What are the parameters needed for surface parametrization of x^2-y^2=1?

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Homework Help Overview

The discussion revolves around finding a surface parametrization for the equation x^2 - y^2 = 1, with specified constraints on x, y, and z. The context is primarily focused on the mathematical representation of a surface in three-dimensional space.

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • Participants explore the representation of x and y using hyperbolic functions and question the necessity of including a z component in the parametrization. There is discussion about whether z should be treated as a parameter or if it should be incorporated into the equation.

Discussion Status

The discussion is ongoing, with participants raising questions about the inclusion of z in the surface parametrization and considering different approaches to define the parameters needed for the hyperbola. Some guidance has been offered regarding the use of hyperbolic functions.

Contextual Notes

There is uncertainty regarding the role of z in the parametrization, as the original equation does not explicitly include it. Participants are also navigating the constraints imposed on the variables x, y, and z.

Tony11235
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My problem is finding a surface parametrization of the surface x^2-y^2=1, where x>0, -1<=y<=1 and 0<=z<=1.

I know that x and y in x^2-y^2=1, can be represented as cosh(u) and sinh(u), but I'm not sure what to do for the z part. Any quick help?
 
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Tony11235 said:
My problem is finding a surface parametrization of the surface x^2-y^2=1, where x>0, -1<=y<=1 and 0<=z<=1.
Did you intend your equation not to contain any z terms?
 
EnumaElish said:
Did you intend your equation not to contain any z terms?

Should it contain a z component? I was assuming it should because it is a surface parametrization but maybe not.
 
Does a line on the XY plane contain a y component? Unless it's a vertical line, it does.
 
A parametrization of a surface, a 2 dimensional figure, necessarily involves 2 parameters. Since there is no "z" in your equations, you might consider taking z itself as a parameter or say z= v where v is a parameter, then look for a parametrization of the hyperbola x2- y[/sup]2[/sup]. Since it is a hyperbola, the hyperbolic functions leap to mind!
 

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