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Parametric equation of a parabola rotating on its axis

Give the equation of the surfaceThe $y$ coordinate when rotating around the $x$-axis.Surface area of lateral section of paraboloidEquation of a coneWrite the parametric equation of the revolution surface generated by the line when it rotates around the axis $Oz$.Convert Surface of Revolution to Parametric EquationsRotating a conic section to form a 3d shapeEquation of a parabola, given the vertex and the axisParametric Equations of an Ellipsoidal HelixParametric equations for the surface of revolution generated by rotating the hyperbola $z^2−y^2=2$ about the $y$-axis

0

$begingroup$

Write the parametric equation of the surface generated by a parabola
rotating around its axis.

I guess it's simply getting from the parabola equation to the parametric equations of a generic paraboloid. But I don't know how to get to the param. equations of that surface of revolution.

Any help would be really appreciated.

geometry surfaces

share|cite|improve this question

asked Mar 10 at 20:03

Mandelbrot Jr.Mandelbrot Jr.

175

$endgroup$

add a comment | 

0

$begingroup$

Write the parametric equation of the surface generated by a parabola
rotating around its axis.

I guess it's simply getting from the parabola equation to the parametric equations of a generic paraboloid. But I don't know how to get to the param. equations of that surface of revolution.

Any help would be really appreciated.

geometry surfaces

share|cite|improve this question

asked Mar 10 at 20:03

Mandelbrot Jr.Mandelbrot Jr.

175

$endgroup$

add a comment | 

$begingroup$

Write the parametric equation of the surface generated by a parabola
rotating around its axis.

I guess it's simply getting from the parabola equation to the parametric equations of a generic paraboloid. But I don't know how to get to the param. equations of that surface of revolution.

Any help would be really appreciated.

geometry surfaces

share|cite|improve this question

asked Mar 10 at 20:03

Mandelbrot Jr.Mandelbrot Jr.

175

$endgroup$

share|cite|improve this question

asked Mar 10 at 20:03

Mandelbrot Jr.Mandelbrot Jr.

175

share|cite|improve this question

asked Mar 10 at 20:03

Mandelbrot Jr.Mandelbrot Jr.

175

asked Mar 10 at 20:03

Mandelbrot Jr.Mandelbrot Jr.

175

asked Mar 10 at 20:03

Mandelbrot Jr.Mandelbrot Jr.

175

1 Answer
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1

$begingroup$

Like in other cases where the former $x$ axis becomes a rotating axis, it is natural to swith from $x$ to $r$ and to name the fixed vertical axis $z$ in replacement of $y$, i.e., transform $y=x^2$ into $y=r^2$ i.e., finally :

$$z=x^2+y^2tag1$$

which is the cartesian equation of the paraboloid.

If you want parametric equations from (1), just take :

$$(x,y,z)=(x,y,x^2+y^2)tag2$$

but if one prefers to take polar coordinates in the horizontal plane $x=r cos theta, y=r sin theta$, we obtain another parametric set of equations :

$$(x,y,z)=(r cos theta, r sin theta, r^2)tag3$$

share|cite|improve this answer

edited Mar 10 at 20:59

answered Mar 10 at 20:50

Jean MarieJean Marie

30.7k42154

$endgroup$

add a comment | 

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1

$begingroup$

Like in other cases where the former $x$ axis becomes a rotating axis, it is natural to swith from $x$ to $r$ and to name the fixed vertical axis $z$ in replacement of $y$, i.e., transform $y=x^2$ into $y=r^2$ i.e., finally :

$$z=x^2+y^2tag1$$

which is the cartesian equation of the paraboloid.

If you want parametric equations from (1), just take :

$$(x,y,z)=(x,y,x^2+y^2)tag2$$

but if one prefers to take polar coordinates in the horizontal plane $x=r cos theta, y=r sin theta$, we obtain another parametric set of equations :

$$(x,y,z)=(r cos theta, r sin theta, r^2)tag3$$

share|cite|improve this answer

edited Mar 10 at 20:59

answered Mar 10 at 20:50

Jean MarieJean Marie

30.7k42154

$endgroup$

add a comment | 

1

$begingroup$

Like in other cases where the former $x$ axis becomes a rotating axis, it is natural to swith from $x$ to $r$ and to name the fixed vertical axis $z$ in replacement of $y$, i.e., transform $y=x^2$ into $y=r^2$ i.e., finally :

$$z=x^2+y^2tag1$$

which is the cartesian equation of the paraboloid.

If you want parametric equations from (1), just take :

$$(x,y,z)=(x,y,x^2+y^2)tag2$$

but if one prefers to take polar coordinates in the horizontal plane $x=r cos theta, y=r sin theta$, we obtain another parametric set of equations :

$$(x,y,z)=(r cos theta, r sin theta, r^2)tag3$$

share|cite|improve this answer

edited Mar 10 at 20:59

answered Mar 10 at 20:50

Jean MarieJean Marie

30.7k42154

$endgroup$

add a comment | 

$begingroup$

Like in other cases where the former $x$ axis becomes a rotating axis, it is natural to swith from $x$ to $r$ and to name the fixed vertical axis $z$ in replacement of $y$, i.e., transform $y=x^2$ into $y=r^2$ i.e., finally :

$$z=x^2+y^2tag1$$

which is the cartesian equation of the paraboloid.

If you want parametric equations from (1), just take :

$$(x,y,z)=(x,y,x^2+y^2)tag2$$

but if one prefers to take polar coordinates in the horizontal plane $x=r cos theta, y=r sin theta$, we obtain another parametric set of equations :

$$(x,y,z)=(r cos theta, r sin theta, r^2)tag3$$

share|cite|improve this answer

edited Mar 10 at 20:59

answered Mar 10 at 20:50

Jean MarieJean Marie

30.7k42154

$endgroup$

share|cite|improve this answer

edited Mar 10 at 20:59

answered Mar 10 at 20:50

Jean MarieJean Marie

30.7k42154

answered Mar 10 at 20:50

Jean MarieJean Marie

30.7k42154

answered Mar 10 at 20:50

Jean MarieJean Marie

30.7k42154