Holomorphic bijection from intersection of two circles to a region between two rays
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
995416
add a comment |
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
995416
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 |
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
995416
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
995416
asked Nov 22 at 22:01
Cute Brownie
995416
asked Nov 22 at 22:01
Cute Brownie
995416
asked Nov 22 at 22:01
Cute Brownie
995416
asked Nov 22 at 22:01
Cute Brownie
995416
995416
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 |
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3009716%2fholomorphic-bijection-from-intersection-of-two-circles-to-a-region-between-two-r%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3009716%2fholomorphic-bijection-from-intersection-of-two-circles-to-a-region-between-two-r%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
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