Trace $x=y$ when plotting a paraboloid

Question

When plotting regions in $\mathbb{R}^{3}$, it is a common strategy to draw the region for some constant parameter (like $y=0$ or $x=0$ or $z=0$). When plotting $z=y^{2}+x^{2}$, I tried to plot for when $x=y$ to see what happened on this plane (or at least that was my intention). In this case, $z= 2x^{2}$, but this is relatively to the x and z-axis, because I know that the figure drawn in the $x=y$ plane is a normal parabola ($z="x"^{2}$). So is there a way that allows us to see the figures as they happen and not distorted by the axis representation? (I don't know if I'm being explicit enough?)

Answer 1

In this case you could think about transforming $(x,y)$ in to radial coordinates, $(r,\theta)$. For $x = y$, you have $(r,\theta) = (\sqrt{2}x,\frac{\pi}{4})$.

$z = r^2$

$z$ is independent of the angle.

Answer 2

Call $w$ the axis in the direction $x=y$. One unit along $w$ projects to $\dfrac1{\sqrt 2}$ unit along $x$. Hence the parabola is

$$z=w^2.$$