Plotting a 3-d surface in spherical co-ordinates

In summary, the conversation discusses an equation of a 3-d surface in spherical coordinates and the process of plotting it on a 3-d space. The equation is converted into Cartesian coordinates and the resulting equation is discussed. The conversation also touches upon the projection of the surface to the y=0 plane and the type of curve that is formed when the surface is rotated around the z axis. The curve is identified as the radiation pattern of a Hertz Dipole Antenna.
  • #1
ask_LXXXVI
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Homework Statement


The following is an equation of a 3-d surface in spherical co-ordinates.
r = f(theta,phi) = sin(theta)
Plot it on 3-d space.

Homework Equations


r = [tex]\sqrt{x^2 + y^2 + z^2 }[/tex]
theta = arccos (z/ [tex]\sqrt{x^2 + y^2 + z^2 }[/tex])

The Attempt at a Solution


I converted the equation of the surface into cartesian co-ordinates , and got the following equation of a 3-d surface

(x^2 + y^2 + z^2)^2 = (x^2 + y^2)
is the above equation in cartesian co-ordinates right ?
and is it a right approach to plot it?

I think the plot is as shown in image given below ,
but how to arrive at the plot given below I am unable to think.Please help

[PLAIN]http://img689.imageshack.us/img689/7079/capturevug.jpg
 
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  • #2
The surface has axial symmetry around the z axis. What is the projection of the surface to the y=0 plane? You get the 3D surface by rotating this curve around z. What kind of curve is this? ehild
 
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  • #3
ehild said:
The surface has axial symmetry around the z axis. What is the projection of the surface to the y=0 plane? You get the 3D surface by rotating this curve around z.


ehild

Thanks for helping me in visualising how to plot the curve.

ehild said:
What kind of curve is this?


ehild

It is the radiation pattern of a Hertz Dipole Antenna.For more you can readhttp://books.google.co.in/books?id=...R K shevgaonkar&pg=PA379#v=onepage&q&f=false"from google books
 
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1. How do I convert cartesian coordinates to spherical coordinates?

To convert from cartesian coordinates (x, y, z) to spherical coordinates (r, θ, φ), you can use the following equations:
r = √(x² + y² + z²)
θ = arctan(y/x)
φ = arccos(z/r)

2. What is the equation for a 3-d surface in spherical coordinates?

The equation for a 3-d surface in spherical coordinates is ρ = f(θ, φ), where ρ represents the distance from the origin, θ represents the azimuthal angle, and φ represents the polar angle.

3. How do I plot a 3-d surface in spherical coordinates using software?

To plot a 3-d surface in spherical coordinates, you can use software such as MATLAB or Mathematica. These programs have built-in functions for plotting 3-d surfaces in spherical coordinates, such as sphereplot in MATLAB and SphericalPlot3D in Mathematica.

4. How does the shape of a surface change when plotted in spherical coordinates compared to cartesian coordinates?

The shape of a surface can appear different when plotted in spherical coordinates compared to cartesian coordinates. This is because spherical coordinates represent distance from the origin and angles, while cartesian coordinates represent distance along the x, y, and z axes. For example, a flat plane in cartesian coordinates may appear as a curved surface in spherical coordinates.

5. Can I plot a 3-d surface in spherical coordinates without using software?

Yes, it is possible to plot a 3-d surface in spherical coordinates without using software. This can be done by hand using a graphing calculator or by creating a 3-d model using physical materials. However, using software can make the process easier and more accurate.

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