Normal Curves of Ellipses












6












$begingroup$


Consider a graph of a lot of different ellipses with foci $(-1,0)$ and $(1,0)$



These ellipses take on the form



$$sqrt{(x+1)^2 + y^2} + sqrt{(x-1)^2+y^2}= K $$



where $2 < K < infty$ (the case of $K = 2$ is a degenerate case).



You can graph these for several K such as $K=2.4, 2.3, 2.2, 2.1, 2.05, ...$ on a graph calculator of your choice and basically the ellipses nest nicely into each other (see here).



Now I want to generalize the following image. Which is basically depicts how lines from the origin form "normal" curves w.r.t circles (i.e. they always intersect at a right angle from the tangent of the circle at the point of intersect around the origin)



What are the "normal curves" of the collection of ellipses i've listed?



Work:



One route is to try to describe the ellipses in standard form (which looks somewhat algebraically messy) and then try to characterize their derivatives and proceed but this feels like it would quickly get out of hand in terms of sheer amount of work so i'm hoping someone here with better intuition can highlight what the general form of the curves are.






geometry conic-sections







share|cite|improve this question









$endgroup$








  • 3




    $begingroup$
    i am suspicious that the answer will be the set of hyperbolas with the same foci. Some intuition tells me those curves fit very well
    $endgroup$
    – frogeyedpeas
    Dec 27 '18 at 3:07








  • 1




    $begingroup$
    The answer here: en.wikipedia.org/wiki/…
    $endgroup$
    – Aretino
    Dec 28 '18 at 16:09










  • $begingroup$
    en.wikipedia.org/wiki/Confocal_conic_sections
    $endgroup$
    – Yves Daoust
    Dec 31 '18 at 11:40
















6












$begingroup$


Consider a graph of a lot of different ellipses with foci $(-1,0)$ and $(1,0)$



These ellipses take on the form



$$sqrt{(x+1)^2 + y^2} + sqrt{(x-1)^2+y^2}= K $$



where $2 < K < infty$ (the case of $K = 2$ is a degenerate case).



You can graph these for several K such as $K=2.4, 2.3, 2.2, 2.1, 2.05, ...$ on a graph calculator of your choice and basically the ellipses nest nicely into each other (see here).



Now I want to generalize the following image. Which is basically depicts how lines from the origin form "normal" curves w.r.t circles (i.e. they always intersect at a right angle from the tangent of the circle at the point of intersect around the origin)



What are the "normal curves" of the collection of ellipses i've listed?



Work:



One route is to try to describe the ellipses in standard form (which looks somewhat algebraically messy) and then try to characterize their derivatives and proceed but this feels like it would quickly get out of hand in terms of sheer amount of work so i'm hoping someone here with better intuition can highlight what the general form of the curves are.






geometry conic-sections







share|cite|improve this question









$endgroup$








  • 3




    $begingroup$
    i am suspicious that the answer will be the set of hyperbolas with the same foci. Some intuition tells me those curves fit very well
    $endgroup$
    – frogeyedpeas
    Dec 27 '18 at 3:07








  • 1




    $begingroup$
    The answer here: en.wikipedia.org/wiki/…
    $endgroup$
    – Aretino
    Dec 28 '18 at 16:09










  • $begingroup$
    en.wikipedia.org/wiki/Confocal_conic_sections
    $endgroup$
    – Yves Daoust
    Dec 31 '18 at 11:40














6












6








6


1



$begingroup$


Consider a graph of a lot of different ellipses with foci $(-1,0)$ and $(1,0)$



These ellipses take on the form



$$sqrt{(x+1)^2 + y^2} + sqrt{(x-1)^2+y^2}= K $$



where $2 < K < infty$ (the case of $K = 2$ is a degenerate case).



You can graph these for several K such as $K=2.4, 2.3, 2.2, 2.1, 2.05, ...$ on a graph calculator of your choice and basically the ellipses nest nicely into each other (see here).



Now I want to generalize the following image. Which is basically depicts how lines from the origin form "normal" curves w.r.t circles (i.e. they always intersect at a right angle from the tangent of the circle at the point of intersect around the origin)



What are the "normal curves" of the collection of ellipses i've listed?



Work:



One route is to try to describe the ellipses in standard form (which looks somewhat algebraically messy) and then try to characterize their derivatives and proceed but this feels like it would quickly get out of hand in terms of sheer amount of work so i'm hoping someone here with better intuition can highlight what the general form of the curves are.






geometry conic-sections







share|cite|improve this question









$endgroup$




Consider a graph of a lot of different ellipses with foci $(-1,0)$ and $(1,0)$



These ellipses take on the form



$$sqrt{(x+1)^2 + y^2} + sqrt{(x-1)^2+y^2}= K $$



where $2 < K < infty$ (the case of $K = 2$ is a degenerate case).



You can graph these for several K such as $K=2.4, 2.3, 2.2, 2.1, 2.05, ...$ on a graph calculator of your choice and basically the ellipses nest nicely into each other (see here).



Now I want to generalize the following image. Which is basically depicts how lines from the origin form "normal" curves w.r.t circles (i.e. they always intersect at a right angle from the tangent of the circle at the point of intersect around the origin)



What are the "normal curves" of the collection of ellipses i've listed?



Work:



One route is to try to describe the ellipses in standard form (which looks somewhat algebraically messy) and then try to characterize their derivatives and proceed but this feels like it would quickly get out of hand in terms of sheer amount of work so i'm hoping someone here with better intuition can highlight what the general form of the curves are.






geometry conic-sections




geometry conic-sections






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Dec 27 '18 at 2:11









frogeyedpeasfrogeyedpeas

7,58972053




7,58972053








  • 3




    $begingroup$
    i am suspicious that the answer will be the set of hyperbolas with the same foci. Some intuition tells me those curves fit very well
    $endgroup$
    – frogeyedpeas
    Dec 27 '18 at 3:07








  • 1




    $begingroup$
    The answer here: en.wikipedia.org/wiki/…
    $endgroup$
    – Aretino
    Dec 28 '18 at 16:09










  • $begingroup$
    en.wikipedia.org/wiki/Confocal_conic_sections
    $endgroup$
    – Yves Daoust
    Dec 31 '18 at 11:40














  • 3




    $begingroup$
    i am suspicious that the answer will be the set of hyperbolas with the same foci. Some intuition tells me those curves fit very well
    $endgroup$
    – frogeyedpeas
    Dec 27 '18 at 3:07








  • 1




    $begingroup$
    The answer here: en.wikipedia.org/wiki/…
    $endgroup$
    – Aretino
    Dec 28 '18 at 16:09










  • $begingroup$
    en.wikipedia.org/wiki/Confocal_conic_sections
    $endgroup$
    – Yves Daoust
    Dec 31 '18 at 11:40








3




3




$begingroup$
i am suspicious that the answer will be the set of hyperbolas with the same foci. Some intuition tells me those curves fit very well
$endgroup$
– frogeyedpeas
Dec 27 '18 at 3:07






$begingroup$
i am suspicious that the answer will be the set of hyperbolas with the same foci. Some intuition tells me those curves fit very well
$endgroup$
– frogeyedpeas
Dec 27 '18 at 3:07






1




1




$begingroup$
The answer here: en.wikipedia.org/wiki/…
$endgroup$
– Aretino
Dec 28 '18 at 16:09




$begingroup$
The answer here: en.wikipedia.org/wiki/…
$endgroup$
– Aretino
Dec 28 '18 at 16:09












$begingroup$
en.wikipedia.org/wiki/Confocal_conic_sections
$endgroup$
– Yves Daoust
Dec 31 '18 at 11:40




$begingroup$
en.wikipedia.org/wiki/Confocal_conic_sections
$endgroup$
– Yves Daoust
Dec 31 '18 at 11:40










1 Answer
1






active

oldest

votes


















3












$begingroup$

It may not be mathematically rigorous, but look at the reflection laws of conic sections.



In your ellipses a light beam from one focus is reflected into the other focus. Now imagine turning the reflector 90 degrees, then the reflected beam is turned 180 degrees so now it goes directly away from the second focus. Which is ... the reflection law for a hyperbola.



We see, then, that the orthogonal curves to confocal ellipses are confocal hyperbolas with the same pair of foci.






share|cite|improve this answer











$endgroup$













    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
    });


    }
    });














    draft saved

    draft discarded


















    StackExchange.ready(
    function () {
    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3053528%2fnormal-curves-of-ellipses%23new-answer', 'question_page');
    }
    );

    Post as a guest















    Required, but never shown

























    1 Answer
    1






    active

    oldest

    votes








    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    3












    $begingroup$

    It may not be mathematically rigorous, but look at the reflection laws of conic sections.



    In your ellipses a light beam from one focus is reflected into the other focus. Now imagine turning the reflector 90 degrees, then the reflected beam is turned 180 degrees so now it goes directly away from the second focus. Which is ... the reflection law for a hyperbola.



    We see, then, that the orthogonal curves to confocal ellipses are confocal hyperbolas with the same pair of foci.






    share|cite|improve this answer











    $endgroup$


















      3












      $begingroup$

      It may not be mathematically rigorous, but look at the reflection laws of conic sections.



      In your ellipses a light beam from one focus is reflected into the other focus. Now imagine turning the reflector 90 degrees, then the reflected beam is turned 180 degrees so now it goes directly away from the second focus. Which is ... the reflection law for a hyperbola.



      We see, then, that the orthogonal curves to confocal ellipses are confocal hyperbolas with the same pair of foci.






      share|cite|improve this answer











      $endgroup$
















        3












        3








        3





        $begingroup$

        It may not be mathematically rigorous, but look at the reflection laws of conic sections.



        In your ellipses a light beam from one focus is reflected into the other focus. Now imagine turning the reflector 90 degrees, then the reflected beam is turned 180 degrees so now it goes directly away from the second focus. Which is ... the reflection law for a hyperbola.



        We see, then, that the orthogonal curves to confocal ellipses are confocal hyperbolas with the same pair of foci.






        share|cite|improve this answer











        $endgroup$



        It may not be mathematically rigorous, but look at the reflection laws of conic sections.



        In your ellipses a light beam from one focus is reflected into the other focus. Now imagine turning the reflector 90 degrees, then the reflected beam is turned 180 degrees so now it goes directly away from the second focus. Which is ... the reflection law for a hyperbola.



        We see, then, that the orthogonal curves to confocal ellipses are confocal hyperbolas with the same pair of foci.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Dec 31 '18 at 11:32









        user376343

        3,9584829




        3,9584829










        answered Dec 27 '18 at 3:11









        Oscar LanziOscar Lanzi

        13.2k12136




        13.2k12136






























            draft saved

            draft discarded




















































            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.




            draft saved


            draft discarded














            StackExchange.ready(
            function () {
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3053528%2fnormal-curves-of-ellipses%23new-answer', 'question_page');
            }
            );

            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







            -

            Popular posts from this blog

            To store a contact into the json file from server.js file using a class in NodeJS

            Image
            0 I have created three files -contactPresentation.js, contactService.js, contacts.json. I will run the contactService.js file. And it will navigate to contactPresentation.js file that will only choose an option to either add or display a file. If add(), then add the data into add of service file.And the data will be added to the contacts.json file . And if display(), then display the display() of service file. And the display will take the data from contacts.json file. However, i know my code is flawed. Please correct it . I am also getting an error - this.contacts.push(new ContactService(userName, contactNumber, emailId)); ^ TypeError: Cannot read property 'push' of undefined at ContactService.add (e:NodeJSProjectcontactManager2contactService.js:57:23) at new C
            Read more

            Redirect URL with Chrome Remote Debugging Android Devices

            Image
            1 How can i redirect an URL with Chrome Remote Debugging Android Devices? like Requestly Chrome extension on desktop. google-chrome debugging url redirect testing share | improve this question edited Nov 24 '18 at 8:16 sachinjain024 14.7k 23 73 129 asked Nov 23 '18 at 16:09 Momen Momen 91 10
            Read more

            Dieringhausen

            Image
            Dieringhausen Stadt Gummersbach 50.982777777778 7.5316666666667 165 Koordinaten: 50° 58′ 58″  N , 7° 31′ 54″  O Höhe: 165  (160–245)  m Einwohner: 5055  (30. Jun. 2016) Postleitzahl: 51645 Vorwahl: 02261 Lage von Dieringhausen in Gummersbach Im Hohl ca. 1927–1930 Ansicht von Südwesten bei Bünghausen Dieringhausen (im örtlichen Dialekt Dierkesen ) ist ein Ortsteil von Gummersbach im Oberbergischen Kreis, Regierungsbezirk Köln, Nordrhein-Westfalen (Deutschland). Inhaltsverzeichnis 1 Geographie 2 Geschichte 3 Kultur 3.1 Sehenswürdigkeiten 3.2 Regelmäßige Veranstaltungen 4 Infrastruktur und Wirtschaft 4.1 Verkehr 4.1.1 Schienen- und Busverkehr 4.1.2 Straßen 4.2 Öffentliche Einrichtungen 4.2.1 Schulen und Bildungseinrichtungen 4.2.2 Kirchliche Einrichtungen 5 Persönlichkeiten 5.1 In Dieringhausen geboren 5.2 In Dieringhausen gelebt 6 Einzelnachweise 7
            Read more