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In summary, an equation of (x-a)^2 + (y-b)^2 = R^2 defines a circle in three dimensions, while an equation of x^2 + y^2 + z^2 = R^2 defines a shell/sphere in three dimensions.

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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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Pengwuino

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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')

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Nano-Passion

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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.

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Pengwuino

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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.

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Nano-Passion

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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

## Related to How Does the Standard Equation Define a Cylinder?

## 1. What is the standard equation of a cylinder?

The standard equation of a cylinder is *x ^{2} + y^{2} = r^{2}*, where

*r*is the radius of the base of the cylinder. This equation represents a circular cross-section at any given height along the length of the cylinder.

## 2. How is the standard equation of a cylinder different from other equations of a cylinder?

The standard equation of a cylinder is unique in that it represents a perfect circular cross-section at any given height, while other equations may represent an ellipse or other shapes.

## 3. How can the standard equation of a cylinder be used to find the volume of a cylinder?

The volume of a cylinder can be found by multiplying the area of the base, given by *πr ^{2}*, by the height of the cylinder. Therefore, using the standard equation of a cylinder, the volume can be calculated as

*V = πr*.

^{2}h## 4. Can the standard equation of a cylinder be used to represent a tilted or oblique cylinder?

No, the standard equation of a cylinder only represents a cylinder with a base that is parallel to the xy-plane. Tilted or oblique cylinders require a different equation, taking into account the angle of the base with respect to the xy-plane.

## 5. How can the standard equation of a cylinder be graphed?

To graph the standard equation of a cylinder, the values of *x* and *y* can be chosen arbitrarily, and then the corresponding values of *z* can be calculated using the equation. These points can then be plotted in three-dimensional space to form a cylinder. Alternatively, a graphing calculator or software can be used to plot the equation directly.

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