What do we mean by asymptotes of a conicoid ?
The asymptotes of below conicoid $$ax^2+by^2+cz^2=1$$ is the cone $$ax^2+by^2+cz^2=0$$
We can derive this by the general definition of a asymptotes . Mathematics seems correct here. However , i cannot visualize how a cone can meet a fixed surface(in this case a conicoid) at infinity ? What do we mean by asymptote of a cone ,cylinder etc. ?
Perhaps this will help:
The outer surface is the hyperboloid $x^{2} + y^{2} - z^{2} = 1$, and the inner surface is the cone $x^{2} + y^{2} - z^{2} = 0$. You can see that the surfaces get closer together as $z$ goes to $\pm\infty$.