Parametric representation of paraboloid cylinder

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SUMMARY

The discussion focuses on the parametric representation of a paraboloid cylinder defined by the equation z = y^3. The participants establish that for the parameters, x can be represented as u, y as v, and z as v^3, leading to the parametric form r(u, v) = [u, v, v^3]. They also highlight that while there is a general method for determining r(u, v) for functions like z(x,y), alternative approaches may be necessary for more complex surfaces, including rearranging equations or changing coordinate systems.

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geft
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The equation is z = y^3. I know how to do normal planes and spheres, but I don't know what to set for r(u,v) when it comes to paraboloid cylinders.
 
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geft said:
The equation is z = y^3. I know how to do normal planes and spheres, but I don't know what to set for r(u,v) when it comes to paraboloid cylinders.

Since x can be anything, make it one of your variables (either u, or v).
 
x = u
y = v
z = v^3

r(u, v) = [u, v, v^3]?

Is there a formula for the r(u, v)?
 
geft said:
x = u
y = v
z = v^3

r(u, v) = [u, v, v^3]?

Is there a formula for the r(u, v)?

If you're asking if there is a formulaic method for determining what r(u,v) should be, then the answer is "sorta". In the case of something along the lines of z(x,y) then you let u and v be x and y, and then just have the function as your z parameter. However you don't always get things defined by functions like that, in which case you need to get a little creative. Either re-arranging things, or even jumping coordinate systems to make stuff easier.
 

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