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Why is $w$ real if z is on the circle through $z_1,z_2,z_3$?
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Question on answer to Show that $z$ is on the circle passing through $z_1$, $z_2$ and $z_3$ if and only if [$z$, $z_1$, $z_2$, $z_3$] is real or ∞*, a duplicate of Four complex numbers $z_1,z_2,z_3,z_4$ lie on a generalized circle if and only if they have a real cross ratio $[z_1,z_2,z_3,z_4]inmathbbR$ and related to The cross-ratio of four points is a real number exactly when the four points lye on a line or a circle
*Exer 3.24 in A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka
Let me see if I understand the answer correctly:
Based on Exer 2.14 or Exer 3.10, $T(z)$ refers to an arbitrary Möbius of the form $$fracaz+bcz+d, ad-bc ne 0.$$
$forall T$, Möbius, $$[z,z_1,z_2,z_3] = [T(z),T(z_1),T(z_2),T(z_3)]$$
Recall that the cross ratio itself is Möbius by Prop 3.12. Choose $T(z)$ to be $[z,z_1,z_2,z_3]$. Observe that $$T(z_1)=0, T(z_2)=1, T(z_3)=infty,$$ which is pointed out in Prop 3.12.
Denote $w:=T(z)$. Observe that $$w = T(z) = [z,z_1,z_2,z_3] = [T(z),T(z_1),T(z_2),T(z_3)] = [w,0,1,infty].$$
By (4), $$w in mathbb R iff [z,z_1,z_2,z_3] in mathbb R$$
Recall that $forall T$, Möbius, $T$ is bijective and maps clines to clines by Prop 3.1 and Thm 3.4.
By (6), $$w in mathbb R iff z in C(z_1,z_2,z_3)$$
By (5) and (7), $$therefore, [z,z_1,z_2,z_3] in mathbb R iff z in C(z_1,z_2,z_3)$$ QED
Questions:
Which part of the paraphrasing is wrong, and why?
Why is (7) true please?
I believe this is the crux of my misunderstanding. I don't believe I understand the explanations in the answers in the linked questions.
- Actually, it seems like we needed to have proved that $$[z,z_1,z_2,z_3] in mathbb R cup infty iff z in C(z_1,z_2,z_3).$$ Am I wrong? Was that done? If $[z,z_1,z_2,z_3] = infty$, then $z=z_3 in C(z_1,z_2,z_3)$. Not sure about converse. Oh actually, I think what may have been meant by 'real line' is $mathbb R cup infty$? In which case, I think #4 in the answer should have been 'iff w is on the real line' rather than 'w is real' ?
Btw, can we get somewhere with using Cauchy-Riemann? I'm thinking if $f$ is real-valued then it is constant wherever it is holomorphic by Exer 2.19. Then the cross ratio $[z,z_1,z_2,z_3]$ equals some constant real number $c$. If we would solve for $z$ in terms of $c,z_1,z_2,z_3$, then could we do something with it? I mean, other than manually computing the equation of the circle and then plugging $z$ into it.
complex-analysis holomorphic-functions mobius-transformation cauchy-riemann-equation
asked Jul 28 at 13:36
BCLC
6,99421973
add a comment |Â
up vote
0
down vote
favorite
Question on answer to Show that $z$ is on the circle passing through $z_1$, $z_2$ and $z_3$ if and only if [$z$, $z_1$, $z_2$, $z_3$] is real or ∞*, a duplicate of Four complex numbers $z_1,z_2,z_3,z_4$ lie on a generalized circle if and only if they have a real cross ratio $[z_1,z_2,z_3,z_4]inmathbbR$ and related to The cross-ratio of four points is a real number exactly when the four points lye on a line or a circle
*Exer 3.24 in A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka
Let me see if I understand the answer correctly:
Based on Exer 2.14 or Exer 3.10, $T(z)$ refers to an arbitrary Möbius of the form $$fracaz+bcz+d, ad-bc ne 0.$$
$forall T$, Möbius, $$[z,z_1,z_2,z_3] = [T(z),T(z_1),T(z_2),T(z_3)]$$
Recall that the cross ratio itself is Möbius by Prop 3.12. Choose $T(z)$ to be $[z,z_1,z_2,z_3]$. Observe that $$T(z_1)=0, T(z_2)=1, T(z_3)=infty,$$ which is pointed out in Prop 3.12.
Denote $w:=T(z)$. Observe that $$w = T(z) = [z,z_1,z_2,z_3] = [T(z),T(z_1),T(z_2),T(z_3)] = [w,0,1,infty].$$
By (4), $$w in mathbb R iff [z,z_1,z_2,z_3] in mathbb R$$
Recall that $forall T$, Möbius, $T$ is bijective and maps clines to clines by Prop 3.1 and Thm 3.4.
By (6), $$w in mathbb R iff z in C(z_1,z_2,z_3)$$
By (5) and (7), $$therefore, [z,z_1,z_2,z_3] in mathbb R iff z in C(z_1,z_2,z_3)$$ QED
Questions:
Which part of the paraphrasing is wrong, and why?
Why is (7) true please?
I believe this is the crux of my misunderstanding. I don't believe I understand the explanations in the answers in the linked questions.
- Actually, it seems like we needed to have proved that $$[z,z_1,z_2,z_3] in mathbb R cup infty iff z in C(z_1,z_2,z_3).$$ Am I wrong? Was that done? If $[z,z_1,z_2,z_3] = infty$, then $z=z_3 in C(z_1,z_2,z_3)$. Not sure about converse. Oh actually, I think what may have been meant by 'real line' is $mathbb R cup infty$? In which case, I think #4 in the answer should have been 'iff w is on the real line' rather than 'w is real' ?
Btw, can we get somewhere with using Cauchy-Riemann? I'm thinking if $f$ is real-valued then it is constant wherever it is holomorphic by Exer 2.19. Then the cross ratio $[z,z_1,z_2,z_3]$ equals some constant real number $c$. If we would solve for $z$ in terms of $c,z_1,z_2,z_3$, then could we do something with it? I mean, other than manually computing the equation of the circle and then plugging $z$ into it.
complex-analysis holomorphic-functions mobius-transformation cauchy-riemann-equation
asked Jul 28 at 13:36
BCLC
6,99421973
1
Functions of the form that $T$ has send the set of circles and lines to the set of circles and lines. Since $z$ is in the circle passing through $z_1,z_2,z_3$, and these three points are sent to the real line, to the points $0,1,infty$, then the point $z$ would also be sent to that line.
– user578878
Jul 28 at 13:49
@nextpuzzle thanks....soooo if the 3 points' image is in R then the circle between them has an image in R and so if z is on the circle, then the image of z is real valued? Why/why not? And what about the converse? If w is in R, then why is the preimage in the circle passing through the preimages of 0,1,infty?
– BCLC
Jul 28 at 17:14
@nextpuzzle I think I got it. Is my answer wrong please?
– BCLC
Jul 29 at 5:37
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Question on answer to Show that $z$ is on the circle passing through $z_1$, $z_2$ and $z_3$ if and only if [$z$, $z_1$, $z_2$, $z_3$] is real or ∞*, a duplicate of Four complex numbers $z_1,z_2,z_3,z_4$ lie on a generalized circle if and only if they have a real cross ratio $[z_1,z_2,z_3,z_4]inmathbbR$ and related to The cross-ratio of four points is a real number exactly when the four points lye on a line or a circle
*Exer 3.24 in A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka
Let me see if I understand the answer correctly:
Based on Exer 2.14 or Exer 3.10, $T(z)$ refers to an arbitrary Möbius of the form $$fracaz+bcz+d, ad-bc ne 0.$$
$forall T$, Möbius, $$[z,z_1,z_2,z_3] = [T(z),T(z_1),T(z_2),T(z_3)]$$
Recall that the cross ratio itself is Möbius by Prop 3.12. Choose $T(z)$ to be $[z,z_1,z_2,z_3]$. Observe that $$T(z_1)=0, T(z_2)=1, T(z_3)=infty,$$ which is pointed out in Prop 3.12.
Denote $w:=T(z)$. Observe that $$w = T(z) = [z,z_1,z_2,z_3] = [T(z),T(z_1),T(z_2),T(z_3)] = [w,0,1,infty].$$
By (4), $$w in mathbb R iff [z,z_1,z_2,z_3] in mathbb R$$
Recall that $forall T$, Möbius, $T$ is bijective and maps clines to clines by Prop 3.1 and Thm 3.4.
By (6), $$w in mathbb R iff z in C(z_1,z_2,z_3)$$
By (5) and (7), $$therefore, [z,z_1,z_2,z_3] in mathbb R iff z in C(z_1,z_2,z_3)$$ QED
Questions:
Which part of the paraphrasing is wrong, and why?
Why is (7) true please?
I believe this is the crux of my misunderstanding. I don't believe I understand the explanations in the answers in the linked questions.
- Actually, it seems like we needed to have proved that $$[z,z_1,z_2,z_3] in mathbb R cup infty iff z in C(z_1,z_2,z_3).$$ Am I wrong? Was that done? If $[z,z_1,z_2,z_3] = infty$, then $z=z_3 in C(z_1,z_2,z_3)$. Not sure about converse. Oh actually, I think what may have been meant by 'real line' is $mathbb R cup infty$? In which case, I think #4 in the answer should have been 'iff w is on the real line' rather than 'w is real' ?
Btw, can we get somewhere with using Cauchy-Riemann? I'm thinking if $f$ is real-valued then it is constant wherever it is holomorphic by Exer 2.19. Then the cross ratio $[z,z_1,z_2,z_3]$ equals some constant real number $c$. If we would solve for $z$ in terms of $c,z_1,z_2,z_3$, then could we do something with it? I mean, other than manually computing the equation of the circle and then plugging $z$ into it.
complex-analysis holomorphic-functions mobius-transformation cauchy-riemann-equation
asked Jul 28 at 13:36
BCLC
6,99421973
Question on answer to Show that $z$ is on the circle passing through $z_1$, $z_2$ and $z_3$ if and only if [$z$, $z_1$, $z_2$, $z_3$] is real or ∞*, a duplicate of Four complex numbers $z_1,z_2,z_3,z_4$ lie on a generalized circle if and only if they have a real cross ratio $[z_1,z_2,z_3,z_4]inmathbbR$ and related to The cross-ratio of four points is a real number exactly when the four points lye on a line or a circle
*Exer 3.24 in A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka
Let me see if I understand the answer correctly:
Based on Exer 2.14 or Exer 3.10, $T(z)$ refers to an arbitrary Möbius of the form $$fracaz+bcz+d, ad-bc ne 0.$$
$forall T$, Möbius, $$[z,z_1,z_2,z_3] = [T(z),T(z_1),T(z_2),T(z_3)]$$
Recall that the cross ratio itself is Möbius by Prop 3.12. Choose $T(z)$ to be $[z,z_1,z_2,z_3]$. Observe that $$T(z_1)=0, T(z_2)=1, T(z_3)=infty,$$ which is pointed out in Prop 3.12.
Denote $w:=T(z)$. Observe that $$w = T(z) = [z,z_1,z_2,z_3] = [T(z),T(z_1),T(z_2),T(z_3)] = [w,0,1,infty].$$
By (4), $$w in mathbb R iff [z,z_1,z_2,z_3] in mathbb R$$
Recall that $forall T$, Möbius, $T$ is bijective and maps clines to clines by Prop 3.1 and Thm 3.4.
By (6), $$w in mathbb R iff z in C(z_1,z_2,z_3)$$
By (5) and (7), $$therefore, [z,z_1,z_2,z_3] in mathbb R iff z in C(z_1,z_2,z_3)$$ QED
Questions:
Which part of the paraphrasing is wrong, and why?
Why is (7) true please?
I believe this is the crux of my misunderstanding. I don't believe I understand the explanations in the answers in the linked questions.
- Actually, it seems like we needed to have proved that $$[z,z_1,z_2,z_3] in mathbb R cup infty iff z in C(z_1,z_2,z_3).$$ Am I wrong? Was that done? If $[z,z_1,z_2,z_3] = infty$, then $z=z_3 in C(z_1,z_2,z_3)$. Not sure about converse. Oh actually, I think what may have been meant by 'real line' is $mathbb R cup infty$? In which case, I think #4 in the answer should have been 'iff w is on the real line' rather than 'w is real' ?
Btw, can we get somewhere with using Cauchy-Riemann? I'm thinking if $f$ is real-valued then it is constant wherever it is holomorphic by Exer 2.19. Then the cross ratio $[z,z_1,z_2,z_3]$ equals some constant real number $c$. If we would solve for $z$ in terms of $c,z_1,z_2,z_3$, then could we do something with it? I mean, other than manually computing the equation of the circle and then plugging $z$ into it.
complex-analysis holomorphic-functions mobius-transformation cauchy-riemann-equation
asked Jul 28 at 13:36
BCLC
6,99421973
edited Jul 28 at 13:41
asked Jul 28 at 13:36
BCLC
6,99421973
asked Jul 28 at 13:36
BCLC
6,99421973
asked Jul 28 at 13:36
BCLC
6,99421973
6,99421973
1
Functions of the form that $T$ has send the set of circles and lines to the set of circles and lines. Since $z$ is in the circle passing through $z_1,z_2,z_3$, and these three points are sent to the real line, to the points $0,1,infty$, then the point $z$ would also be sent to that line.
– user578878
Jul 28 at 13:49
@nextpuzzle thanks....soooo if the 3 points' image is in R then the circle between them has an image in R and so if z is on the circle, then the image of z is real valued? Why/why not? And what about the converse? If w is in R, then why is the preimage in the circle passing through the preimages of 0,1,infty?
– BCLC
Jul 28 at 17:14
@nextpuzzle I think I got it. Is my answer wrong please?
– BCLC
Jul 29 at 5:37
add a comment |Â
1
Functions of the form that $T$ has send the set of circles and lines to the set of circles and lines. Since $z$ is in the circle passing through $z_1,z_2,z_3$, and these three points are sent to the real line, to the points $0,1,infty$, then the point $z$ would also be sent to that line.
– user578878
Jul 28 at 13:49
@nextpuzzle thanks....soooo if the 3 points' image is in R then the circle between them has an image in R and so if z is on the circle, then the image of z is real valued? Why/why not? And what about the converse? If w is in R, then why is the preimage in the circle passing through the preimages of 0,1,infty?
– BCLC
Jul 28 at 17:14
@nextpuzzle I think I got it. Is my answer wrong please?
– BCLC
Jul 29 at 5:37
1
1
Functions of the form that $T$ has send the set of circles and lines to the set of circles and lines. Since $z$ is in the circle passing through $z_1,z_2,z_3$, and these three points are sent to the real line, to the points $0,1,infty$, then the point $z$ would also be sent to that line.
– user578878
Jul 28 at 13:49
Functions of the form that $T$ has send the set of circles and lines to the set of circles and lines. Since $z$ is in the circle passing through $z_1,z_2,z_3$, and these three points are sent to the real line, to the points $0,1,infty$, then the point $z$ would also be sent to that line.
– user578878
Jul 28 at 13:49
@nextpuzzle thanks....soooo if the 3 points' image is in R then the circle between them has an image in R and so if z is on the circle, then the image of z is real valued? Why/why not? And what about the converse? If w is in R, then why is the preimage in the circle passing through the preimages of 0,1,infty?
– BCLC
Jul 28 at 17:14
@nextpuzzle thanks....soooo if the 3 points' image is in R then the circle between them has an image in R and so if z is on the circle, then the image of z is real valued? Why/why not? And what about the converse? If w is in R, then why is the preimage in the circle passing through the preimages of 0,1,infty?
– BCLC
Jul 28 at 17:14
@nextpuzzle I think I got it. Is my answer wrong please?
– BCLC
Jul 29 at 5:37
@nextpuzzle I think I got it. Is my answer wrong please?
– BCLC
Jul 29 at 5:37
add a comment |Â
1 Answer
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I think I got it: w is the image of z under T. Thus...
$leftarrow$
If z is on the circle, then the image under T is on the real line. Why? T maps 3 distinct points to 2 finite points and $infty$. Thus, T maps the points to a subset of the real line. I conclude this is the whole real line either because $infty$ is there or because an empty or proper subset would not be a line. Now because T maps clines to clines bijectively, it maps the entire circle passing through the 3 points to entire real line. $therefore$, if z is in the circle, its image is on the real line. But the image under T is the cross ratio.
$rightarrow$
Now suppose the cross ratio, which is the image of z under T, is on the real line. We know the whole circle is mapped to the whole real line under T. Since T is bijective and maps clines to clines, the preimage of T(z), w/c is z, is on the circle.
QED
answered Jul 29 at 5:36
BCLC
6,99421973
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
accepted
I think I got it: w is the image of z under T. Thus...
$leftarrow$
If z is on the circle, then the image under T is on the real line. Why? T maps 3 distinct points to 2 finite points and $infty$. Thus, T maps the points to a subset of the real line. I conclude this is the whole real line either because $infty$ is there or because an empty or proper subset would not be a line. Now because T maps clines to clines bijectively, it maps the entire circle passing through the 3 points to entire real line. $therefore$, if z is in the circle, its image is on the real line. But the image under T is the cross ratio.
$rightarrow$
Now suppose the cross ratio, which is the image of z under T, is on the real line. We know the whole circle is mapped to the whole real line under T. Since T is bijective and maps clines to clines, the preimage of T(z), w/c is z, is on the circle.
QED
answered Jul 29 at 5:36
BCLC
6,99421973
add a comment |Â
up vote
0
down vote
accepted
I think I got it: w is the image of z under T. Thus...
$leftarrow$
If z is on the circle, then the image under T is on the real line. Why? T maps 3 distinct points to 2 finite points and $infty$. Thus, T maps the points to a subset of the real line. I conclude this is the whole real line either because $infty$ is there or because an empty or proper subset would not be a line. Now because T maps clines to clines bijectively, it maps the entire circle passing through the 3 points to entire real line. $therefore$, if z is in the circle, its image is on the real line. But the image under T is the cross ratio.
$rightarrow$
Now suppose the cross ratio, which is the image of z under T, is on the real line. We know the whole circle is mapped to the whole real line under T. Since T is bijective and maps clines to clines, the preimage of T(z), w/c is z, is on the circle.
QED
answered Jul 29 at 5:36
BCLC
6,99421973
add a comment |Â
up vote
0
down vote
accepted
up vote
0
down vote
accepted
I think I got it: w is the image of z under T. Thus...
$leftarrow$
If z is on the circle, then the image under T is on the real line. Why? T maps 3 distinct points to 2 finite points and $infty$. Thus, T maps the points to a subset of the real line. I conclude this is the whole real line either because $infty$ is there or because an empty or proper subset would not be a line. Now because T maps clines to clines bijectively, it maps the entire circle passing through the 3 points to entire real line. $therefore$, if z is in the circle, its image is on the real line. But the image under T is the cross ratio.
$rightarrow$
Now suppose the cross ratio, which is the image of z under T, is on the real line. We know the whole circle is mapped to the whole real line under T. Since T is bijective and maps clines to clines, the preimage of T(z), w/c is z, is on the circle.
QED
answered Jul 29 at 5:36
BCLC
6,99421973
I think I got it: w is the image of z under T. Thus...
$leftarrow$
If z is on the circle, then the image under T is on the real line. Why? T maps 3 distinct points to 2 finite points and $infty$. Thus, T maps the points to a subset of the real line. I conclude this is the whole real line either because $infty$ is there or because an empty or proper subset would not be a line. Now because T maps clines to clines bijectively, it maps the entire circle passing through the 3 points to entire real line. $therefore$, if z is in the circle, its image is on the real line. But the image under T is the cross ratio.
$rightarrow$
Now suppose the cross ratio, which is the image of z under T, is on the real line. We know the whole circle is mapped to the whole real line under T. Since T is bijective and maps clines to clines, the preimage of T(z), w/c is z, is on the circle.
QED
answered Jul 29 at 5:36
BCLC
6,99421973
answered Jul 29 at 5:36
BCLC
6,99421973
answered Jul 29 at 5:36
BCLC
6,99421973
answered Jul 29 at 5:36
BCLC
6,99421973
6,99421973
add a comment |Â
add a comment |Â
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1
Functions of the form that $T$ has send the set of circles and lines to the set of circles and lines. Since $z$ is in the circle passing through $z_1,z_2,z_3$, and these three points are sent to the real line, to the points $0,1,infty$, then the point $z$ would also be sent to that line.
– user578878
Jul 28 at 13:49
@nextpuzzle thanks....soooo if the 3 points' image is in R then the circle between them has an image in R and so if z is on the circle, then the image of z is real valued? Why/why not? And what about the converse? If w is in R, then why is the preimage in the circle passing through the preimages of 0,1,infty?
– BCLC
Jul 28 at 17:14
@nextpuzzle I think I got it. Is my answer wrong please?
– BCLC
Jul 29 at 5:37