Ellipse on a Circular Cylinder in Cylindrical Coordinates












1














Is there an equation in cylindrical coordinates for an ellipse (tilted at some angle) on the surface of a right circular cylinder of radius r? For simplicity, I envision the cylinder to be coincident with the x-axis.



I am aware that the cylinder could be "unwrapped" into a plane, which would result in the ellipse becoming a sine curve. I am just not sure how that information ties into cylindrical coordinates.



EDIT: I have realized that I am looking for a parametric equation. For an ellipse on the surface of a cylinder of radius r which has a certain angle of inclination, is there a parameterization where I can calculate the axial coordinate seperately from the azimuth angle for a given t?



Thank you.



Ellipse on a cylinder










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  • You need a pair of implicit Cartesian equations to describe a curve in 3D. If you want a single equation, it’s not too hard to come up with a parameterization of the curve.
    – amd
    Nov 28 '18 at 21:58










  • That is a very good point; In my mind I am looking for a parameterization of the curve that takes into account the angle of inclination, but I failed to put that in my original question. For a given cylinder radius, I believe I need two parameters: an expression for the axial coordinate and an expression for the azimuth. Could you help me with that parameterization of the ellipse?
    – kreeser1
    Nov 28 '18 at 23:35










  • You can choose a cylindrical coordinate system $(rho,phi,z)$ such that the ellipse is $rho = r, z = r tanalpha sinphi$, where $alpha$ is the angle between the plane normal and cylinder axis. You’ve already accepted an answer, but I can expand on this if you like.
    – amd
    Nov 30 '18 at 22:36


















1














Is there an equation in cylindrical coordinates for an ellipse (tilted at some angle) on the surface of a right circular cylinder of radius r? For simplicity, I envision the cylinder to be coincident with the x-axis.



I am aware that the cylinder could be "unwrapped" into a plane, which would result in the ellipse becoming a sine curve. I am just not sure how that information ties into cylindrical coordinates.



EDIT: I have realized that I am looking for a parametric equation. For an ellipse on the surface of a cylinder of radius r which has a certain angle of inclination, is there a parameterization where I can calculate the axial coordinate seperately from the azimuth angle for a given t?



Thank you.



Ellipse on a cylinder










share|cite|improve this question
























  • You need a pair of implicit Cartesian equations to describe a curve in 3D. If you want a single equation, it’s not too hard to come up with a parameterization of the curve.
    – amd
    Nov 28 '18 at 21:58










  • That is a very good point; In my mind I am looking for a parameterization of the curve that takes into account the angle of inclination, but I failed to put that in my original question. For a given cylinder radius, I believe I need two parameters: an expression for the axial coordinate and an expression for the azimuth. Could you help me with that parameterization of the ellipse?
    – kreeser1
    Nov 28 '18 at 23:35










  • You can choose a cylindrical coordinate system $(rho,phi,z)$ such that the ellipse is $rho = r, z = r tanalpha sinphi$, where $alpha$ is the angle between the plane normal and cylinder axis. You’ve already accepted an answer, but I can expand on this if you like.
    – amd
    Nov 30 '18 at 22:36
















1












1








1







Is there an equation in cylindrical coordinates for an ellipse (tilted at some angle) on the surface of a right circular cylinder of radius r? For simplicity, I envision the cylinder to be coincident with the x-axis.



I am aware that the cylinder could be "unwrapped" into a plane, which would result in the ellipse becoming a sine curve. I am just not sure how that information ties into cylindrical coordinates.



EDIT: I have realized that I am looking for a parametric equation. For an ellipse on the surface of a cylinder of radius r which has a certain angle of inclination, is there a parameterization where I can calculate the axial coordinate seperately from the azimuth angle for a given t?



Thank you.



Ellipse on a cylinder










share|cite|improve this question















Is there an equation in cylindrical coordinates for an ellipse (tilted at some angle) on the surface of a right circular cylinder of radius r? For simplicity, I envision the cylinder to be coincident with the x-axis.



I am aware that the cylinder could be "unwrapped" into a plane, which would result in the ellipse becoming a sine curve. I am just not sure how that information ties into cylindrical coordinates.



EDIT: I have realized that I am looking for a parametric equation. For an ellipse on the surface of a cylinder of radius r which has a certain angle of inclination, is there a parameterization where I can calculate the axial coordinate seperately from the azimuth angle for a given t?



Thank you.



Ellipse on a cylinder







conic-sections






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edited Nov 28 '18 at 23:39







kreeser1

















asked Nov 28 '18 at 17:36









kreeser1kreeser1

10110




10110












  • You need a pair of implicit Cartesian equations to describe a curve in 3D. If you want a single equation, it’s not too hard to come up with a parameterization of the curve.
    – amd
    Nov 28 '18 at 21:58










  • That is a very good point; In my mind I am looking for a parameterization of the curve that takes into account the angle of inclination, but I failed to put that in my original question. For a given cylinder radius, I believe I need two parameters: an expression for the axial coordinate and an expression for the azimuth. Could you help me with that parameterization of the ellipse?
    – kreeser1
    Nov 28 '18 at 23:35










  • You can choose a cylindrical coordinate system $(rho,phi,z)$ such that the ellipse is $rho = r, z = r tanalpha sinphi$, where $alpha$ is the angle between the plane normal and cylinder axis. You’ve already accepted an answer, but I can expand on this if you like.
    – amd
    Nov 30 '18 at 22:36




















  • You need a pair of implicit Cartesian equations to describe a curve in 3D. If you want a single equation, it’s not too hard to come up with a parameterization of the curve.
    – amd
    Nov 28 '18 at 21:58










  • That is a very good point; In my mind I am looking for a parameterization of the curve that takes into account the angle of inclination, but I failed to put that in my original question. For a given cylinder radius, I believe I need two parameters: an expression for the axial coordinate and an expression for the azimuth. Could you help me with that parameterization of the ellipse?
    – kreeser1
    Nov 28 '18 at 23:35










  • You can choose a cylindrical coordinate system $(rho,phi,z)$ such that the ellipse is $rho = r, z = r tanalpha sinphi$, where $alpha$ is the angle between the plane normal and cylinder axis. You’ve already accepted an answer, but I can expand on this if you like.
    – amd
    Nov 30 '18 at 22:36


















You need a pair of implicit Cartesian equations to describe a curve in 3D. If you want a single equation, it’s not too hard to come up with a parameterization of the curve.
– amd
Nov 28 '18 at 21:58




You need a pair of implicit Cartesian equations to describe a curve in 3D. If you want a single equation, it’s not too hard to come up with a parameterization of the curve.
– amd
Nov 28 '18 at 21:58












That is a very good point; In my mind I am looking for a parameterization of the curve that takes into account the angle of inclination, but I failed to put that in my original question. For a given cylinder radius, I believe I need two parameters: an expression for the axial coordinate and an expression for the azimuth. Could you help me with that parameterization of the ellipse?
– kreeser1
Nov 28 '18 at 23:35




That is a very good point; In my mind I am looking for a parameterization of the curve that takes into account the angle of inclination, but I failed to put that in my original question. For a given cylinder radius, I believe I need two parameters: an expression for the axial coordinate and an expression for the azimuth. Could you help me with that parameterization of the ellipse?
– kreeser1
Nov 28 '18 at 23:35












You can choose a cylindrical coordinate system $(rho,phi,z)$ such that the ellipse is $rho = r, z = r tanalpha sinphi$, where $alpha$ is the angle between the plane normal and cylinder axis. You’ve already accepted an answer, but I can expand on this if you like.
– amd
Nov 30 '18 at 22:36






You can choose a cylindrical coordinate system $(rho,phi,z)$ such that the ellipse is $rho = r, z = r tanalpha sinphi$, where $alpha$ is the angle between the plane normal and cylinder axis. You’ve already accepted an answer, but I can expand on this if you like.
– amd
Nov 30 '18 at 22:36












2 Answers
2






active

oldest

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2














For greater simplicity, is better that cylinder axis $=$ $z$-axis. Let be $R$ the cylinder radius, $ax + by + cz + d = 0$ the plane containing the ellipse. Then, $r = R$ and
$$a Rcostheta + b Rsintheta + cz = d.$$






share|cite|improve this answer





















  • I think that it’s worth emphasizing that this is a pair of implicit Cartesian equations. Without the constraint $r=R$, you get an intersecting-looking surface, not an ellipse.
    – amd
    Nov 28 '18 at 21:57










  • @amd, correct. The ellipse is the intersection of the cylinder $x^2 + y^2 = R^2$ and the plane $ax+by+cz+d=0$.
    – Martín-Blas Pérez Pinilla
    Nov 29 '18 at 15:39










  • That was meant to be “interesting,” not “intersecting.” My fingers have minds of their own at times.
    – amd
    Nov 29 '18 at 19:27



















0














As stated before assume the (normalized) equation of the plane is



$$ frac{a x + b y + c z}{sqrt{a^2+b^2+c^2}} = d $$



and the parametric equation of the cylinder



$$ pmatrix{x & y & z}= f(varphi,z) = pmatrix{ R cosvarphi, Rsinvarphi, z} $$



Where the two intersect you have your ellipse in cartesian coordinates



$$ z(varphi) = frac{d sqrt{a^2+b^2+c^2}-R (a cosvarphi+b sinvarphi) }{c} $$



or



$$vec{r}_{rm curve}(varphi) = pmatrix{x\y\z} = pmatrix{ R cosvarphi \ R sinvarphi \ frac{d sqrt{a^2+b^2+c^2}-R (a cosvarphi+b sinvarphi) }{c} } $$



Now let's find the properties of this ellipse.



The center of the ellipse is at $vec{r}_{rm cen} = pmatrix{0 & 0 & frac{d sqrt{a^2+b^2+c^2}}{c}} $



The ellipse in polar coordinates is



$$ r(varphi) = | vec{r}_{rm curve}-vec{r}_{rm cen} | = sqrt{R^2 + frac{R^2}{c^2} left( frac{a^2+b^2}{2} + a b sin(2 varphi) + frac{a^2-b^2}{2} cos(2varphi) right)} $$



This allows us to find the major and minor radii



$$ begin{aligned}
r_{rm major} & = R frac{ sqrt{a^2+b^2+c^2}}{c} \
r_{rm minor} & = R
end{aligned} $$



The principal axes of the ellipse are on



$$ varphi = frac{1}{2} {rm atan}left( frac{2 a b}{a^2-b^2} right) + n frac{pi}{2} $$






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    2 Answers
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    active

    oldest

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    2 Answers
    2






    active

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    active

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    active

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    2














    For greater simplicity, is better that cylinder axis $=$ $z$-axis. Let be $R$ the cylinder radius, $ax + by + cz + d = 0$ the plane containing the ellipse. Then, $r = R$ and
    $$a Rcostheta + b Rsintheta + cz = d.$$






    share|cite|improve this answer





















    • I think that it’s worth emphasizing that this is a pair of implicit Cartesian equations. Without the constraint $r=R$, you get an intersecting-looking surface, not an ellipse.
      – amd
      Nov 28 '18 at 21:57










    • @amd, correct. The ellipse is the intersection of the cylinder $x^2 + y^2 = R^2$ and the plane $ax+by+cz+d=0$.
      – Martín-Blas Pérez Pinilla
      Nov 29 '18 at 15:39










    • That was meant to be “interesting,” not “intersecting.” My fingers have minds of their own at times.
      – amd
      Nov 29 '18 at 19:27
















    2














    For greater simplicity, is better that cylinder axis $=$ $z$-axis. Let be $R$ the cylinder radius, $ax + by + cz + d = 0$ the plane containing the ellipse. Then, $r = R$ and
    $$a Rcostheta + b Rsintheta + cz = d.$$






    share|cite|improve this answer





















    • I think that it’s worth emphasizing that this is a pair of implicit Cartesian equations. Without the constraint $r=R$, you get an intersecting-looking surface, not an ellipse.
      – amd
      Nov 28 '18 at 21:57










    • @amd, correct. The ellipse is the intersection of the cylinder $x^2 + y^2 = R^2$ and the plane $ax+by+cz+d=0$.
      – Martín-Blas Pérez Pinilla
      Nov 29 '18 at 15:39










    • That was meant to be “interesting,” not “intersecting.” My fingers have minds of their own at times.
      – amd
      Nov 29 '18 at 19:27














    2












    2








    2






    For greater simplicity, is better that cylinder axis $=$ $z$-axis. Let be $R$ the cylinder radius, $ax + by + cz + d = 0$ the plane containing the ellipse. Then, $r = R$ and
    $$a Rcostheta + b Rsintheta + cz = d.$$






    share|cite|improve this answer












    For greater simplicity, is better that cylinder axis $=$ $z$-axis. Let be $R$ the cylinder radius, $ax + by + cz + d = 0$ the plane containing the ellipse. Then, $r = R$ and
    $$a Rcostheta + b Rsintheta + cz = d.$$







    share|cite|improve this answer












    share|cite|improve this answer



    share|cite|improve this answer










    answered Nov 28 '18 at 17:46









    Martín-Blas Pérez PinillaMartín-Blas Pérez Pinilla

    34.1k42771




    34.1k42771












    • I think that it’s worth emphasizing that this is a pair of implicit Cartesian equations. Without the constraint $r=R$, you get an intersecting-looking surface, not an ellipse.
      – amd
      Nov 28 '18 at 21:57










    • @amd, correct. The ellipse is the intersection of the cylinder $x^2 + y^2 = R^2$ and the plane $ax+by+cz+d=0$.
      – Martín-Blas Pérez Pinilla
      Nov 29 '18 at 15:39










    • That was meant to be “interesting,” not “intersecting.” My fingers have minds of their own at times.
      – amd
      Nov 29 '18 at 19:27


















    • I think that it’s worth emphasizing that this is a pair of implicit Cartesian equations. Without the constraint $r=R$, you get an intersecting-looking surface, not an ellipse.
      – amd
      Nov 28 '18 at 21:57










    • @amd, correct. The ellipse is the intersection of the cylinder $x^2 + y^2 = R^2$ and the plane $ax+by+cz+d=0$.
      – Martín-Blas Pérez Pinilla
      Nov 29 '18 at 15:39










    • That was meant to be “interesting,” not “intersecting.” My fingers have minds of their own at times.
      – amd
      Nov 29 '18 at 19:27
















    I think that it’s worth emphasizing that this is a pair of implicit Cartesian equations. Without the constraint $r=R$, you get an intersecting-looking surface, not an ellipse.
    – amd
    Nov 28 '18 at 21:57




    I think that it’s worth emphasizing that this is a pair of implicit Cartesian equations. Without the constraint $r=R$, you get an intersecting-looking surface, not an ellipse.
    – amd
    Nov 28 '18 at 21:57












    @amd, correct. The ellipse is the intersection of the cylinder $x^2 + y^2 = R^2$ and the plane $ax+by+cz+d=0$.
    – Martín-Blas Pérez Pinilla
    Nov 29 '18 at 15:39




    @amd, correct. The ellipse is the intersection of the cylinder $x^2 + y^2 = R^2$ and the plane $ax+by+cz+d=0$.
    – Martín-Blas Pérez Pinilla
    Nov 29 '18 at 15:39












    That was meant to be “interesting,” not “intersecting.” My fingers have minds of their own at times.
    – amd
    Nov 29 '18 at 19:27




    That was meant to be “interesting,” not “intersecting.” My fingers have minds of their own at times.
    – amd
    Nov 29 '18 at 19:27











    0














    As stated before assume the (normalized) equation of the plane is



    $$ frac{a x + b y + c z}{sqrt{a^2+b^2+c^2}} = d $$



    and the parametric equation of the cylinder



    $$ pmatrix{x & y & z}= f(varphi,z) = pmatrix{ R cosvarphi, Rsinvarphi, z} $$



    Where the two intersect you have your ellipse in cartesian coordinates



    $$ z(varphi) = frac{d sqrt{a^2+b^2+c^2}-R (a cosvarphi+b sinvarphi) }{c} $$



    or



    $$vec{r}_{rm curve}(varphi) = pmatrix{x\y\z} = pmatrix{ R cosvarphi \ R sinvarphi \ frac{d sqrt{a^2+b^2+c^2}-R (a cosvarphi+b sinvarphi) }{c} } $$



    Now let's find the properties of this ellipse.



    The center of the ellipse is at $vec{r}_{rm cen} = pmatrix{0 & 0 & frac{d sqrt{a^2+b^2+c^2}}{c}} $



    The ellipse in polar coordinates is



    $$ r(varphi) = | vec{r}_{rm curve}-vec{r}_{rm cen} | = sqrt{R^2 + frac{R^2}{c^2} left( frac{a^2+b^2}{2} + a b sin(2 varphi) + frac{a^2-b^2}{2} cos(2varphi) right)} $$



    This allows us to find the major and minor radii



    $$ begin{aligned}
    r_{rm major} & = R frac{ sqrt{a^2+b^2+c^2}}{c} \
    r_{rm minor} & = R
    end{aligned} $$



    The principal axes of the ellipse are on



    $$ varphi = frac{1}{2} {rm atan}left( frac{2 a b}{a^2-b^2} right) + n frac{pi}{2} $$






    share|cite|improve this answer


























      0














      As stated before assume the (normalized) equation of the plane is



      $$ frac{a x + b y + c z}{sqrt{a^2+b^2+c^2}} = d $$



      and the parametric equation of the cylinder



      $$ pmatrix{x & y & z}= f(varphi,z) = pmatrix{ R cosvarphi, Rsinvarphi, z} $$



      Where the two intersect you have your ellipse in cartesian coordinates



      $$ z(varphi) = frac{d sqrt{a^2+b^2+c^2}-R (a cosvarphi+b sinvarphi) }{c} $$



      or



      $$vec{r}_{rm curve}(varphi) = pmatrix{x\y\z} = pmatrix{ R cosvarphi \ R sinvarphi \ frac{d sqrt{a^2+b^2+c^2}-R (a cosvarphi+b sinvarphi) }{c} } $$



      Now let's find the properties of this ellipse.



      The center of the ellipse is at $vec{r}_{rm cen} = pmatrix{0 & 0 & frac{d sqrt{a^2+b^2+c^2}}{c}} $



      The ellipse in polar coordinates is



      $$ r(varphi) = | vec{r}_{rm curve}-vec{r}_{rm cen} | = sqrt{R^2 + frac{R^2}{c^2} left( frac{a^2+b^2}{2} + a b sin(2 varphi) + frac{a^2-b^2}{2} cos(2varphi) right)} $$



      This allows us to find the major and minor radii



      $$ begin{aligned}
      r_{rm major} & = R frac{ sqrt{a^2+b^2+c^2}}{c} \
      r_{rm minor} & = R
      end{aligned} $$



      The principal axes of the ellipse are on



      $$ varphi = frac{1}{2} {rm atan}left( frac{2 a b}{a^2-b^2} right) + n frac{pi}{2} $$






      share|cite|improve this answer
























        0












        0








        0






        As stated before assume the (normalized) equation of the plane is



        $$ frac{a x + b y + c z}{sqrt{a^2+b^2+c^2}} = d $$



        and the parametric equation of the cylinder



        $$ pmatrix{x & y & z}= f(varphi,z) = pmatrix{ R cosvarphi, Rsinvarphi, z} $$



        Where the two intersect you have your ellipse in cartesian coordinates



        $$ z(varphi) = frac{d sqrt{a^2+b^2+c^2}-R (a cosvarphi+b sinvarphi) }{c} $$



        or



        $$vec{r}_{rm curve}(varphi) = pmatrix{x\y\z} = pmatrix{ R cosvarphi \ R sinvarphi \ frac{d sqrt{a^2+b^2+c^2}-R (a cosvarphi+b sinvarphi) }{c} } $$



        Now let's find the properties of this ellipse.



        The center of the ellipse is at $vec{r}_{rm cen} = pmatrix{0 & 0 & frac{d sqrt{a^2+b^2+c^2}}{c}} $



        The ellipse in polar coordinates is



        $$ r(varphi) = | vec{r}_{rm curve}-vec{r}_{rm cen} | = sqrt{R^2 + frac{R^2}{c^2} left( frac{a^2+b^2}{2} + a b sin(2 varphi) + frac{a^2-b^2}{2} cos(2varphi) right)} $$



        This allows us to find the major and minor radii



        $$ begin{aligned}
        r_{rm major} & = R frac{ sqrt{a^2+b^2+c^2}}{c} \
        r_{rm minor} & = R
        end{aligned} $$



        The principal axes of the ellipse are on



        $$ varphi = frac{1}{2} {rm atan}left( frac{2 a b}{a^2-b^2} right) + n frac{pi}{2} $$






        share|cite|improve this answer












        As stated before assume the (normalized) equation of the plane is



        $$ frac{a x + b y + c z}{sqrt{a^2+b^2+c^2}} = d $$



        and the parametric equation of the cylinder



        $$ pmatrix{x & y & z}= f(varphi,z) = pmatrix{ R cosvarphi, Rsinvarphi, z} $$



        Where the two intersect you have your ellipse in cartesian coordinates



        $$ z(varphi) = frac{d sqrt{a^2+b^2+c^2}-R (a cosvarphi+b sinvarphi) }{c} $$



        or



        $$vec{r}_{rm curve}(varphi) = pmatrix{x\y\z} = pmatrix{ R cosvarphi \ R sinvarphi \ frac{d sqrt{a^2+b^2+c^2}-R (a cosvarphi+b sinvarphi) }{c} } $$



        Now let's find the properties of this ellipse.



        The center of the ellipse is at $vec{r}_{rm cen} = pmatrix{0 & 0 & frac{d sqrt{a^2+b^2+c^2}}{c}} $



        The ellipse in polar coordinates is



        $$ r(varphi) = | vec{r}_{rm curve}-vec{r}_{rm cen} | = sqrt{R^2 + frac{R^2}{c^2} left( frac{a^2+b^2}{2} + a b sin(2 varphi) + frac{a^2-b^2}{2} cos(2varphi) right)} $$



        This allows us to find the major and minor radii



        $$ begin{aligned}
        r_{rm major} & = R frac{ sqrt{a^2+b^2+c^2}}{c} \
        r_{rm minor} & = R
        end{aligned} $$



        The principal axes of the ellipse are on



        $$ varphi = frac{1}{2} {rm atan}left( frac{2 a b}{a^2-b^2} right) + n frac{pi}{2} $$







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Dec 1 '18 at 0:22









        ja72ja72

        7,42212044




        7,42212044






























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