Question

Describe in words the surface whose equation is given \(\phi {\rm{ = \pi /3}}\).

Short answer

The equation is \({\rm{3}}{{\rm{z}}^{\rm{2}}}{\rm{ = }}{{\rm{x}}^{\rm{2}}}{\rm{ + }}{{\rm{y}}^{\rm{2}}}\), the equation shape is \(\phi {\rm{ = \pi /3 > 0}}\) the upper half of the double cone (Half cone).

Step-by-step solution

Given data.

Equation \(\phi {\rm{ = \pi /3}}\)

Equation shape.

Spherical coordinates,

\(\begin{array}{l}{\rm{(1)x = \rho sin}}\phi {\rm{cos\theta }}\\{\rm{(2)y = \rho sin}}\phi {\rm{sin\theta }}\\{\rm{(3)z = \rho cos}}\phi \\{\rm{(4)}}{{\rm{x}}^{\rm{2}}}{\rm{ + }}{{\rm{y}}^{\rm{2}}}{\rm{ + }}{{\rm{z}}^{\rm{2}}}{\rm{ = }}{{\rm{\rho }}^{\rm{2}}}\end{array}\)

Here,

\(\begin{aligned}{\rm{cos}}\phi \rm &= cos(\pi /3)\\\rm &= \frac{{\rm{1}}}{{\rm{2}}}\end{aligned}\)

Therefore,

\(\begin{aligned}{\rm{co}}{{\rm{s}}^{\rm{2}}}\phi \rm &= \frac{{\rm{1}}}{{\rm{4}}}{{\rm{\rho }}^{\rm{2}}}{\rm{co}}{{\rm{s}}^{\rm{2}}}\phi \\ \rm &= \frac{{\rm{1}}}{{\rm{4}}}{{\rm{\rho }}^{\rm{2}}}\end{aligned}\)

So,

\(\begin{aligned}{{\rm{z}}^{\rm{2}}} \rm &= \frac{{\rm{1}}}{{\rm{4}}}\left( {{{\rm{x}}^{\rm{2}}}{\rm{ + }}{{\rm{y}}^{\rm{2}}}{\rm{ + }}{{\rm{z}}^{\rm{2}}}} \right)\\{\rm{4}}{{\rm{z}}^{\rm{2}}} \rm &= {{\rm{x}}^{\rm{2}}}{\rm{ + }}{{\rm{y}}^{\rm{2}}}{\rm{ + }}{{\rm{z}}^{\rm{2}}}\\{\rm{3}}{{\rm{z}}^{\rm{2}}} \rm &= {{\rm{x}}^{\rm{2}}}{\rm{ + }}{{\rm{y}}^{\rm{2}}}\end{aligned}\)

Here,\({\rm{3}}{{\rm{z}}^{\rm{2}}}{\rm{ = }}{{\rm{x}}^{\rm{2}}}{\rm{ + }}{{\rm{y}}^{\rm{2}}}\) this is an equation of double cone. The given equation is \(\phi {\rm{ = \pi /3 > 0}}\).

As the result of the given equation shape is \(\phi {\rm{ = \pi /3 > 0}}\) the upper half of the double cone (Half cone).