Parametric Surfaces — Question 8

Question 8

Two parametrizations are $$\mathbf r(u,v)=\langle u,v,u+v\rangle, \qquad \mathbf s(p,q)=\langle q,p,p+q\rangle.$$

Tasks

1. Show that they have the same image.

2. Compute the ordered normal from each parametrization.

3. Determine whether they induce the same or opposite orientation.

Question 8 – Solution

Strategy. Recognize that the second parametrization swaps the two parameter roles, then compare ordered cross products.

Step 1: Compare images Both satisfy $$x+y=z.$$ Given $(u,v)$ for $\mathbf r$, choosing $(p,q)=(v,u)$ gives $\mathbf s(v,u)=\mathbf r(u,v)$. Hence their images are the same plane.

Step 2: First normal $$\mathbf r_u=\langle 1,0,1\rangle, \qquad \mathbf r_v=\langle 0,1,1\rangle,$$ so $$\mathbf r_u\times\mathbf r_v=\boxed{\langle-1,-1,1\rangle}.$$

Step 3: Second normal $$\mathbf s_p=\langle 0,1,1\rangle, \qquad \mathbf s_q=\langle 1,0,1\rangle,$$ and therefore $$\mathbf s_p\times\mathbf s_q=\boxed{\langle 1,1,-1\rangle}.$$ The normals are negatives, so the parametrizations induce $\boxed{\text{opposite orientations}}$.

Verification Swapping two parameters reverses the order in a cross product, and $\mathbf a\times\mathbf b=-(\mathbf b\times\mathbf a)$.