to $17+3i$ would be the translation $f:\ z\mapsto z+c$ with translation vector $c=17+2i$.

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