Previous sections introduced parametric curves. These are topologically one dimensional relations produced by a single parameter. Section 1.3 introduced planes, a type of surface which is a topologically two dimensional relation that are produced by two parameters. Here we develop other topologically two dimensional relations (surfaces) that are produced by two parameters.
Subsection3.5.1Recognizing Surfaces in Parametric Form
Activity57.
The purpose of this activity is to use the idea that surfaces are “wide” curves (consist of a set of curves) to express them in parametric rather than as a function.
Graph the following parametric surfaces using technology then answer the questions.
(a)
Consider \(P(t,u)=(1,2,2)t+(-1,1,-1)u+(7,2,4).\)
(i)
What is a part of \(P(t,u)\) that is a line?
(ii)
Why does adding \((1,2,2)t\) to \((-1,1,-1)u+(7,2,4)\) turn a line into a plane?
Why does adding \((\cos\theta,\sin\theta,0)\) to the line \((-t,t,-t)+(7,2,4)\) turn a line into a (leaning) cylinder?
(c)
\(F(t,\theta)=(0,\cos\theta,\sin\theta)+(t,t^2,0).\) Why does adding \((0,\cos\theta,\sin\theta)\) to \((t,t^2,0)\) turn this parabola into a pipe segment (parabolic cylinder)?
Subsection3.5.2Tangent Planes for Parametrized Surfaces
In Section 2.3 we learned to calculate planes tangent to a surface at a point when the surface was expressed as a function. Here we use an idea from that section to calculate tangent planes when the surface is parameterized.
Activity58.
The purpose of this activity is to generate an equation for a tangent plane using partials.
Use the parameterized surface \(S(t,u)=(e^t,e^u,tu)\text{.}\)
