Section 6.5 Parametric Surfaces

6.5 Parametric Surfaces

We can describe a curve using parametric curve $\overrightarrow{r(t)}=<x(t),y(t),z(t)>$with a single parameter. It is a curve because the coordinates of $(x,y,z)$on the curve depending on the single parameter $t$. The surface in the space is the natural extension. 

Definition: 

A parameterized surface is given by a description of the form $\overrightarrow{r(u,v)}=<x(u,v),y(u,v),z(u,v)>$.

Example 1: Describe surface $S$parameterized by $\overrightarrow{r(u,v)}=<\text{cos}(u),\text{sin}(u),v>$,$-\mathrm{\infty }<u<\mathrm{\infty }$,$-\mathrm{\infty }<v<\mathrm{\infty }$. Find an equation that describe the surface.

Exercise 1: Describe surface $S$parameterized by $\overrightarrow{r(u,v)}=<\text{cos}(u),v,\text{sin}(u)>$,$-\mathrm{\infty }<u<\mathrm{\infty }$,$-\mathrm{\infty }<v<\mathrm{\infty }$. Find an equation that describe the surface.

Example 2: Describe surface $S$parameterized by $\overrightarrow{r(u,v)}=<v\text{cos}(u),v,v\text{sin}(u)>$,$-\mathrm{\infty }<u<\mathrm{\infty }$,$-\mathrm{\infty }<v<\mathrm{\infty }$. Find an equation that describe the surface.

Exercise 2: Describe surface $S$parameterized by $\overrightarrow{r(u,v)}=<v\text{cos}(u),v\text{sin}(u),v^2>$,$-\mathrm{\infty }<u<\mathrm{\infty }$,$-\mathrm{\infty }<v<\mathrm{\infty }$. Find an equation that describe the surface.

Example 3: Find the parametric representation of the plane passing through $(1,2,3)$and contains the vectors $<-1,3,4>$and $<4,5,6>$.

Exercise 3: Find the parametric representation of the plane passing through $(3,1,2)$and contains the vectors $<1,0,-2>$and $<2,1,0>$.

Example 4: Find the parametric representation of the hyperboloid $4x^2-4y^2-z^2=4$that lies in front of the $yz$-plane. 

Exercise 4:Find the parametric representation of the ellipsoid $x^2+y^2+9z^2=9$that lies in front of the $xz$-plane. 

Definition: 

For a smooth surface $\overrightarrow{r(u,v)}=<x(u,v),y(u,v),z(u,v)>$, the tangent plane is the plane that contains the tangent vectors $\overrightarrow{r_u}=<x_u,y_u,z_u>$and $\overrightarrow{r_v}=<x_v,y_v,z_v>$, and the vector

$\overrightarrow{r_u}\times \overrightarrow{r_v}$is a normal vector to the tangent plane.

Example 5: Find the equation of the tangent plane at $(2,3,0)$of the surface $\overrightarrow{r(u,v)}=<u+v,3u^2,u-v>$.

Exercise 5: Find the equation of the tangent plane at $(1,1,0)$of the surface $\overrightarrow{r(u,v)}=<u+v^2,u^2+v,v>$.

Example 6: Find the parametric representation of the sphere $x^2+y^2+z^2=4$that lies above the cone $z=\sqrt{x^2+y^2}$.

Group work:

1. Find the parametric representation of the cylinder $x^2+z^2=25$ that lies above the $xy$-plane and between the planes $y=5$and $y=-5$.

2. Find the parametric representation of the sphere $x^2+y^2+z^2=64$ that lies between $z=0$and $z=\sqrt{32}$. 

3. Find the parametric representation of the plane $z=x+4$ that lies inside the cylinder $x^2+y^2=4$.

4. Find the equation of the tangent plane at $(0,4,1)$ of the surface $\overrightarrow{r(u,v)}=<u^3-v^3,v+3u,2u-v>$.