# Find the parametric equation of the curve to be the intersection of the paraboloid $z=x^{2}+y^{2}$ and the plane $y=z$

A space-curve $C$ is defined to be the intersection of the paraboloid $z=x^{2}+y^{2}$ and the plane $y=z$. How should one try to find the parametric equation of the curve? It seems natural to let $x=(t-t^{2})^{\frac{1}{2}}$, $y=t$, z=$t$. However, rearranging the equations I got $\frac{1}{4}=x^{2}+\left ( y-\frac{1}{2} \right )^{2}$, which calls for the substitution $x=\cos(t)$, $y=\frac{1}{2}+\sin(t)$, $z=\frac{1}{2}+\sin(t)$. Which one do you suggest? Any tips on generalizing parametrization?

• This is a perfectly good parameterization, and probably the first one I'd produce, but as is usually the case, which is best depends on your purposes. – Travis Jun 6 '15 at 16:03
• My goal is to find the equations for the vectors of the Frenet frame. – Shemafied Jun 6 '15 at 16:10

The issue with your first parameterization is that it implies $x \ge 0$ while if $(x,y,z)$ is a point of the paraboloid, $(-x,y,z)$ also.