Drawing Quadratic Surfaces: Tips for Rendering Sheets by Hand

nameVoid
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z=y^2-x^2

Trying to render these sheets by hand is very difficult for me. I can conceptualize the sheet in general by observing that the trace in z=y^2-k^2 follows the trace of z=k^2-x^2 as z tends to negative infinity. The opposite is also true as z tends to infinity. This information combined with the hyperbolic traces gives me a general idea of the sheet. Any tips in how to draw this.
 
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You can use softwares like Maple and MatLab.
There is one online too.Check here.
 
Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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