How Does the Standard Equation Define a Cylinder?

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Discussion Overview

The discussion centers around the interpretation of the equation (x-a)² + (y-b)² = R² in three-dimensional space and its relation to defining a cylinder. Participants explore the implications of the absence of the variable 'z' in the equation and how this relates to the concept of a cylinder extending infinitely along the z-axis.

Discussion Character

  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants assert that the equation represents a circle in the xy-plane and question how it can define a cylinder in three dimensions without explicitly including 'z'.
  • Others argue that the lack of restrictions on 'z' allows any value of 'z' to satisfy the equation, thus forming a cylinder that extends infinitely along the z-axis.
  • A participant expresses frustration over the perceived lack of clarity in the explanation of the concept, feeling that the absence of 'z' in the equation was not adequately addressed.
  • One participant contrasts the cylinder's equation with that of a sphere, noting that the latter imposes restrictions on 'z', which is not the case for the cylinder.

Areas of Agreement / Disagreement

Participants do not reach a consensus; there are competing views regarding the interpretation of the equation and its implications for three-dimensional geometry.

Contextual Notes

Some participants express uncertainty about the conceptual transition from a two-dimensional circle to a three-dimensional cylinder, highlighting a potential gap in understanding the implications of the equation's structure.

Nano-Passion
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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?
 
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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')
 
Pengwuino said:
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.
 
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.
 
Pengwuino said:
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.

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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