The cone and the sphere intersect in a circle whose radius is 1, whose center is at $\left(0\,0\,1\right)$, and which lies in the plane $z\=1$. This is established by solving the equations $z\=\sqrt{x^2\+y^2}$ and $x^2\+y^2\+{\left(z-1\right)}^2\=1$ simultaneously. Since these equations imply $z^2\+{\left(z-1\right)}^2\=1$, the solutions are easily seen to be $z\=0$ and $z\=1$. Each generator of the cone therefore has length $\sqrt{2}$.

The sphere is given in spherical coordinates by $\mathrm{\ρ}\=2 \cos \left(\mathrm{\φ}\right)$, so the plot data-structure $p_1$ houses the graph of the upper hemisphere. The cone is given in spherical coordinates by $\mathrm{\φ}\=\mathrm{\π}\/4$, so the plot data-structure $p_2$ houses the graph of the cone.