Parametric Surfaces — Question 7

Question 7

Consider $$\mathbf r(u,v)=\langle u^2,v,u^3\rangle, \qquad -1\le u\le 1,\qquad -1\le v\le 1.$$

Tasks

1. Eliminate $u$ to find an implicit relation between $x$ and $z$.

2. Determine where the parametrization is regular.

3. Explain the geometric significance of the failure set.

Question 7 – Solution

Strategy. Compare $x=u^2$ and $z=u^3$, then locate zeros of the tangent cross product.

Step 1: Eliminate the parameter $$z^2=u^6=(u^2)^3=x^3, \qquad x\ge 0.$$ The image is the patch of the cuspidal cylinder $z^2=x^3$ with $0\le x\le 1$ and $-1\le y\le 1$.

Step 2: Test regularity $$\mathbf r_u=\langle 2u,0,3u^2\rangle, \qquad \mathbf r_v=\langle 0,1,0\rangle,$$ so $$\mathbf r_u\times\mathbf r_v=\langle-3u^2,0,2u\rangle.$$ This vector vanishes exactly when $u=0$. Therefore the parametrization is regular for $u\ne 0$ and singular along $$\boxed{\mathbf r(0,v)=\langle 0,v,0\rangle}.$$

Step 3: Interpret That line is the sharp cuspidal edge. At it, the $u$-direction tangent collapses to zero, so the two tangent vectors cannot span a tangent plane.

Verification For $u\ne 0$, the third component $2u$ of the cross product is nonzero, proving regularity away from the edge.