Prove that a harmonic homology preserves the conic
$begingroup$
I came across a question in the book by Judith N. Cederberg and I’m learning about projective geometry.
One of the question was “Show that a harmonic homology whose centre and axis are pole and polar with respect to a point conic $mathscr C$ keeps $mathscr C$ invariant.” A harmonic homology just means that the cross ratio is $-1$.
I am unsure on how I can even start the proof. I took on some simple examples. For example. I let the center be $Z(0,0,1)$ and axis be $z(0,0,1)$. I am able to find that the matrix is a symmetric diagonal matrix.
Anyone has any hints on how I can show that it keeps the conic $mathscr C$ invariant?
geometry analytic-geometry projective-geometry
asked Dec 10 '18 at 23:03
IcycarusIcycarus
4801314
$endgroup$
add a comment |
$begingroup$
I came across a question in the book by Judith N. Cederberg and I’m learning about projective geometry.
One of the question was “Show that a harmonic homology whose centre and axis are pole and polar with respect to a point conic $mathscr C$ keeps $mathscr C$ invariant.” A harmonic homology just means that the cross ratio is $-1$.
I am unsure on how I can even start the proof. I took on some simple examples. For example. I let the center be $Z(0,0,1)$ and axis be $z(0,0,1)$. I am able to find that the matrix is a symmetric diagonal matrix.
Anyone has any hints on how I can show that it keeps the conic $mathscr C$ invariant?
geometry analytic-geometry projective-geometry
asked Dec 10 '18 at 23:03
IcycarusIcycarus
4801314
$endgroup$
add a comment |
$begingroup$
I came across a question in the book by Judith N. Cederberg and I’m learning about projective geometry.
One of the question was “Show that a harmonic homology whose centre and axis are pole and polar with respect to a point conic $mathscr C$ keeps $mathscr C$ invariant.” A harmonic homology just means that the cross ratio is $-1$.
I am unsure on how I can even start the proof. I took on some simple examples. For example. I let the center be $Z(0,0,1)$ and axis be $z(0,0,1)$. I am able to find that the matrix is a symmetric diagonal matrix.
Anyone has any hints on how I can show that it keeps the conic $mathscr C$ invariant?
geometry analytic-geometry projective-geometry
asked Dec 10 '18 at 23:03
IcycarusIcycarus
4801314
$endgroup$
I came across a question in the book by Judith N. Cederberg and I’m learning about projective geometry.
One of the question was “Show that a harmonic homology whose centre and axis are pole and polar with respect to a point conic $mathscr C$ keeps $mathscr C$ invariant.” A harmonic homology just means that the cross ratio is $-1$.
I am unsure on how I can even start the proof. I took on some simple examples. For example. I let the center be $Z(0,0,1)$ and axis be $z(0,0,1)$. I am able to find that the matrix is a symmetric diagonal matrix.
Anyone has any hints on how I can show that it keeps the conic $mathscr C$ invariant?
geometry analytic-geometry projective-geometry
geometry analytic-geometry projective-geometry
asked Dec 10 '18 at 23:03
IcycarusIcycarus
4801314
asked Dec 10 '18 at 23:03
IcycarusIcycarus
4801314
asked Dec 10 '18 at 23:03
IcycarusIcycarus
4801314
asked Dec 10 '18 at 23:03
IcycarusIcycarus
4801314
asked Dec 10 '18 at 23:03
IcycarusIcycarus
4801314
4801314
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%2f3034614%2fprove-that-a-harmonic-homology-preserves-the-conic%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%2f3034614%2fprove-that-a-harmonic-homology-preserves-the-conic%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
