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Math Help - Sketching Quadric Surfaces

  1. #1
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    Sketching Quadric Surfaces

    I don't struggle so much with the analytical side of things, but when it comes to visualizing and sketching 3D shapes, I have trouble conceptualizing the darn things. There's an array of problems in the section our class is currently covering that involves sketching the region bounded by the graphs of given equations.

    I have Maple 11 student version, but I haven't learned much in the way of using it. I figured I would try to plot the quadric surfaces in Maple to get a better understanding, but that has turned out to be more pleasant in theory than in practice. Trying to learn how to use this software has been costing me more time than actually just plowing through the material.

    Anyway, the actual problem in question I can't seem to understand involves two equations:


    z = \sqrt{4-x^2}, y = \sqrt{4-x^2}, x=0, y=0, z=0

    I'm supposed to sketch the region bound by these equations. I set either equation equal to zero and end up with x=2, and that's where I hit the wall. Any insight is greatly appreciated.
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by LongTermStudent View Post
    I don't struggle so much with the analytical side of things, but when it comes to visualizing and sketching 3D shapes, I have trouble conceptualizing the darn things. There's an array of problems in the section our class is currently covering that involves sketching the region bounded by the graphs of given equations.

    I have Maple 11 student version, but I haven't learned much in the way of using it. I figured I would try to plot the quadric surfaces in Maple to get a better understanding, but that has turned out to be more pleasant in theory than in practice. Trying to learn how to use this software has been costing me more time than actually just plowing through the material.

    Anyway, the actual problem in question I can't seem to understand involves two equations:


    z = \sqrt{4-x^2}, y = \sqrt{4-x^2}, x=0, y=0, z=0

    I'm supposed to sketch the region bound by these equations. I set either equation equal to zero and end up with x=2, and that's where I hit the wall. Any insight is greatly appreciated.
    Note that z=\sqrt{4-x^2} is a cylinder along the y axis, and y=\sqrt{4-x^2} is a cylinder along the z axis.

    Can you try to plot it now?

    --Chris
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  3. #3
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    Ok, what I sketched on my paper is a cylinder on the z-axis and one on the y-axis, and they meet at the origin. It looks kind of like an L-shaped tube (I only went in the positive direction). I would imagine that the radius of the cylinder is 2. Am I on the right track?
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  4. #4
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by LongTermStudent View Post
    Ok, what I sketched on my paper is a cylinder on the z-axis and one on the y-axis, and they meet at the origin. It looks kind of like an L-shaped tube (I only went in the positive direction). I would imagine that the radius of the cylinder is 2. Am I on the right track?
    It seems like you're on the right track.

    Isn't it hard to imagine this stuff? I'm having a hard time myself!

    There is a possibility I did this wrong...but this is what I got it to look like:



    I hope this helps.

    --Chris
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  5. #5
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    Quote Originally Posted by Chris L T521 View Post
    Isn't it hard to imagine this stuff? I'm having a hard time myself!

    You could say that again. My professor says that we're not being tested on our ability to sketch quadric surfaces, but I just know that being able to visualize them will be key in problems we're going to have further down the road. I had a chance to scan what I have. It doesn't look at all like what you have. Is there something I'm leaving out? I don't recall incorporating x=0, y=0, z=0 anywhere in my sketching.

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  6. #6
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    This is what it looks like to me assuming the x=0, y=0,z=0 are planes on the coordinate axes:
    Attached Thumbnails Attached Thumbnails Sketching Quadric Surfaces-cylinders.jpg  
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  7. #7
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by shawsend View Post
    This is what it looks like to me assuming the x=0, y=0,z=0 are planes on the coordinate axes:
    One of the cylinders are correct [the one along the y axis], but the other cylinder you have is along the x-axis, when it should be along the z axis.

    BTW, how did you do that in Mathematica? I'm still trying to figure out how to graph something like that using this software...

    --Chris
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  8. #8
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    Ok. Thanks. I think this is it then. It's the best I can do for now: I'm unable to remove that top section of the curve so I just made it transparent (used Opacity[0.4]). Here's the code:

    Code:
    poly1 = Graphics3D[Polygon[{{3, 3, 0}, 
         {-3, 3, 0}, {-3, -3, 0}, {3, -3, 0}}]]
    poly2 = Graphics3D[Polygon[{{0, 3, 3}, 
         {0, -3, 3}, {0, -3, -3}, {0, 3, -3}}]]
    poly3 = Graphics3D[Polygon[{{3, 0, 3}, 
         {-3, 0, 3}, {-3, 0, -3}, {3, 0, -3}}]]
    p1 = Plot3D[{Sqrt[4 - x^2]}, {x, 0, 2}, 
       {y, -2, 0}, BoxRatios -> {1, 1, 1}, 
       PlotRange -> {{-3, 3}, {-3, 3}, {-3, 3}}, 
       RegionFunction -> Function[{x, y}, 
         Abs[y] <= Sqrt[4 - x^2]]]
    cp1 = ParametricPlot3D[{t, -Sqrt[4 - t^2], 
        Sqrt[4 - z^2]}, {t, 0, 2}, {z, 0, 2}, 
       PlotStyle -> Opacity[0.4]]
    Show[{p1, cp1, poly1, poly2, poly3}, 
      PlotRange -> {{-3, 3}, {-3, 3}, {-3, 3}}]
    Attached Thumbnails Attached Thumbnails Sketching Quadric Surfaces-cylinders2.jpg  
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