Section 6.5 Parametric Surfaces
6.5 Parametric Surfaces
We can describe a curve using parametric curve [latex]\overrightarrow{r(t)}=\lt x(t),y(t),z(t)\gt[/latex]with a single parameter. It is a curve because the coordinates of [latex](x,y,z)[/latex]on the curve depending on the single parameter [latex]t[/latex]. The surface in the space is the natural extension.
Definition:
A parameterized surface is given by a description of the form [latex]\overrightarrow{r(u,v)}=\lt x(u,v),y(u,v),z(u,v)\gt[/latex].
Example 1: Describe surface [latex]S[/latex]parameterized by [latex]\overrightarrow{r(u,v)}=\lt \text{cos}(u),\text{sin}(u),v\gt[/latex],[latex]-\infty\lt u\lt \infty[/latex],[latex]-\infty\lt v\lt \infty[/latex]. Find an equation that describe the surface.
Exercise 1: Describe surface [latex]S[/latex]parameterized by [latex]\overrightarrow{r(u,v)}=\lt \text{cos}(u),v,\text{sin}(u)\gt[/latex],[latex]-\infty\lt u\lt \infty[/latex],[latex]-\infty\lt v\lt \infty[/latex]. Find an equation that describe the surface.
Example 2: Describe surface [latex]S[/latex]parameterized by [latex]\overrightarrow{r(u,v)}=\lt v\text{cos}(u),v,v\text{sin}(u)\gt[/latex],[latex]-\infty\lt u\lt \infty[/latex],[latex]-\infty\lt v\lt \infty[/latex]. Find an equation that describe the surface.
Exercise 2: Describe surface [latex]S[/latex]parameterized by [latex]\overrightarrow{r(u,v)}=\lt v\text{cos}(u),v\text{sin}(u),v^{2}\gt[/latex],[latex]-\infty\lt u\lt \infty[/latex],[latex]-\infty\lt v\lt \infty[/latex]. Find an equation that describe the surface.
Example 3: Find the parametric representation of the plane passing through [latex](1,2,3)[/latex]and contains the vectors [latex]\lt -1,3,4\gt[/latex]and [latex]\lt 4,5,6\gt[/latex].
Exercise 3: Find the parametric representation of the plane passing through [latex](3,1,2)[/latex]and contains the vectors [latex]\lt 1,0,-2\gt[/latex]and [latex]\lt 2,1,0\gt[/latex].
Example 4: Find the parametric representation of the hyperboloid [latex]4x^{2}-4y^{2}-z^{2}=4[/latex]that lies in front of the [latex]yz[/latex]-plane.
Exercise 4:Find the parametric representation of the ellipsoid [latex]x^{2}+y^{2}+9z^{2}=9[/latex]that lies in front of the [latex]xz[/latex]-plane.
Definition:
For a smooth surface [latex]\overrightarrow{r(u,v)}=\lt x(u,v),y(u,v),z(u,v)\gt[/latex], the tangent plane is the plane that contains the tangent vectors [latex]\overrightarrow{r_{u}}=\lt x_{u},y_{u},z_{u}\gt[/latex]and [latex]\overrightarrow{r_{v}}=\lt x_{v},y_{v},z_{v}\gt[/latex], and the vector
[latex]\overrightarrow{r_{u}}\times\overrightarrow{r_{v}}[/latex]is a normal vector to the tangent plane.
Example 5: Find the equation of the tangent plane at [latex](2,3,0)[/latex]of the surface [latex]\overrightarrow{r(u,v)}=\lt u+v,3u^{2},u-v\gt[/latex].
Exercise 5: Find the equation of the tangent plane at [latex](1,1,0)[/latex]of the surface [latex]\overrightarrow{r(u,v)}=\lt u+v^{2},u^{2}+v,v\gt[/latex].
Example 6: Find the parametric representation of the sphere [latex]x^{2}+y^{2}+z^{2}=4[/latex]that lies above the cone [latex]z=\sqrt{x^{2}+y^{2}}[/latex].
Group work:
1. Find the parametric representation of the cylinder [latex]x^{2}+z^{2}=25[/latex] that lies above the [latex]xy[/latex]-plane and between the planes [latex]y=5[/latex]and [latex]y=-5[/latex].
2. Find the parametric representation of the sphere [latex]x^{2}+y^{2}+z^{2}=64[/latex] that lies between [latex]z=0[/latex]and [latex]z=\sqrt{32}[/latex].
3. Find the parametric representation of the plane [latex]z=x+4[/latex] that lies inside the cylinder [latex]x^{2}+y^{2}=4[/latex].
4. Find the equation of the tangent plane at [latex](0,4,1)[/latex] of the surface [latex]\overrightarrow{r(u,v)}=\lt u^{3}-v^{3},v+3u,2u-v\gt[/latex].