What Are the Hyperbolic Characteristics of the Quadratic Surface Z=x²-y²?

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The discussion centers on the hyperbolic characteristics of the quadratic surface defined by the equation Z = x² - y². It is established that the trace for Z = 0 results in the crossed lines y = ±x, which serve as asymptotes for the hyperbolas formed by other values of Z. The curves for Z ≠ 0 are confirmed to be hyperbolas fitting between these asymptotes. Additionally, the lines y = ±x are classified as degenerate hyperbolas in this context.

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Z=x^2-y^2
The book is showing the trace for z=0 to be a hyperbola however I see y=x and y=-x
 
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hi nameVoid! :smile:

(try using the X2 button just above the Reply box :wink:)
nameVoid said:
Z=x^2-y^2
The book is showing the trace for z=0 to be a hyperbola however I see y=x and y=-x

what book? :confused:

yes, Z = 0 is the crossed lines y = ±x

the curves for all other values of Z will be hyperbolas, fitting between y = ±x
 
Also, since the lines ##y=\pm x## are the asymptotes for the family of level curves for that surface, they are sometimes considered to be degenerate hyperbolas.
 

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