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Can someone please explain trigonometry in Equations (2) to (8) of:

PoincareDisk ?

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Narasimham
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  • there it is not clear to me what do they mean by $\theta_2$ ?? – Anubhav Mukherjee Aug 29 '15 at 13:43
  • it's not clear to me what the calculations on the page your link points to are supposed to mean. There is no clear statement what they intend to demonstrate or illustrate, and the it is not explained what the quantities they are using are representing. What exactly is it you want to have explained? – Thomas Aug 29 '15 at 13:59
  • I agree, It appears that the horizontal dashed line is the x-axis. Also $ \phi +d\theta = \pi /2$. – DanielWainfleet Aug 29 '15 at 15:20

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It is just basic geometry (nothing hyperbolic)

  • the boundary circle is the unit circle
  • $\phi_1$ and $\phi_2$ are the angles from the x-axis to the ideal points of the hyperbolic line. (the points where the arc meets the boundary circle)

Then:

  • $\phi$ is the angle from the x-axis to the centre of the orthogonal circle.

  • $d\phi$ is the angle from $\phi_1$ or $\phi_2$ to the angle $\phi$.

  • $r$ is the distance from $\phi_1$ or $\phi_2$ to the centre of the orthogonal circle. (and thus the radius of the orthogonal circle)

  • $R$ is the distance from the origin to the centre of the orthogonal circle.

nothing really interesting I thought

Willemien
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  • Are 7 tagged equations needed to show orthogonal circle angles complementary? – Narasimham Aug 29 '15 at 16:05
  • 3 and 8 are obvious (8 could be better formulated as complementary angle to $d\phi$. 4 and 6 are interesting, I was thinking to add them to the wikipedia article, https://en.wikipedia.org/wiki/Poincar%C3%A9_disk_model but there was no logical place to put them (there is no mention of ideal points in the whole article, yet) – Willemien Aug 29 '15 at 17:05