# Hyperbolic GeometryWatch

#1
I am having a bit of a problem with the question below. I have done question (i) and found the answer to be 3+2i. I am having a very hard time trying to understand what is required of question (ii). Any help would be greatly appreciated.

Define a Mobius transformation, M, of the hyperbolic upper half plane by
M(z) = (5z-13)/(z-1)

(i) Find the unique fixed point, P, of M in the hyperbolic plane and show that it lies on the h-line
'n' given by mod(z-6)=sqrt13
(ii) Determine the image, M(n), of the h-line n.
(iii) Find the angle at P between the positive ray n+ and its image, M^(n+). (Positive
here means emanating from P in the direction of increasing real part, x.)
0
#2
(Original post by nuodai)
is given by the set of points , and the transformation sends . Let and find in terms of . Then you can substitute into the expression to find the image.
Does that mean I find the inverse of the Mobius transformation? I got z=(w-13)/(w-5)
Where do I substitute the expression?
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#3
(Original post by nuodai)
Sorry I just realised I made a mistake. Woops. You don't need to find in terms of .

The transformation maps , and so the line maps to the line . You just need to find this in terms of .
Sorry, now I'm a bit confused. How do I do that exactly?
0
7 years ago
#4
(Original post by lalyala)
Sorry, now I'm a bit confused. How do I do that exactly?
I don't know what's wrong with me today... forget everything I said in my last two posts -- I was right the first time, but I'll go from the top.

A Möbius transformation is just a function which maps complex numbers to complex number. Here your Möbius transformation is . The image of a point is the point , and the image of a set of points is the set of images of all the points. That is, . Here, your set is , which is the hyperbolic line given by the set of complex numbers for which . This means that .

Now, if you write then , and so substituting this into the above we get

So you need to substitute into .
0
#5
(Original post by nuodai)
I don't know what's wrong with me today... forget everything I said in my last two posts -- I was right the first time, but I'll go from the top.

A Möbius transformation is just a function which maps complex numbers to complex number. Here your Möbius transformation is . The image of a point is the point , and the image of a set of points is the set of images of all the points. That is, . Here, your set is , which is the hyperbolic line given by the set of complex numbers for which . This means that .

Now, if you write then , and so substituting this into the above we get

So you need to substitute into .
|(w-13)/(w-5) - 6| = sqrt(13)
Am I on the right track here?
0
7 years ago
#6
(Original post by lalyala)
|(w-13)/(w-5) - 6| = sqrt(13)
Am I on the right track here?
Yup. Now you can rearrange to simplify the expression (and interpret it geometrically if necessary... which would make sense if this is a geometry question).
0
#7
(Original post by nuodai)
Yup. Now you can rearrange to simplify the expression (and interpret it geometrically if necessary... which would make sense if this is a geometry question).
Am I correct to substitute w=x+iy and rearrange until its a complex number? Or would that just complicate things further? Sorry need a bit more guidance. Really appreciate your help.
0
7 years ago
#8
(Original post by lalyala)
Am I correct to substitute w=x+iy and rearrange until its a complex number? Or would that just complicate things further? Sorry need a bit more guidance. Really appreciate your help.
It really depends what's easiest for you. You can rearrange to put it in the form , which defines either a line or a circle depending on the values of ... but if you've not come across this form before, then write and find a Cartesian equation for the new hyperbolic line (i.e. line or circle). You can then convert it back to complex numbers using the fact that a circle with centre and radius is given by .
0
#9
(Original post by nuodai)
It really depends what's easiest for you. You can rearrange to put it in the form , which defines either a line or a circle depending on the values of ... but if you've not come across this form before, then write and find a Cartesian equation for the new hyperbolic line (i.e. line or circle). You can then convert it back to complex numbers using the fact that a circle with centre and radius is given by .
Okay, I think I'll do the latter as it seems like something the syllabus covered in comparison to the first option.
Is the answer supposed to be in the form of |z-a|=r or x+iy or... something else?
0
7 years ago
#10
(Original post by lalyala)
Okay, I think I'll do the latter as it seems like something the syllabus covered in comparison to the first option.
Is the answer supposed to be in the form of |z-a|=r or x+iy or... something else?
I don't suppose it matters what form you leave it in. Since this is a geometry course, I'd recommend describing it geometrically (i.e. "the hyperbolic line corresponding to the Euclidean circle with centre a and radius r").
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