Sketching a parametrised cone and a geodesic lying on it.

Question

I just started a new module at University and I am having some trouble with parametrisation.

I am given a parametrisation of a geodesic lying on a cone in notation $r(t)=x(t){\bf i}+y(t){\bf j}+z(t){\bf k}$. Is this the same as $r(t)=(x(t),y(t),z(t))$ ?

They want me to sketch the cone and the curve lying on it. How should I approach this?

Answer 1

Start by noticing that $z=1-\sqrt{\alpha^2-1}\sqrt{x^2+y^2}$, which is the equation of a cone with vertex at $(0,0,1)$, whose intersection with the $(x,y)$ plane is a circle of radius $1/\sqrt{\alpha^2-1}$.