In which cases is there a need for U and V in parametric equations

Question

I'm am reviewing parametric equations (to get a better grasp over how they are used to make shapes in computer graphics) and currently I have an understanding of how the parameter $t$ is used to essentially scale a position vector where $t \in R$ and another vector can be added to $t *v$ to get a point on another line as is explained in this video. What I don't understand is how parameters of $u$ and $v$ could be used within parametric equations. I think that if I could see an example of how $u$ and $v$ are used to make a 3d cylinder for example, that would really help provide some intuition on the subject. Any general overview of applications of such equations would also be helpful, thanks.

Answer 1

The equation of a plane uses two variables(can be $u$ and $v$) in its parametric equation. In general the equation of a plane is written in it's cartesian form(since a plane is a function $f(x,y) = z$) instead of its parametric form. A line on the otherhand is written in a parametric form in 3space(not so much in 2space) because it is easier to use than its cartesian form(which is a triple equality in the form $\frac{x - x_0}{d_x} = \frac{y - y_0}{d_y} = \frac{z - z_0}{d_z}$). Using $u$ and $v$ is popular when described parametric surfaces. Here is the parametric form of a sphere \begin{align} x &= r\cos(u)\sin(v)\\ y &= r\sin(u)\sin(v)\\ z &= r\cos(v) \end{align} where $r$ is the radius, $0 \leq u \leq 2\pi$ and $0 \leq v \leq \pi$. The reason $u$ and $v$ have restrictions is because the surface is closed. If it were open(like a function) then it wouldn't matter, but if the surface is a function it would be easier to use its cartesian form $z = f(x,y)$. If you use khan acadmeny and are interested in parametric surfaces I think there is a section on it in the multivariable calculus section.

Parametric equations in 3d are important when describing surfaces that are not functions. When you study multivariable calculus and learn about flux and surface integrals, this idea becomes independant. It also makes calculations of integrals easier. Just like in 2d space, its easier to integrate a circle in its parametric form than its cartesian($x^2 + y^2 = r^2$).