Making a cylinder into a circle?

[christian0710]

Hi I don't understand how you can take a cylinder with equation
x²+y²=2x
And rewrite it to (x-1)²+y²=1

And then it suddenly becomes the equation for the base circle of the cylinder.

Would it not usually require that you remove some variable to transform it from 3D to 2-dimensional?

If we have a paraboloid z=1-x²+y² you can insert the plane z=0 and get the circle 1=x²+y² which makes sense because you remove a variable. How does this cylinder example work? I'd love to understand it more clearly.

 

[tiny-tim]

hi christian0710!

Hi I don't understand how you can take a cylinder with equation
x²+y²=2x
And rewrite it to (x-1)²+y²=1

And then it suddenly becomes the equation for the base circle of the cylinder.

Would it not usually require that you remove some variable to transform it from 3D to 2-dimensional?

nooo …

(x-1)²+y²=1 is a three-dimensional equation …

it's the equation for the whole cylinder​

 

[nate9228]

In regards to rewriting it...
take x²+y²=2x and subtract 2x from each side. Then add one to each side. This gives x²-2x+1+y²=1. The first 3 terms in the LHS are equivalent to (x-1)².

 

[Mark44]

Hi I don't understand how you can take a cylinder with equation
x²+y²=2x
And rewrite it to (x-1)²+y²=1

The two equation are equivalent, which means that they have exactly the same solution sets. Geometrically, they describe exactly the same thing.

What you are missing is the context for these equation, which is that they are equations in 3-D space. There is an implied third variable, z, which can take any value, since there is no restriction on it.

If the context for the equations happened to be 2-D space (the plane) then each would represent a circle (the same one).

 

[christian0710]

The two equation are equivalent, which means that they have exactly the same solution sets. Geometrically, they describe exactly the same thing.

What you are missing is the context for these equation, which is that they are equations in 3-D space. There is an implied third variable, z, which can take any value, since there is no restriction on it.

If the context for the equations happened to be 2-D space (the plane) then each would represent a circle (the same one).

Thank you! That was exactly what i needed to understand, when it's 3D we can shift the circle in the xy-plane up and down the z-axis --> cylinder!

Thank you again!