In this preview activity, we will explore the parameterizations of a few familiar surfaces and confirm some of the geometric properties described in the introduction above.
Use the ideas from Section 13.9 to give a parameterization \(\vr(s,t)\) of the following surface. Be sure to specify the bounds on each of your parameters.
Draw a graph of \(S_1\) from the previous part. Label the points that correspond to \((s,t)\) points of \((0,0)\text{,}\)\((0,1)\text{,}\)\((1,0)\text{,}\) and \((2,3)\text{.}\)
For the parameterization from part a, find the value for \(\vr_s\text{,}\)\(\vr_t\text{,}\) and \(\vr_s \times \vr_t\) at the \((s,t)\) points of \((0,0)\text{,}\)\((0,1)\text{,}\)\((1,0)\text{,}\) and \((2,3)\text{.}\)
Draw your vector results from d on your graph and confirm the geometric properties described in the introduction to this section. Namely, \(\vr_s\) and \(\vr_t\) should be tangent to the surface, while \(\vr_s \times \vr_t\) should be orthogonal to the surface (in addition to \(\vr_s\) and \(\vr_t\)).
