4

I have started an Multivariable course, and I'm learning about spherical coordinates. My problem now is learn how to graph this kind of shapes.

This is the problem:

What shapes are described when...?

Solution:

a) $\rho = 1$ : Sphere with radius 1.

b) $\phi = \frac{\pi}{3}$ : Cone with angle $\frac{\pi}{3}$.

c) $\theta = \frac{\pi}{4}$ : Semi-circular cross-section with diameter along z-axis

d) $\rho = \cos{(\phi)}$ : ?

$\rho = \cos{(\phi)}$

e) $\rho = \cos{(2\theta)}$ : ...?

$\rho = \cos{(2\theta)}$

Are they correct? How to describe, verbally, the last two -d) and e).

InfZero
  • 875
  • 2
    The answer depends in part on your definitions of $\theta$ and $\phi$. Mathematicians tend to use $\theta$ to denote longitude and $\phi$ to denote either colatitude or latitude, while physicists often (usually?) use $\phi$ to denote longitude and $\theta$ for colatitude. Could you please clarify your conventions? :) – Andrew D. Hwang Aug 07 '16 at 21:08
  • @AndrewD.Hwang, in a) b) c) the values are constant the another ones vary. For d) and e) I have entered these lines in Wolfram Mathematica: SphericalPlot3D[Cos[phi], {theta, 0, 2 Pi}, {phi, 0, Pi}] and SphericalPlot3D[Cos[2theta], {theta, 0, 2 Pi}, {phi, 0, Pi}]. – InfZero Aug 07 '16 at 21:19
  • @ Andrew D. Hwang apparently it is the mathematicians' conventions that he uses. But he should say us if $\phi$ is the latitude or colatitude. – Jean Marie Aug 07 '16 at 21:56
  • @JeanMarie: Just trying to install good habits, and to ensure the question is as self-contained as possible for posterity. ;) – Andrew D. Hwang Aug 07 '16 at 22:33
  • @Andrew D. Hwang Be sure I completely agree with you ! We denote a common trend, that the conventions are implicitly those of some software, and it's up to you to find which is which. – Jean Marie Aug 07 '16 at 22:45

1 Answers1

2

If $\phi$ is a cone with angle $\pi/3$ then:

d) $\rho=\cos\phi$:

Multiply both terms by $\rho$ and you get $$ \rho^2=\rho\cos\phi \quad \Rightarrow \quad x^2+y^2+z^2=z, $$ which is a sphere of radius $1/2$ centered at $(0,0,1/2)$:enter image description here

e) $\rho=\cos2\theta$:

We could eventually find the cartesian equation here, but it will not be of any help, as it is not a classical surface:

enter image description here

a), b) and c) are correct. To convince yourself, find the cartesian equations.

Kuifje
  • 9,584