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# First year uni maths - geometry watch

1. 2 questions that I got a bit stuck on.

1) 3 hyperbolic lines are given. All are z is an element of H (hyperbolic plane).

a: mod(z-1) =3
b: re(z) = 3
c: mod(z-3) = 2 root(3)

Find the vertices A, B, C.

A:= b intersection c
B:= a intersection c
C:= a intersection b

Note: I was unsure of how to calculate the intersections.

2) A mobius transformation is defined by:

lambda(z) = (10z-26)/(2z-2)

Find the unique fixed point of lambda in H(hyperbolic plane).
Show where it lies on the H line.
2. For #1 the fact that they're in the hyperbolic plane is irrelevant; just find the points of intersection as if you're just working in the upper half-plane with the Euclidean metric.

For #2, set and solve for . You might get more than one possible answer, but one of them won't lie in H.
3. (Original post by nuodai)
For #1 the fact that they're in the hyperbolic plane is irrelevant; just find the points of intersection as if you're just working in the upper half-plane with the Euclidean metric.

For #2, set and solve for . You might get more than one possible answer, but one of them won't lie in H.
I'm unsure as to how to find the intersections
4. (Original post by forever_and_always)
I'm unsure as to how to find the intersections
It's probably easier to work with Cartesian coordinates. For example, to find where a and b intersect note that is a circle with centre 1+0i and radius 3, so we can write this as . The line is precisely the line , and so these intersect whenever , which you can solve for . You get a positive and negative solution for y, so it should be clear which lies in the upper half-plane and which doesn't. You do a similar thing for the other ones.
5. (Original post by nuodai)
It's probably easier to work with Cartesian coordinates. For example, to find where a and b intersect note that is a circle with centre 1+0i and radius 3, so we can write this as . The line is precisely the line , and so these intersect whenever , which you can solve for . You get a positive and negative solution for y, so it should be clear which lies in the upper half-plane and which doesn't. You do a similar thing for the other ones.
The question says that I have to find the complex numbers which represent those vertices. With the method quoted, I end up with plus or minus root 5. But that doesn't give me a complex number
6. (Original post by forever_and_always)
The question says that I have to find the complex numbers which represent those vertices. With the method quoted, I end up with plus or minus root 5. But that doesn't give me a complex number
No, but you know that and , so if then what is ? [All I did was flip between complex number notation and (x,y)-plane notation.]

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Updated: April 12, 2011
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