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Stereographic projection of ellipsoid


Stereographic projection continuous at $infty$Stereographic Projection proofs(pathagorean triples)Showing that stereographic projection is a homeomorphismStereographic Projection from an Arbitrary PointStereographic projection (Theorem that circles on the sphere get mapped to circles on the plane)Proving geometrically that stereographic projection conserves circlesThe inverse map of the stereographic projectionStereographic projection of hyperboloid.Stereographic projection problemHow does the stereographic map from $S^2$ to $mathbb R^2$ “induce” a metric on $mathbb R^2$













0












$begingroup$


I am really new in geometry and especially in working with stereographic projection, so excuse me, please, if my question is too easy.



Given is the ellipsoid: $E = left (x,y,z)in mathbbR^3: fracx^2a^2 + fracy^2b^2 +fracz^2c^2 = 1right $.



I have to find two parametrizations with the following points excluded: $Esetminus(a,0,0)$ and $Esetminus(0,0,-a)$.



OK, we know the definition of the stereographic projection of the unit sphere in $mathbbR^3$ with excluding the north pole $(0,0,1)$. It is given by: $(x,y,z)=left ( frac2xx^2+y^2+1, frac2yx^2+y^2+1, fracx^2+y^2-1x^2+y^2+1 right )$.



I know, the problem takes much time to do the calculations, so i would be very glad if someone could give a hint how to do this calculation, because i really don't get it. Thank you very much in advance.










share|cite|improve this question











$endgroup$
















    0












    $begingroup$


    I am really new in geometry and especially in working with stereographic projection, so excuse me, please, if my question is too easy.



    Given is the ellipsoid: $E = left (x,y,z)in mathbbR^3: fracx^2a^2 + fracy^2b^2 +fracz^2c^2 = 1right $.



    I have to find two parametrizations with the following points excluded: $Esetminus(a,0,0)$ and $Esetminus(0,0,-a)$.



    OK, we know the definition of the stereographic projection of the unit sphere in $mathbbR^3$ with excluding the north pole $(0,0,1)$. It is given by: $(x,y,z)=left ( frac2xx^2+y^2+1, frac2yx^2+y^2+1, fracx^2+y^2-1x^2+y^2+1 right )$.



    I know, the problem takes much time to do the calculations, so i would be very glad if someone could give a hint how to do this calculation, because i really don't get it. Thank you very much in advance.










    share|cite|improve this question











    $endgroup$














      0












      0








      0





      $begingroup$


      I am really new in geometry and especially in working with stereographic projection, so excuse me, please, if my question is too easy.



      Given is the ellipsoid: $E = left (x,y,z)in mathbbR^3: fracx^2a^2 + fracy^2b^2 +fracz^2c^2 = 1right $.



      I have to find two parametrizations with the following points excluded: $Esetminus(a,0,0)$ and $Esetminus(0,0,-a)$.



      OK, we know the definition of the stereographic projection of the unit sphere in $mathbbR^3$ with excluding the north pole $(0,0,1)$. It is given by: $(x,y,z)=left ( frac2xx^2+y^2+1, frac2yx^2+y^2+1, fracx^2+y^2-1x^2+y^2+1 right )$.



      I know, the problem takes much time to do the calculations, so i would be very glad if someone could give a hint how to do this calculation, because i really don't get it. Thank you very much in advance.










      share|cite|improve this question











      $endgroup$




      I am really new in geometry and especially in working with stereographic projection, so excuse me, please, if my question is too easy.



      Given is the ellipsoid: $E = left (x,y,z)in mathbbR^3: fracx^2a^2 + fracy^2b^2 +fracz^2c^2 = 1right $.



      I have to find two parametrizations with the following points excluded: $Esetminus(a,0,0)$ and $Esetminus(0,0,-a)$.



      OK, we know the definition of the stereographic projection of the unit sphere in $mathbbR^3$ with excluding the north pole $(0,0,1)$. It is given by: $(x,y,z)=left ( frac2xx^2+y^2+1, frac2yx^2+y^2+1, fracx^2+y^2-1x^2+y^2+1 right )$.



      I know, the problem takes much time to do the calculations, so i would be very glad if someone could give a hint how to do this calculation, because i really don't get it. Thank you very much in advance.







      geometry geometric-topology






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Mar 15 at 17:06









      MarianD

      1,5291616




      1,5291616










      asked Feb 5 '13 at 19:59









      LullabyLullaby

      917718




      917718




















          1 Answer
          1






          active

          oldest

          votes


















          0












          $begingroup$

          A hint for the first one:



          Any point $P:=(0,u,v)$ in the $(y,z)$-plane determines a line $g_P:=Pvee A$, where $A:=(a,0,0)$. Intersecting
          $$g_P:quad tmapsto bigl((1-t) a,t u,t vbigr)qquad(-infty<t<infty)$$
          with the ellipsoid $E$ you get a quadratic equation for $t$ with one obvious solution $t=0$. The other solution leads you to the point $(x_P,y_P,z_P)in E$ stereographically related to $P$. All in all you will obtain a parametric representation
          $$(u,v)mapstobigl(x_(u,v),y_(u,v),z_(u,v)bigr)in Esetminus(a,0,0) .$$






          share|cite|improve this answer









          $endgroup$












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            1 Answer
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            active

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            0












            $begingroup$

            A hint for the first one:



            Any point $P:=(0,u,v)$ in the $(y,z)$-plane determines a line $g_P:=Pvee A$, where $A:=(a,0,0)$. Intersecting
            $$g_P:quad tmapsto bigl((1-t) a,t u,t vbigr)qquad(-infty<t<infty)$$
            with the ellipsoid $E$ you get a quadratic equation for $t$ with one obvious solution $t=0$. The other solution leads you to the point $(x_P,y_P,z_P)in E$ stereographically related to $P$. All in all you will obtain a parametric representation
            $$(u,v)mapstobigl(x_(u,v),y_(u,v),z_(u,v)bigr)in Esetminus(a,0,0) .$$






            share|cite|improve this answer









            $endgroup$

















              0












              $begingroup$

              A hint for the first one:



              Any point $P:=(0,u,v)$ in the $(y,z)$-plane determines a line $g_P:=Pvee A$, where $A:=(a,0,0)$. Intersecting
              $$g_P:quad tmapsto bigl((1-t) a,t u,t vbigr)qquad(-infty<t<infty)$$
              with the ellipsoid $E$ you get a quadratic equation for $t$ with one obvious solution $t=0$. The other solution leads you to the point $(x_P,y_P,z_P)in E$ stereographically related to $P$. All in all you will obtain a parametric representation
              $$(u,v)mapstobigl(x_(u,v),y_(u,v),z_(u,v)bigr)in Esetminus(a,0,0) .$$






              share|cite|improve this answer









              $endgroup$















                0












                0








                0





                $begingroup$

                A hint for the first one:



                Any point $P:=(0,u,v)$ in the $(y,z)$-plane determines a line $g_P:=Pvee A$, where $A:=(a,0,0)$. Intersecting
                $$g_P:quad tmapsto bigl((1-t) a,t u,t vbigr)qquad(-infty<t<infty)$$
                with the ellipsoid $E$ you get a quadratic equation for $t$ with one obvious solution $t=0$. The other solution leads you to the point $(x_P,y_P,z_P)in E$ stereographically related to $P$. All in all you will obtain a parametric representation
                $$(u,v)mapstobigl(x_(u,v),y_(u,v),z_(u,v)bigr)in Esetminus(a,0,0) .$$






                share|cite|improve this answer









                $endgroup$



                A hint for the first one:



                Any point $P:=(0,u,v)$ in the $(y,z)$-plane determines a line $g_P:=Pvee A$, where $A:=(a,0,0)$. Intersecting
                $$g_P:quad tmapsto bigl((1-t) a,t u,t vbigr)qquad(-infty<t<infty)$$
                with the ellipsoid $E$ you get a quadratic equation for $t$ with one obvious solution $t=0$. The other solution leads you to the point $(x_P,y_P,z_P)in E$ stereographically related to $P$. All in all you will obtain a parametric representation
                $$(u,v)mapstobigl(x_(u,v),y_(u,v),z_(u,v)bigr)in Esetminus(a,0,0) .$$







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Feb 5 '13 at 20:30









                Christian BlatterChristian Blatter

                175k8115327




                175k8115327



























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