Parametric Surfaces — Question 4

Question 4

Consider the standard spherical parametrization $$\mathbf r(\phi,\theta)=\langle 3\sin\phi\cos\theta,3\sin\phi\sin\theta,3\cos\phi\rangle,$$ where $0\le\phi\le\pi$ and $0\le\theta\le 2\pi$.

Tasks

1. Show that the image is the sphere of radius $3$.

2. Compute an outward normal vector from the tangent vectors.

3. Identify where the parametrization fails to be regular or one-to-one.

Question 4 – Solution

Strategy. Verify the implicit equation, then use the ordered product $\mathbf r_\phi\times\mathbf r_\theta$.

Step 1: Identify the image $$x^2+y^2+z^2 =9\sin^2\phi+9\cos^2\phi=9.$$ Thus the image is $x^2+y^2+z^2=9$.

Step 2: Normal Differentiation and crossing give $$\boxed{\mathbf r_\phi\times\mathbf r_\theta =9\sin\phi\langle\sin\phi\cos\theta,\sin\phi\sin\theta,\cos\phi\rangle}.$$ For $0<\phi<\pi$, this is a positive multiple of the outward radial direction.

Step 3: Degeneracies Its magnitude is $9\sin\phi$, which vanishes at $\phi=0$ and $\phi=\pi$. Every value of $\theta$ maps to the same pole there. Also $\theta=0$ and $\theta=2\pi$ describe the same meridian seam.

Verification Away from the poles, $9\sin\phi>0$, so the tangent vectors are independent and the patch is regular locally.