Hyperbolic Circle <=> Euclidean Circle.

In summary, the conversation discusses how to show that the set S={z in H||z-i|=3/5} is a hyperbolic circle for r>0. It also mentions finding sinh(r/2) and w0 and using the unit disk model of Poincare to prove this. The conversation ends with a recommendation for a book on hyperbolic geometry.
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I have this question which is rather simple, basically reiterating a general theorem.

Show that S={z in H||z-i|=3/5} is a hyperbolic circle S={w in H| p(w,w0)=r}
for r>0 and find sinh(r/2) and w0.

Now to show that it's hyperbolic is the easy task, I just want to see if I got my calculations correct for sh(r/2) and w0.

Now it's best to move to the unit disk model of poincare by: f(z)=(z-i)/(z+i) because this is an isometry it keeps the same lengths here.
(D={z in C| |z|<1} so f(S)={u in D||f(z)-f(i)|=|(u-0)|=3/5} so it's a unit eulidean circle around zero, now we have the following relationship between this radius and the hyperbolic radius r=log((1+3/5)/(1-3/5))=2log(2) and sh(r/2)=3/4 and to get back w0 we need to use the fact that f(w0)=(w0-i)/(w0+i)=0 thus w0=i.
Is this approach valid?

thanks in advance.
btw, there's a great book from jim anderson from southhampton university on hyperbolic geometry.
In case someone wants a recommendation (for those who want to learn the subject via models and not by the axiomatic method).
 
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1. What is a hyperbolic circle and a Euclidean circle?

A hyperbolic circle is a circle in hyperbolic geometry, which is a non-Euclidean geometry where the parallel postulate does not hold. A Euclidean circle is a circle in Euclidean geometry, which is a type of geometry where the parallel postulate does hold.

2. How do the equations for a hyperbolic circle and a Euclidean circle differ?

The equation for a hyperbolic circle is x² + y² = r², where r is the radius of the circle. The equation for a Euclidean circle is (x - a)² + (y - b)² = r², where (a,b) is the center of the circle and r is the radius.

3. What is the relationship between a hyperbolic circle and a Euclidean circle?

A hyperbolic circle and a Euclidean circle are both types of circles, but they exist in different types of geometries. In Euclidean geometry, a circle is always a unique shape with a fixed radius, while in hyperbolic geometry, a circle can have an infinite number of sizes and shapes depending on the curvature of the space.

4. How are the properties of a hyperbolic circle and a Euclidean circle different?

Hyperbolic circles have different properties than Euclidean circles due to the different geometries they exist in. For example, in Euclidean geometry, the circumference and area of a circle are related by the formula C = 2πr, while in hyperbolic geometry, the circumference and area are related by the formula C = 2πsinh(r).

5. How are hyperbolic circles and Euclidean circles used in real-world applications?

Both hyperbolic circles and Euclidean circles have applications in various fields, such as physics, engineering, and computer graphics. For example, hyperbolic circles are used in Einstein's theory of general relativity to describe the curvature of space, while Euclidean circles are used in geometry-based computer graphics to create smooth curves and shapes.

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