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Standard equation of a cylinder

  1. Sep 6, 2012 #1
    Okay, so we know that an equation of (x-a)^2 + (y-b)^2 = R^2 is the equation for a circle. Basically where x-a is change in the x-axis and y-b is the change in the y-axis. Here is the thing that I don't get, as quoted from my book.

    This equation in R^3 (three dimension) defines the cylinder of radius R whose central axis is the vertical line through (a,b,0).

    How does it define a cylinder?
  2. jcsd
  3. Sep 6, 2012 #2


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    Are you having a trouble with the fact that 'z' is not accounted for anywhere or do you just note understand the equation?

    Let a = b = 0 and you should clearly see that what you have is a circle centered about the origin (and, in 3-dimensions, a cylinder as those positions satisfy the equation for any 'z')
  4. Sep 6, 2012 #3
    It is the fact that z isn't in the equation at all.

    How does the a cylinder satisfy the equations for any z if there isn't any z? I feel like this is too hand-wavy. For all I'm concerned, the domain is simply that of a two dimensional plane, more precisely a circle of radius R; no three dimensional counterpart is included.
  5. Sep 6, 2012 #4


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    It's best to think of it as there being no restrictions on what z is. For example, the coordinates (x,y,z) = (3, 4, 105) satisfy the equation [itex]x^2 + y^2 = 25[/itex] which means those coordinates will be on the cylinder centered at the origin with a radius of 5. The cylinder extends to infinity because any value of z satisfies that equation.

    Now, if you had something like [itex]x^2 + y^2 + z^2 = R^2[/itex], you do have a restriction on z as well. This is the equation of a shell/sphere in 3-dimensions. The z coordinate has become restricted to the surface of this shell.
  6. Sep 6, 2012 #5
    Well that is odd, I'm used to things being very clearly defined in mathematics. Z isn't anywhere in the domain and isn't anywhere in the set. I wasted a lot of time trying to understand what is going on and then I realize that the concept was just improperly conveyed. /rant
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