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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.
- #1
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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- #2
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')
- #3
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.
- #5
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 x2 + y2 = r2, 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 πr2, by the height of the cylinder. Therefore, using the standard equation of a cylinder, the volume can be calculated as V = πr2h.
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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