Finding the vertex of an elliptic cone

Solution 1:

mostly because there are (distinct) straight lines within the cone that all pass through the vertex. These are lines parametrized by a new variable $t$ $$ (0, 4+t, 4+t) $$

$$ (0, 4-t, 4+t) $$

$$ (t, 4, 4+t) $$

$$ (-t, 4, 4+t) $$

For each one, at $t=0$ it goes through the vertex. The lines do not meet elsewhere. To prove that for a pair, change the variable $t$ to $s$ in one of them, for example $ (t, 4, 4+t) , \; \; (-s, 4, 4+s) . \; \; $ For the coordinates to match you need $t=-s$ but you also need $4+t=4+s. \;$ That happens only when $s=t=0$

using Pythagorean triples, we can find other lines (within the cone) without trig functions.

$$ (4t, 4+3t, 4+5t) $$ $$ (5t, 4+12t, 4+13t) $$ $$ (8t, 4+15t, 4+17t) $$ $$ (20t, 4+21t, 4+29t) $$