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Math Help - Hyperbolic geometry

  1. #1
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    Hyperbolic geometry

    Have the following question:

    1) This question deals with the upper half plane, points are represented by complex numbers.
    Let L denoted the h-line given by Re(z)=0
    Let M be given by the semi-circle with centre 0 and radius 2
    Let N be given by the semi-circle with centre -3 and radius 4

    i) Find
    L intersect N=A
    N intersect M=C
    L intersect M=B

    I have found these fine im just having trouble with the next bit, i have the intersections as A(0,root7), B(0, 2), C(-0.5, root15/2)

    The next part asks:
    What is the value of the angle between the hyperbolic arcs BC and BA at B? And show that the angle between the hyperbolic arcs at CA and CB is tan(theta)=(3root15/11)?


    I was thinking of finding the slopes at the intersection points, hence with the first one we have a vertical h-line therefore one of our angles would be pie/2, then using the slope at at "M" in respect to intersection B, hence having tan(\theta )=0?

    Any help would be most appreciated.
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  2. #2
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    Is this another problem from the MA300 course at the University of Kent??

    You will find some advice on how to tackle a similar question at this thread.
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  3. #3
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    Quote Originally Posted by Opalg View Post
    Is this another problem from the MA300 course at the University of Kent??

    You will find some advice on how to tackle a similar question at this thread.
    Yes i managed everything on the previous thread that was all fairly simple, just can't seem to get tan(theta)=(3root15/11)? For CA and CB, i found tangents at both curves in respect to the intersection point but can't seem to get the correct value for theta?
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  4. #4
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    Quote Originally Posted by breitling View Post
    Have the following question:

    1) This question deals with the upper half plane, points are represented by complex numbers.
    Let L denoted the h-line given by Re(z)=0
    Let M be given by the semi-circle with centre 0 and radius 2
    Let N be given by the semi-circle with centre -3 and radius 4

    i) Find
    L intersect N=A
    N intersect M=C
    L intersect M=B

    I have found these fine im just having trouble with the next bit, i have the intersections as A(0,root7), B(0, 2), C(-0.5, root15/2)
    C is incorrect. The circle with center at -3, radius 4, is (x+3)^2+ y^2= 16 which we can write as x^2+ 6x+ 9+ y^2= 16 or x^2+ 6x+ y^2= 7. The circle with center 0 and radius 2 is x^2+ y^2= 4. Subtracting, 6x= 3 so x= 1/2, not -1/2. I suspect you mistakenly wrote N as (x- 3)^2+ y^2= 16 which would be a circle with center at +3, not -3.

    The next part asks:
    What is the value of the angle between the hyperbolic arcs BC and BA at B? And show that the angle between the hyperbolic arcs at CA and CB is tan(theta)=(3root15/11)?


    I was thinking of finding the slopes at the intersection points, hence with the first one we have a vertical h-line therefore one of our angles would be pie/2, then using the slope at at "M" in respect to intersection B, hence having tan(\theta )=0?

    Any help would be most appreciated.
    AC is a vertical line. BC is the line given by (x+3)^2+ y^2= 16. There 2(x+3)+ 2yy'= 0 so y'= -(x+3)/y which, at B, is -(0+3)/sqrt(7)= -3/sqrt(7). The angle between the lines BC and AC is the arctan of that plus pi/2.
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  5. #5
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    Quote Originally Posted by HallsofIvy View Post
    C is incorrect. The circle with center at -3, radius 4, is (x+3)^2+ y^2= 16 which we can write as x^2+ 6x+ 9+ y^2= 16 or x^2+ 6x+ y^2= 7. The circle with center 0 and radius 2 is x^2+ y^2= 4. Subtracting, 6x= 3 so x= 1/2, not -1/2. I suspect you mistakenly wrote N as (x- 3)^2+ y^2= 16 which would be a circle with center at +3, not -3.


    AC is a vertical line. BC is the line given by (x+3)^2+ y^2= 16. There 2(x+3)+ 2yy'= 0 so y'= -(x+3)/y which, at B, is -(0+3)/sqrt(7)= -3/sqrt(7). The angle between the lines BC and AC is the arctan of that plus pi/2.
    Im not to sure of AC being a vertical line, i know AB is as the lie on the vertical h-line? The intersection of C occurs at N and M, hence i found the slope at BC like you have done above just in in respect to point C to get tan(theta)=root15/3, and then found the slope at CA with respect to point C to get tan(theta)= -root 15/15?
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