Find the conoid with director plane z=0 , director line Oz…

Find the conoid with director plane $z=0$, director line $Oz$ and director curve $c:\ x^2+y^2+z^2=1,\quad x+y+z=0 $.

Step 1: Generators

A generator is horizontal and meets $Oz$ at $(0,0,t)$. It also meets $c$ at some $(x_0,y_0,t)$ with

\[ x_0+y_0=-t,\qquad x_0^2+y_0^2=1-t^2 . \]

Its points are $(sx_0,\ sy_0,\ t)$.

Step 2: Eliminate

For $M(x,y,z)$: $t=z$, $x+y=s(x_0+y_0)=-sz$, and $x^2+y^2=s^2(1-z^2)$. With $s=-\tfrac{x+y}{z}$: $(x^2+y^2)\,z^2=(1-z^2)(x+y)^2$.

Check with a point of $c$ at $z=\tfrac12$: $x_0+y_0=-\tfrac12$ and $x_0^2+y_0^2=\tfrac34$. Then $\tfrac34\cdot\tfrac14=\tfrac34\cdot\tfrac14$. $\checkmark$ The $z$-axis also lies on the surface.

Real generators exist only for $|z|\le\sqrt{2/3}$, the height range of $c$.