March 16th, 2009, 02:47 PM  #1 
Member Joined: Mar 2009 From: San Bernardino, California Posts: 50 Thanks: 0  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 xyplane 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.

March 17th, 2009, 12:23 AM  #2 
Senior Member Joined: Jan 2009 From: Japan Posts: 192 Thanks: 0  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.

March 17th, 2009, 12:42 AM  #3 
Senior Member Joined: Feb 2009 From: Adelaide, Australia Posts: 1,519 Thanks: 3  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 crosssections though. Set x, y, z to be zero in turn and see what shape you have. 
March 17th, 2009, 11:21 AM  #4 
Member Joined: Mar 2009 From: San Bernardino, California Posts: 50 Thanks: 0  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?

March 17th, 2009, 12:44 PM  #5 
Senior Member Joined: Jan 2009 From: Japan Posts: 192 Thanks: 0  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.

March 17th, 2009, 01:21 PM  #6 
Member Joined: Mar 2009 From: San Bernardino, California Posts: 50 Thanks: 0  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 threedimensional 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 xyplane (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 zaxis. Please keep helping me, I really appreciate it I am really trying to get 3D space correct in my head.

March 17th, 2009, 01:25 PM  #7 
Senior Member Joined: Jan 2009 From: Japan Posts: 192 Thanks: 0  Re: Three Dimensional Surface trick?
This is why knowing more than one of the crosssections 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.

March 17th, 2009, 01:48 PM  #8 
Member Joined: Mar 2009 From: San Bernardino, California Posts: 50 Thanks: 0  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 zaxis. 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 threedimensional 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. 
March 17th, 2009, 02:38 PM  #9 
Senior Member Joined: Jan 2009 From: Japan Posts: 192 Thanks: 0  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 twodimensional 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 crosssections 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 crosssection in the xz and yz planes. 
March 17th, 2009, 03:24 PM  #10 
Member Joined: Mar 2009 From: San Bernardino, California Posts: 50 Thanks: 0  Re: Three Dimensional Surface trick?
Ah wow thank you so much things are now instaclear. 

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