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Sketching quadratic surfaces

  1. Jan 23, 2006 #1
    I desperately need some help with sketching cylinders and quadratic surfaces. We did this in first year and I understood it then but now that I look at it, I have no idea where to start. Oh and yes I have been in to talk with my ta quite a few times but I still dont quite understand; I'll go bother my prof tomorrow during his office hours too but for now if someone can just guide me with how to graph these monsters, I'd really appreciate it immensely. This is the only chapter I'm having trouble with. Thanks o:)

    These are just some practice problems from our textbook that don't have to be handed in so if someone could give an explanation of any one of these surfaces, it would be great. I could work on the others if I understood at least one. Lol actually I think 5 is the only one that I get :blushing:

    1. y^2+4z^2=4
    2. z=4-x^2
    3. x-y^2=0
    4. yz=4
    5. z=cosx
    6. x^2-y^2=1

    So ummm yah, thanks for any help:frown:

    Oh and something else, I had another account here but for some reason it doesn't let me post anything anymore? Anybody know why this might be so?
     
  2. jcsd
  3. Jan 23, 2006 #2

    benorin

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    Homework Helper

    quadratic surfaces

    In R3, equations of surfaces which are "missing" a variable (i.e., are of the form F(x,y)=0, or F(x,z)=0, or F(y,z)=0,) describe surfaces which are paralell to the axis of the missing variable and are called cylinders. To graph them, just plot the given equation in the plane of the two given variables and "drag" it paralell to the axis of the third variable. Examples of such surfaces are #'s 1,2,3,4,5, and 6, nice. On with it...

    1. y^2+4z^2=4, note that this is of the form F(y,z)=0 (and hence the above applies,) rearrange to get

    [tex]\left( \frac{y}{2}\right)^2+z^2=1,[/tex] an ellipse in the yz-plane with semi-major axis=2, semi-minor axis=1; plot that and drag it paralell to the x-axis to trace out an elliptic cylinder (curled around the x-axis.)

    2. z=4-x^2 is of the form F(x,z)=0 and is a parabola in the xz-plane which opens downwards with its vertex at (0,0,4); drag this paralell to the y-axis to trace out a parabolic sheet looking something like an infinitely wide (and long) towel draped over a dowel 4 units above the y-axis.

    3. x-y^2=0, of the form F(x,y)=0 and is a parabola in the xy-plane which opens forwards with its vertex at (0,0,0); drag this paralell to the z-axis to trace out a parabolic sheet which is symmetric about the xz-plane.

    4. yz=4 is of the form F(y,z)=0 and is a hyperbola in the yz-plane, drag it paralell to the x-axis...

    5. z=cosx, is of the form F(x,z)=0 and is the cosine curve in the xz-plane, dragging it paralell to the y-axis gives a surface that resembles that wavy sheet metal stuff that is sometimes used to as roofing material for cheep wharehouses.

    6. x^2-y^2=1, is of the form F(x,y)=0 and is a hyperbola in the...
     
  4. Jan 23, 2006 #3
    Ohhhhhhhh yes. I had entirely forgotten about the whole "equations of surfaces which are "missing" a variable describe surfaces which are paralell to the axis of the missing variable" deal. No wonder I had no idea about where in space to actually put my surfaces :blushing: .

    Hahahah that makes so much more sense now. I guess I'll go retry some more questions keeping that point in mind. Thanks a lot :wink:
     
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