How Does the Standard Equation Define a Cylinder?

[Nano-Passion]

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?

 

[Pengwuino]

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')

 

[Nano-Passion]

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')

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.

 

[Pengwuino]

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 $x^2 + y^2 = 25$ 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 $x^2 + y^2 + z^2 = R^2$, 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.

 

[Nano-Passion]

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 $x^2 + y^2 = 25$ 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 $x^2 + y^2 + z^2 = R^2$, 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.

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