Holomorphic bijection from intersection of two circles to a region between two rays
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What is a holomorphic map from the nonempty intersection of two circles with "tip points" $a$ (below) and $b$ (neither of which are included in one another) to the region $A$ between two rays?
Pictue is below:
Here is my work so far: by translation and rotation we may assume that the tip of $A$ is the origin one of the rays id the positive real axis, while the region is found by counterclockwise rotation. The professor suggested a map of the form $frac{z-a}{z-b}$, but I am not sure why such a map works. Certainly it maps $a$ to $0$ and $b$ to $infty$ but we have no idea where in the region between two rays $b$ gets mapped to. Also, the angle $alpha$ must be mentioned somehow...
complex-analysis geometry analytic-geometry mobius-transformation
asked Nov 22 at 22:01
Cute Brownie
980316
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up vote
0
down vote
favorite
What is a holomorphic map from the nonempty intersection of two circles with "tip points" $a$ (below) and $b$ (neither of which are included in one another) to the region $A$ between two rays?
Pictue is below:
Here is my work so far: by translation and rotation we may assume that the tip of $A$ is the origin one of the rays id the positive real axis, while the region is found by counterclockwise rotation. The professor suggested a map of the form $frac{z-a}{z-b}$, but I am not sure why such a map works. Certainly it maps $a$ to $0$ and $b$ to $infty$ but we have no idea where in the region between two rays $b$ gets mapped to. Also, the angle $alpha$ must be mentioned somehow...
complex-analysis geometry analytic-geometry mobius-transformation
asked Nov 22 at 22:01
Cute Brownie
980316
The map works because the two circles are mapped to two generalized circles that have two common points $0$ and $infty$, that is, two straight lines through the origin. All such Mobius transformations will have the form either $k (z - a)/(z - b)$ or $k (z - b)/(z - a)$, with arbitrary $k neq 0$. The angle $A$ is the same as the angle between the circles. Varying $arg k$ rotates the sector around the origin.
– Maxim
Nov 26 at 2:56
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
What is a holomorphic map from the nonempty intersection of two circles with "tip points" $a$ (below) and $b$ (neither of which are included in one another) to the region $A$ between two rays?
Pictue is below:
Here is my work so far: by translation and rotation we may assume that the tip of $A$ is the origin one of the rays id the positive real axis, while the region is found by counterclockwise rotation. The professor suggested a map of the form $frac{z-a}{z-b}$, but I am not sure why such a map works. Certainly it maps $a$ to $0$ and $b$ to $infty$ but we have no idea where in the region between two rays $b$ gets mapped to. Also, the angle $alpha$ must be mentioned somehow...
complex-analysis geometry analytic-geometry mobius-transformation
asked Nov 22 at 22:01
Cute Brownie
980316
What is a holomorphic map from the nonempty intersection of two circles with "tip points" $a$ (below) and $b$ (neither of which are included in one another) to the region $A$ between two rays?
Pictue is below:
Here is my work so far: by translation and rotation we may assume that the tip of $A$ is the origin one of the rays id the positive real axis, while the region is found by counterclockwise rotation. The professor suggested a map of the form $frac{z-a}{z-b}$, but I am not sure why such a map works. Certainly it maps $a$ to $0$ and $b$ to $infty$ but we have no idea where in the region between two rays $b$ gets mapped to. Also, the angle $alpha$ must be mentioned somehow...
complex-analysis geometry analytic-geometry mobius-transformation
complex-analysis geometry analytic-geometry mobius-transformation
asked Nov 22 at 22:01
Cute Brownie
980316
asked Nov 22 at 22:01
Cute Brownie
980316
asked Nov 22 at 22:01
Cute Brownie
980316
asked Nov 22 at 22:01
Cute Brownie
980316
asked Nov 22 at 22:01
Cute Brownie
980316
980316
The map works because the two circles are mapped to two generalized circles that have two common points $0$ and $infty$, that is, two straight lines through the origin. All such Mobius transformations will have the form either $k (z - a)/(z - b)$ or $k (z - b)/(z - a)$, with arbitrary $k neq 0$. The angle $A$ is the same as the angle between the circles. Varying $arg k$ rotates the sector around the origin.
– Maxim
Nov 26 at 2:56
add a comment |
The map works because the two circles are mapped to two generalized circles that have two common points $0$ and $infty$, that is, two straight lines through the origin. All such Mobius transformations will have the form either $k (z - a)/(z - b)$ or $k (z - b)/(z - a)$, with arbitrary $k neq 0$. The angle $A$ is the same as the angle between the circles. Varying $arg k$ rotates the sector around the origin.
– Maxim
Nov 26 at 2:56
The map works because the two circles are mapped to two generalized circles that have two common points $0$ and $infty$, that is, two straight lines through the origin. All such Mobius transformations will have the form either $k (z - a)/(z - b)$ or $k (z - b)/(z - a)$, with arbitrary $k neq 0$. The angle $A$ is the same as the angle between the circles. Varying $arg k$ rotates the sector around the origin.
– Maxim
Nov 26 at 2:56
The map works because the two circles are mapped to two generalized circles that have two common points $0$ and $infty$, that is, two straight lines through the origin. All such Mobius transformations will have the form either $k (z - a)/(z - b)$ or $k (z - b)/(z - a)$, with arbitrary $k neq 0$. The angle $A$ is the same as the angle between the circles. Varying $arg k$ rotates the sector around the origin.
– Maxim
Nov 26 at 2:56
add a comment |
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The map works because the two circles are mapped to two generalized circles that have two common points $0$ and $infty$, that is, two straight lines through the origin. All such Mobius transformations will have the form either $k (z - a)/(z - b)$ or $k (z - b)/(z - a)$, with arbitrary $k neq 0$. The angle $A$ is the same as the angle between the circles. Varying $arg k$ rotates the sector around the origin.
– Maxim
Nov 26 at 2:56