Visualizing a strange quadric surface

Woland

Hi all,

I am trying to visualize the following quadric surface:
x^2 + y^2 + z^2 = z

Could someone please help me understand what this surface looks like? I could not find an example of this one on the internet.

Thank you.

Hurkyl

Staff Emeritus
Gold Member
Do some algebra to simplify it. (e.g. things like collecting similar terms, completing the square, factoring, change of variable, etc)

maze

Consider what the surface
(x-a)^2 + (y-b)^2 + (z-c)^2 = d

looks like

Woland

(x-a)^2 + (y-b)^2 + (z-c)^2 = d is a sphere centered at a, b, c with radius square root of d.

Now regarding the first post, I can bring the z over to the left hand side:

x^2 + y^2 + z^2 - z = 0

Then complete the square

x^2 + y^2 + (z - 1/2)^2 -1/4 = 0
[(z - 1/2)^2 = z^2 - 2*1/2 z + 1/4]

Then

x^2 + y^2 + (z-1/2)^2 = 1/4

This means that this is a sphere centered at 0,0, 1/2. Is that correct?

mathman

This means that this is a sphere centered at 0,0, 1/2. Is that correct?
Yes and the radius is 1/2.

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