How to plot $x^{2}=y^{2}-z^{2}$?

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I have plotted this equation $x^{2}=y^{2}-z^{2}$ using Wolfram|Alpha and I got this graph:

I have made these changes to the equation:

First equation solution:

$y=-\sqrt{y^{2}-x^{2}}$

Second equation solution:

$y=\sqrt{y^{2}-x^{2}}$

I want to make this by hand. How do I can do it? Which coordinate system I need to use to graph it?

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3 Answers

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Write it as $x^2 + z^2 = y^2$. Note that y is the hypotenuse of a triangle with length x and height z. So, this forms a circular cone opening as you increase in y or decrease in y.

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This figure is the (double) cone of equation $x^2=y^2-z^2$.

The gray plane is the plane $(x,y)$.

You can see that it is a cone noting that for any $y=a$ the projection of the surface on the plane $(x,z)$ is a circumference of radius $a$ with equation $z^2+x^2=a^2$.

Note that $z=\sqrt{y^2-x^2}$ is the semi-cone with $z>0$, i.e. above the plane $(x,y)$ and $z=-\sqrt{y^2-x^2}$ is the semi-cone below this plane.

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y^2 = x^2 + z^2 has the form of an equation for a circle. So, you are stacking, in the y direction, circles of increasing radius, one on top of the other.

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