Surfaces of revolution

Question

my question is how can i determine the equation of a surface of revolution obtained from the revolution of the z-axis around the line r:x=y=z. What i tried to do first i transform them to a parametric equation. Where the line r=(0,0,0)+(1,1,1)t. And z-axis has a direction vector ( 0,0,1)t, i pick a generic point p (0,0,t )from z-axis where i can construct equation of a circle, and a plane that contains point p and orthogonal to the axes of revolution r( 1,1,1) $\pi$ : x+y+z-t=0,i stuck here,how can i find the center of Circle and continue this problem, thanks in advance

Answer 1

Observe that the surface obtained from the revolution of the z-axis around the line $x=y=z$ is a cone, with $x=y=z$ as its symmetry line.

It is then convenient to parametrize this cone in the $uvw$-coordinates where $x=y=z$ is the $w$-axis. In the $uvw$-coordinates, the equation of the cone is given by,

$$u^2+v^2=2w^2\tag{1}$$

The transformation (or rotation) between the $uvw$- and $xyz$-coordinates are as follow,

$$w=x+y+z; \>\>\> u=x-y+z; \>\>\> v=x+y-z\tag{2}$$

Plug (2) into (1) the obtain the equation of the surface in the $xyz$-coordinates,

$$(x+y-z)^2+(x-y+z)^2=2(x+y+z)^2$$