Mixing
Find support function of the involutes of circles
$begingroup$
So I have a problem on differential geometry that I am trying to work through.
Determine the support function of the involutes of a circle. Hint: To avoid computations, use the prior exercises.
In the prior exercises I found that if $n(phi)=(cos(phi), sin(phi))$, then $c(phi)=n'(phi)-n(phi)$ parameterizes a circle of radius $sqrt{2}$ centered at the origin. I also found that the evolute is $eta(phi)=h'(phi)n'(phi)-h''(phi)n(phi)$ where $h(phi)$ is the support function.
So I know that the evolute of the involute of a curve $c(t)$ is $c(t)$. And I think that if the support function is a constant, then the curve is a circle but I am not sure how to demonstrate this.
differential-geometry
asked Jan 30 at 20:27
geoplantedgeoplanted
617
$endgroup$
add a comment |
$begingroup$
So I have a problem on differential geometry that I am trying to work through.
Determine the support function of the involutes of a circle. Hint: To avoid computations, use the prior exercises.
In the prior exercises I found that if $n(phi)=(cos(phi), sin(phi))$, then $c(phi)=n'(phi)-n(phi)$ parameterizes a circle of radius $sqrt{2}$ centered at the origin. I also found that the evolute is $eta(phi)=h'(phi)n'(phi)-h''(phi)n(phi)$ where $h(phi)$ is the support function.
So I know that the evolute of the involute of a curve $c(t)$ is $c(t)$. And I think that if the support function is a constant, then the curve is a circle but I am not sure how to demonstrate this.
differential-geometry
asked Jan 30 at 20:27
geoplantedgeoplanted
617
$endgroup$
add a comment |
$begingroup$
So I have a problem on differential geometry that I am trying to work through.
Determine the support function of the involutes of a circle. Hint: To avoid computations, use the prior exercises.
In the prior exercises I found that if $n(phi)=(cos(phi), sin(phi))$, then $c(phi)=n'(phi)-n(phi)$ parameterizes a circle of radius $sqrt{2}$ centered at the origin. I also found that the evolute is $eta(phi)=h'(phi)n'(phi)-h''(phi)n(phi)$ where $h(phi)$ is the support function.
So I know that the evolute of the involute of a curve $c(t)$ is $c(t)$. And I think that if the support function is a constant, then the curve is a circle but I am not sure how to demonstrate this.
differential-geometry
asked Jan 30 at 20:27
geoplantedgeoplanted
617
$endgroup$
So I have a problem on differential geometry that I am trying to work through.
Determine the support function of the involutes of a circle. Hint: To avoid computations, use the prior exercises.
In the prior exercises I found that if $n(phi)=(cos(phi), sin(phi))$, then $c(phi)=n'(phi)-n(phi)$ parameterizes a circle of radius $sqrt{2}$ centered at the origin. I also found that the evolute is $eta(phi)=h'(phi)n'(phi)-h''(phi)n(phi)$ where $h(phi)$ is the support function.
So I know that the evolute of the involute of a curve $c(t)$ is $c(t)$. And I think that if the support function is a constant, then the curve is a circle but I am not sure how to demonstrate this.
differential-geometry
differential-geometry
asked Jan 30 at 20:27
geoplantedgeoplanted
617
asked Jan 30 at 20:27
geoplantedgeoplanted
617
asked Jan 30 at 20:27
geoplantedgeoplanted
617
asked Jan 30 at 20:27
geoplantedgeoplanted
617
asked Jan 30 at 20:27
geoplantedgeoplanted
617
617
add a comment |
add a comment |
0
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%2f3094062%2ffind-support-function-of-the-involutes-of-circles%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
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.
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%2f3094062%2ffind-support-function-of-the-involutes-of-circles%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
