Finding parametrization of the curve of intersection












0












$begingroup$


Given 2 equations $z = x^2 - y^2$ and $z = x^2 +xy -1$, find a parametrization of the curve of the intersection of the surfaces.



By equating them together, I get $y^2 +xy -1 =0$.
Letting $x=t$, I substitute into the following equation and get $y^2 +ty-1=0$. By solving the equation via completing the squares, I get $y=(frac{t+sqrt (t^2+4)}{2})$ or $y=(frac{t-sqrt (t^2+4)}{2})$. Is it right to derive that the parametrized equations are:



$x=t$



$y=(frac{t+sqrt (t^2+4)}{2})$ or $y=(frac{t-sqrt (t^2+4)}{2})$






vector-spaces surfaces parametrization







share|cite|improve this question









$endgroup$

















    0












    $begingroup$


    Given 2 equations $z = x^2 - y^2$ and $z = x^2 +xy -1$, find a parametrization of the curve of the intersection of the surfaces.



    By equating them together, I get $y^2 +xy -1 =0$.
    Letting $x=t$, I substitute into the following equation and get $y^2 +ty-1=0$. By solving the equation via completing the squares, I get $y=(frac{t+sqrt (t^2+4)}{2})$ or $y=(frac{t-sqrt (t^2+4)}{2})$. Is it right to derive that the parametrized equations are:



    $x=t$



    $y=(frac{t+sqrt (t^2+4)}{2})$ or $y=(frac{t-sqrt (t^2+4)}{2})$






    vector-spaces surfaces parametrization







    share|cite|improve this question









    $endgroup$















      0












      0








      0





      $begingroup$


      Given 2 equations $z = x^2 - y^2$ and $z = x^2 +xy -1$, find a parametrization of the curve of the intersection of the surfaces.



      By equating them together, I get $y^2 +xy -1 =0$.
      Letting $x=t$, I substitute into the following equation and get $y^2 +ty-1=0$. By solving the equation via completing the squares, I get $y=(frac{t+sqrt (t^2+4)}{2})$ or $y=(frac{t-sqrt (t^2+4)}{2})$. Is it right to derive that the parametrized equations are:



      $x=t$



      $y=(frac{t+sqrt (t^2+4)}{2})$ or $y=(frac{t-sqrt (t^2+4)}{2})$






      vector-spaces surfaces parametrization







      share|cite|improve this question









      $endgroup$




      Given 2 equations $z = x^2 - y^2$ and $z = x^2 +xy -1$, find a parametrization of the curve of the intersection of the surfaces.



      By equating them together, I get $y^2 +xy -1 =0$.
      Letting $x=t$, I substitute into the following equation and get $y^2 +ty-1=0$. By solving the equation via completing the squares, I get $y=(frac{t+sqrt (t^2+4)}{2})$ or $y=(frac{t-sqrt (t^2+4)}{2})$. Is it right to derive that the parametrized equations are:



      $x=t$



      $y=(frac{t+sqrt (t^2+4)}{2})$ or $y=(frac{t-sqrt (t^2+4)}{2})$






      vector-spaces surfaces parametrization




      vector-spaces surfaces parametrization






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Jan 26 at 13:30









      Cheryl Cheryl

      926




      926






















          1 Answer
          1






          active

          oldest

          votes


















          1












          $begingroup$

          Firstly, there is a sign error in your computation ; it should be



          $$y=frac{color{red}{-}tpmsqrt{t^2+4}}{2}$$



          with a minus sign in front of $t$.



          Besides, as your curve is a space curve, you need a third equation $z=...$ expressed too as a function of parameter $t$. this equation will be obtained by plugging the expressions of $x$ and $y$ you have obtained in either of the two surface equations.



          In order to have a concrete understanding of the result, here is a graphical representation of the two surfaces which both are 2 kinds of Hyperbolic Paraboloids (http://mathworld.wolfram.com/HyperbolicParaboloid.html) and their intersection curve



          enter image description here






          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%2f3088245%2ffinding-parametrization-of-the-curve-of-intersection%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









            1












            $begingroup$

            Firstly, there is a sign error in your computation ; it should be



            $$y=frac{color{red}{-}tpmsqrt{t^2+4}}{2}$$



            with a minus sign in front of $t$.



            Besides, as your curve is a space curve, you need a third equation $z=...$ expressed too as a function of parameter $t$. this equation will be obtained by plugging the expressions of $x$ and $y$ you have obtained in either of the two surface equations.



            In order to have a concrete understanding of the result, here is a graphical representation of the two surfaces which both are 2 kinds of Hyperbolic Paraboloids (http://mathworld.wolfram.com/HyperbolicParaboloid.html) and their intersection curve



            enter image description here






            share|cite|improve this answer











            $endgroup$


















              1












              $begingroup$

              Firstly, there is a sign error in your computation ; it should be



              $$y=frac{color{red}{-}tpmsqrt{t^2+4}}{2}$$



              with a minus sign in front of $t$.



              Besides, as your curve is a space curve, you need a third equation $z=...$ expressed too as a function of parameter $t$. this equation will be obtained by plugging the expressions of $x$ and $y$ you have obtained in either of the two surface equations.



              In order to have a concrete understanding of the result, here is a graphical representation of the two surfaces which both are 2 kinds of Hyperbolic Paraboloids (http://mathworld.wolfram.com/HyperbolicParaboloid.html) and their intersection curve



              enter image description here






              share|cite|improve this answer











              $endgroup$
















                1












                1








                1





                $begingroup$

                Firstly, there is a sign error in your computation ; it should be



                $$y=frac{color{red}{-}tpmsqrt{t^2+4}}{2}$$



                with a minus sign in front of $t$.



                Besides, as your curve is a space curve, you need a third equation $z=...$ expressed too as a function of parameter $t$. this equation will be obtained by plugging the expressions of $x$ and $y$ you have obtained in either of the two surface equations.



                In order to have a concrete understanding of the result, here is a graphical representation of the two surfaces which both are 2 kinds of Hyperbolic Paraboloids (http://mathworld.wolfram.com/HyperbolicParaboloid.html) and their intersection curve



                enter image description here






                share|cite|improve this answer











                $endgroup$



                Firstly, there is a sign error in your computation ; it should be



                $$y=frac{color{red}{-}tpmsqrt{t^2+4}}{2}$$



                with a minus sign in front of $t$.



                Besides, as your curve is a space curve, you need a third equation $z=...$ expressed too as a function of parameter $t$. this equation will be obtained by plugging the expressions of $x$ and $y$ you have obtained in either of the two surface equations.



                In order to have a concrete understanding of the result, here is a graphical representation of the two surfaces which both are 2 kinds of Hyperbolic Paraboloids (http://mathworld.wolfram.com/HyperbolicParaboloid.html) and their intersection curve



                enter image description here







                share|cite|improve this answer














                share|cite|improve this answer



                share|cite|improve this answer








                edited Jan 27 at 0:18

























                answered Jan 26 at 19:13









                Jean MarieJean Marie

                30.9k42155




                30.9k42155






























                    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%2f3088245%2ffinding-parametrization-of-the-curve-of-intersection%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

                    MongoDB - Not Authorized To Execute Command

                    Image
                    .everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty,.everyoneloves__bot-mid-leaderboard:empty{ height:90px;width:728px;box-sizing:border-box; } 0 I have successfully enabled authorization on MongoDB and I have created an account on the admin database and then I created an account for my database called test. The following connection string to connect to my test database works successfully: mongo --host 192.168.17.52 --port 27017 -u user1 -p password --authenticationDatabase test Only problem I have now is, I cannot execute commands such as: show dbs. I get the following error when I try to do so: "errmsg" : "not authorized on admin to execute command { listDatabases: 1.0, lsid: { id: UUID("a1d5bc0d-bc58-485e-b232-270758a89455") }, $db: "admin...
                    Read more

                    Npm cannot find a required file even through it is in the searched directory

                    Image
                    up vote 0 down vote favorite I got that error: users-Air:pdfuploader user$ npm run start client@0.1.0 start /Users/user/Documents/pdfuploader react-scripts start Could not find a required file. Name: index.html Searched in: /Users/user/Documents/pdfuploader/public npm ERR! code ELIFECYCLE npm ERR! errno 1 npm ERR! client@0.1.0 start: `react-scripts start` npm ERR! Exit status 1 npm ERR! npm ERR! Failed at the client@0.1.0 start script. npm ERR! This is probably not a problem with npm. There is likely additional logging output above. npm ERR! A complete log of this run can be found in: npm ERR! /Users/user/.npm/_logs/2018-11-19T10_19_14_555Z-debug.log users-Air:pdfuploader user$ And the file is in /Users/user/Documents/pdfuploader/public as you can see in my github repository: https://github.c...
                    Read more

                    in spring boot 2.1 many test slices are not allowed anymore due to multiple @BootstrapWith

                    Image
                    1 I tried to upgrade a yummy sandwich made of two test slices (@JsonTest and @JdbcTest in my case, crunchy test code in between) adding spring boot 2.1 flavour to it. But it seems it was not much of a success. I cannot annotate my tests with many @...Test since they are now each bringing their own XxxTestContextBootstrapper. It used to work when they all used same SpringBootTestContextBootstrapper. @RunWith(SpringRunner.class) @JdbcTest @JsonTest public class Test { @Test public void test() { System.out.printn("Hello, World !"); } } The error I get from BootstrapUtils is illegalStateException : Configuration error: found multiple declarations of @BootstrapWith for test class I understand I might be doing something wrong here but is there an easy way I could load both Json and Jdbc contex...
                    Read more