Finding the norm of $w + frac{1 - |w|^2}{|w - z|^2}(w - z)$, where $w$ and $z$ are in $mathbb{R}^n$
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I'm trying to show a result from Manfred Stoll's book on Hyperbolic geometry. Consider the function
$$M_w(z) = w + frac{1 - |w|^2}{|w - z|^2}(w - z)$$ where $w,z in mathbb{R}^n$ and $|w| < 1$. (I am using different variable names than Stoll). This function generalizes the Möbius transformations on the complex numbers.
Stoll writes that
$$|M_w(z)|^2 = frac{|w - z|^2 + (1 - |w|^2)(1-|z|^2)}{|w - z|^2}$$
How do you show this? I tried computing $langle M_w(z), M_w(z) rangle$, but I'm not getting any closer.
This result ultimately shows that $M_w(z)$ maps into the unit ball.
hyperbolic-geometry mobius-transformation
edited Dec 3 '18 at 2:36
Blue
48.5k870154
asked Dec 2 '18 at 21:47
dinstructiondinstruction
554423
$endgroup$
add a comment |
$begingroup$
I'm trying to show a result from Manfred Stoll's book on Hyperbolic geometry. Consider the function
$$M_w(z) = w + frac{1 - |w|^2}{|w - z|^2}(w - z)$$ where $w,z in mathbb{R}^n$ and $|w| < 1$. (I am using different variable names than Stoll). This function generalizes the Möbius transformations on the complex numbers.
Stoll writes that
$$|M_w(z)|^2 = frac{|w - z|^2 + (1 - |w|^2)(1-|z|^2)}{|w - z|^2}$$
How do you show this? I tried computing $langle M_w(z), M_w(z) rangle$, but I'm not getting any closer.
This result ultimately shows that $M_w(z)$ maps into the unit ball.
hyperbolic-geometry mobius-transformation
edited Dec 3 '18 at 2:36
Blue
48.5k870154
asked Dec 2 '18 at 21:47
dinstructiondinstruction
554423
$endgroup$
add a comment |
$begingroup$
I'm trying to show a result from Manfred Stoll's book on Hyperbolic geometry. Consider the function
$$M_w(z) = w + frac{1 - |w|^2}{|w - z|^2}(w - z)$$ where $w,z in mathbb{R}^n$ and $|w| < 1$. (I am using different variable names than Stoll). This function generalizes the Möbius transformations on the complex numbers.
Stoll writes that
$$|M_w(z)|^2 = frac{|w - z|^2 + (1 - |w|^2)(1-|z|^2)}{|w - z|^2}$$
How do you show this? I tried computing $langle M_w(z), M_w(z) rangle$, but I'm not getting any closer.
This result ultimately shows that $M_w(z)$ maps into the unit ball.
hyperbolic-geometry mobius-transformation
edited Dec 3 '18 at 2:36
Blue
48.5k870154
asked Dec 2 '18 at 21:47
dinstructiondinstruction
554423
$endgroup$
I'm trying to show a result from Manfred Stoll's book on Hyperbolic geometry. Consider the function
$$M_w(z) = w + frac{1 - |w|^2}{|w - z|^2}(w - z)$$ where $w,z in mathbb{R}^n$ and $|w| < 1$. (I am using different variable names than Stoll). This function generalizes the Möbius transformations on the complex numbers.
Stoll writes that
$$|M_w(z)|^2 = frac{|w - z|^2 + (1 - |w|^2)(1-|z|^2)}{|w - z|^2}$$
How do you show this? I tried computing $langle M_w(z), M_w(z) rangle$, but I'm not getting any closer.
This result ultimately shows that $M_w(z)$ maps into the unit ball.
hyperbolic-geometry mobius-transformation
hyperbolic-geometry mobius-transformation
edited Dec 3 '18 at 2:36
Blue
48.5k870154
asked Dec 2 '18 at 21:47
dinstructiondinstruction
554423
edited Dec 3 '18 at 2:36
Blue
48.5k870154
asked Dec 2 '18 at 21:47
dinstructiondinstruction
554423
edited Dec 3 '18 at 2:36
Blue
48.5k870154
edited Dec 3 '18 at 2:36
Blue
48.5k870154
edited Dec 3 '18 at 2:36
Blue
48.5k870154
48.5k870154
asked Dec 2 '18 at 21:47
dinstructiondinstruction
554423
asked Dec 2 '18 at 21:47
dinstructiondinstruction
554423
asked Dec 2 '18 at 21:47
dinstructiondinstruction
554423
554423
add a comment |
add a comment |
1 Answer
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$begingroup$
$$begin{align}
left|M_w(z)right|^2 &=
|w|^2+frac{left(1-|w|^2right)^2}{|w-z|^4}|w-z|^2+2frac{1-|w|^2}{|w-z|^2};wcdot(w-z) \[8pt]
|w-z|^2;left|M_w(z)right|^2 &=
|w|^2|w-z|^2+left(1-|w|^2right)^2+2left(1-|w|^2right);left(|w|^2-wcdot zright) \[8pt]
&=
phantom{+2}|w|^2left(|w|^2+|z|^2-2wcdot zright)\
&phantom{=}+phantom{2}left(1-2|w|^2+|w|^4right)\
&phantom{=}+2left(|w|^2-wcdot z-|w|^4+|w|^2wcdot z right) \[8pt]
&=
left(-2wcdot zright)+ left(1 + |w|^2|z|^2right)\[6pt]
&=
left(|w|^2+|z|^2-2wcdot zright)+ left(1 -|w|^2-|z|^2+ |w|^2|z|^2right)\[6pt]
&=
|w-z|^2+ left(1 -|w|^2right)left(1-|z|^2right)\
end{align}$$
answered Dec 3 '18 at 2:33
BlueBlue
48.5k870154
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1 Answer
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1 Answer
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active
oldest
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active
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active
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$begingroup$
$$begin{align}
left|M_w(z)right|^2 &=
|w|^2+frac{left(1-|w|^2right)^2}{|w-z|^4}|w-z|^2+2frac{1-|w|^2}{|w-z|^2};wcdot(w-z) \[8pt]
|w-z|^2;left|M_w(z)right|^2 &=
|w|^2|w-z|^2+left(1-|w|^2right)^2+2left(1-|w|^2right);left(|w|^2-wcdot zright) \[8pt]
&=
phantom{+2}|w|^2left(|w|^2+|z|^2-2wcdot zright)\
&phantom{=}+phantom{2}left(1-2|w|^2+|w|^4right)\
&phantom{=}+2left(|w|^2-wcdot z-|w|^4+|w|^2wcdot z right) \[8pt]
&=
left(-2wcdot zright)+ left(1 + |w|^2|z|^2right)\[6pt]
&=
left(|w|^2+|z|^2-2wcdot zright)+ left(1 -|w|^2-|z|^2+ |w|^2|z|^2right)\[6pt]
&=
|w-z|^2+ left(1 -|w|^2right)left(1-|z|^2right)\
end{align}$$
answered Dec 3 '18 at 2:33
BlueBlue
48.5k870154
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$begingroup$
$$begin{align}
left|M_w(z)right|^2 &=
|w|^2+frac{left(1-|w|^2right)^2}{|w-z|^4}|w-z|^2+2frac{1-|w|^2}{|w-z|^2};wcdot(w-z) \[8pt]
|w-z|^2;left|M_w(z)right|^2 &=
|w|^2|w-z|^2+left(1-|w|^2right)^2+2left(1-|w|^2right);left(|w|^2-wcdot zright) \[8pt]
&=
phantom{+2}|w|^2left(|w|^2+|z|^2-2wcdot zright)\
&phantom{=}+phantom{2}left(1-2|w|^2+|w|^4right)\
&phantom{=}+2left(|w|^2-wcdot z-|w|^4+|w|^2wcdot z right) \[8pt]
&=
left(-2wcdot zright)+ left(1 + |w|^2|z|^2right)\[6pt]
&=
left(|w|^2+|z|^2-2wcdot zright)+ left(1 -|w|^2-|z|^2+ |w|^2|z|^2right)\[6pt]
&=
|w-z|^2+ left(1 -|w|^2right)left(1-|z|^2right)\
end{align}$$
answered Dec 3 '18 at 2:33
BlueBlue
48.5k870154
$endgroup$
add a comment |
$begingroup$
$$begin{align}
left|M_w(z)right|^2 &=
|w|^2+frac{left(1-|w|^2right)^2}{|w-z|^4}|w-z|^2+2frac{1-|w|^2}{|w-z|^2};wcdot(w-z) \[8pt]
|w-z|^2;left|M_w(z)right|^2 &=
|w|^2|w-z|^2+left(1-|w|^2right)^2+2left(1-|w|^2right);left(|w|^2-wcdot zright) \[8pt]
&=
phantom{+2}|w|^2left(|w|^2+|z|^2-2wcdot zright)\
&phantom{=}+phantom{2}left(1-2|w|^2+|w|^4right)\
&phantom{=}+2left(|w|^2-wcdot z-|w|^4+|w|^2wcdot z right) \[8pt]
&=
left(-2wcdot zright)+ left(1 + |w|^2|z|^2right)\[6pt]
&=
left(|w|^2+|z|^2-2wcdot zright)+ left(1 -|w|^2-|z|^2+ |w|^2|z|^2right)\[6pt]
&=
|w-z|^2+ left(1 -|w|^2right)left(1-|z|^2right)\
end{align}$$
answered Dec 3 '18 at 2:33
BlueBlue
48.5k870154
$endgroup$
$$begin{align}
left|M_w(z)right|^2 &=
|w|^2+frac{left(1-|w|^2right)^2}{|w-z|^4}|w-z|^2+2frac{1-|w|^2}{|w-z|^2};wcdot(w-z) \[8pt]
|w-z|^2;left|M_w(z)right|^2 &=
|w|^2|w-z|^2+left(1-|w|^2right)^2+2left(1-|w|^2right);left(|w|^2-wcdot zright) \[8pt]
&=
phantom{+2}|w|^2left(|w|^2+|z|^2-2wcdot zright)\
&phantom{=}+phantom{2}left(1-2|w|^2+|w|^4right)\
&phantom{=}+2left(|w|^2-wcdot z-|w|^4+|w|^2wcdot z right) \[8pt]
&=
left(-2wcdot zright)+ left(1 + |w|^2|z|^2right)\[6pt]
&=
left(|w|^2+|z|^2-2wcdot zright)+ left(1 -|w|^2-|z|^2+ |w|^2|z|^2right)\[6pt]
&=
|w-z|^2+ left(1 -|w|^2right)left(1-|z|^2right)\
end{align}$$
answered Dec 3 '18 at 2:33
BlueBlue
48.5k870154
answered Dec 3 '18 at 2:33
BlueBlue
48.5k870154
answered Dec 3 '18 at 2:33
BlueBlue
48.5k870154
answered Dec 3 '18 at 2:33
BlueBlue
48.5k870154
48.5k870154
add a comment |
add a comment |
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