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

x^{2}=y^{2}-z^{2}

I have made these changes to the equation:

First equation solution:

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

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

Second equation solution:

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

\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?

InfZero
  • 875
  • There are three variables, so the solutions are a subset of $\mathbb{R}^3$ (we can also imagine the solutions existing in $\mathbb{RP}^2$); either way you need three axes to graph it (not sure if this answers your question?) – MT_ Aug 11 '16 at 19:43

3 Answers3

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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. enter image description here

Emilio Novati
  • 62,675
1

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.

Kaynex
  • 2,448
0

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.

Tomek Dobrzynski
  • 155