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March 16th, 2009, 02:47 PM   #1
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Three Dimensional Surface trick?

Right now I am reviewing for a midterm and trying to memorize the equations for the quadric surfaces like the ellipsoid, elliptic cone, and parabolic cylinder but I also recall my professor mentioning a "math trick" that can give you the equation of the three dimensional surface given the equation of its two dimensional xy-plane based counterparts (eg. use the equation of the ellipse to find the equation of an ellipsoid). But unfortunately I can not find it in my notes anywhere. The only thing I can remember from it is something along the lines of replacing x with or something like that. Does anyone have an idea of what the trick is? I'd appreciate the help if anyone knows it so that I can just derive these surface formulas rather than memorizing them.
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March 17th, 2009, 12:23 AM   #2
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Re: Three Dimensional Surface trick?

It sounds like you're talking about a surface of revolution. This can be found by revolving a closed curve and considering the surface swept out. If you're more specific about what you want, we can be more helpful.
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March 17th, 2009, 12:42 AM   #3
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Re: Three Dimensional Surface trick?

Given an even function y=f(x), you can sometimes quickly make a surface of revolution by replacing y with z, and x with polar r = .

Parabola:



Elliptical (well, circular) paraboloid:



Hyperbola:



Hyperboloid of one sheet:



I don't see how this is easier than taking cross-sections though. Set x, y, z to be zero in turn and see what shape you have.
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March 17th, 2009, 11:21 AM   #4
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Re: Three Dimensional Surface trick?

That's exactly what my teacher proposed. As for setting x, y, and z equal to zero, doesn't that just give you the "cylinder" with the base in the shape of the cross section? Or am I missing the point you are trying to make?
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March 17th, 2009, 12:44 PM   #5
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Re: Three Dimensional Surface trick?

Setting x, y, and z to zero 'in turn' would produce cross sections in the yz, xz, and xy planes respectively.
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March 17th, 2009, 01:21 PM   #6
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Re: Three Dimensional Surface trick?

Well yes I see that the cross section in the coordinate planes would constitute the the equation of the surface with the "missing" variable in relation to the particular coordinate plane but I thought that in three-dimensional space, that those particular cross section equations from 2D represent a cylindrical shape based off of that cross section? For example the equation of an ellipse in the xy-plane (your standard ellipse equation) in three dimensional space represents an elliptical cylinder because with z missing, z varies indefinitely and the shape is technically an ellipse "dragged" up and down the z-axis. Please keep helping me, I really appreciate it I am really trying to get 3D space correct in my head.
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March 17th, 2009, 01:25 PM   #7
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Re: Three Dimensional Surface trick?

This is why knowing more than one of the cross-sections can be helpful. One possibility for generating a 3 dimensional surface from an ellipse would indeed be to 'drag it up'. You could, instead, spin it around an axis, and this would produce an ellipsoid. Imagine a spinning coin; with a circular cross section, it produces a sphere by revolution.
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March 17th, 2009, 01:48 PM   #8
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Re: Three Dimensional Surface trick?

So correct me if I am wrong but:

Elliptic Cylinder: , because z varies indefinitely and is therefore and ellipse "dragged" along the z-axis.

Ellipsoid: , because an ellipsoid is and ellipse rotated in space whose size depends on x and y and there fore the size of z depends on x and y.


Assuming those are correct thus far, then what is the three-dimensional equation of the cross section of each of these with a plane, an ellipse??? In the case that z is not present (elliptic cylinder) then it must vary indefinitely and therefore form a cylinder with the base of the cross section. In the case that z is limited in terms of x and y (ellipsoid) then it forms a surface analog of the cross section. My guess is that since z=0 for an xy cross section then the equation of an ellipse is
but that is an elliptic cylinder.
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March 17th, 2009, 02:38 PM   #9
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Re: Three Dimensional Surface trick?

You're talking about two different types of equations here and not making the distinction clear. There's a qualitative difference between "for all z,
" and "for z = 0" the same equation. The first is a relationship on and the second on . The two-dimensional equivalent would be saying that is a horizontal line since at x = 0, y = 1. Given only "when x = 0, y = 1", there are an uncountably infinite number of functions which fulfill this condition.

An ellipsoid centered at the origin will have elliptical cross-sections in the xy, xz, and yz planes. A cylinder with an elliptical base centered at the origin in the xy plane will have two parallel lines as a cross-section in the xz and yz planes.
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March 17th, 2009, 03:24 PM   #10
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Re: Three Dimensional Surface trick?

Ah wow thank you so much things are now insta-clear.
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