Curves & Surfaces and their Parameterizations#
Revised
03 Mar 2023
Shell (sphere) in \(\mathbb{R}^3\)#
A shell (sphere) in \(\mathbb{R}^3\) is just the 2D surface which is \(R\) units from the center.
\( (x-a)^2+(y-b)^2+(z-c)^2=R^2 \)
Ball (solid sphere) in \(\mathbb{R}^3\)#
A ball (solid sphere) in \(\mathbb{R}^3\) is
\( (x-a)^2+(y-b)^2+(z-c)^2\le R^2 \)
Formula for Dimension#
Dimension of the system of equations = total number of variables - number of independent equations
Each equation makes one variable depedent, but inequalities do not force a dependence.
Vector Function#
A vector function is just a vector that has variables and that can change.
Example
\( \begin{aligned} \vec{r}(t)&=\langle2t,3t-1,5-t\rangle\\ \vec{r}(0)&=\langle0,-1,5\rangle\\ \vec{r}(1)&=\langle2,2,4\rangle\\ \vec{r}(2)&=\langle4,5,3\rangle\\ \end{aligned} \)
A vector function may be used to describe a changing position.
If a vector function with one variable produces position vectors, then the graph will be a curve (i.e., a one-dimensional path) which follows the path of an object moving along those positions.
A vector function which describes a path is called a parameterization of the path where the parameter \(t\) is an independent variable of the vector function.
A parameterization may be rewritten as a system of equations.
\( \begin{aligned} \vec{r}(t)=\langle2t,3t-1,5-t\rangle \iff \begin{aligned} x&=2t\\ y&=3t-1\\ z&=5-t\\ \end{aligned} \end{aligned} \)
Lines in \(\mathbb{R}^3\) and their parameterization#
Let a line in \(\mathbb{R}^3\) go through the point \((x,y,z)\) and move in the direction \(\langle a,b,c\rangle\).
Any point on the line can be written
\( \langle x,y,z\rangle+t\langle a,b,c\rangle \)
A line in \(\mathbb{R}^3\) which goes through point \(\vec{P}\) and has a direction vector \(\vec{v}\) has the parameterization
\( \vec{r}(t)=\vec{P}+t\vec{v} \)
\( \begin{aligned} t=0&\implies\vec{r}(0)=\langle x_0,y_0,z_0\rangle\\ t=1&\implies\vec{r}(1)=\langle x_1,y_1,z_1\rangle+\langle a,b,c\rangle\\ t=2&\implies\vec{r}(2)=\langle x_2,y_2,z_2\rangle+2\langle a,b,c\rangle\\ \end{aligned} \)
Parameterizations of lines are functions whose outputs are position vectors which describe the points on the line.
\( \vec{r}(t)=\vec{r}_0+t\vec{v} \)
\( \vec{r}(t)=\underbrace{\vec{r}_0}_{\text{starting point}}+t\underbrace{\vec{v}}_{\text{position vector}} \)
Lines in \(\mathbb{R}^2\) have slope.
Lines in \(\mathbb{R}^3\) don’t have slope; they have direction.
The domain of \(t\) may be restricted such that the line is a line segment.
\( \vec{r}(t)=\vec{r}_0+t\vec{v} \) where \(\,a\le t\le b\)
Then \(\vec{r}(a)\) and \(\vec{r}(b)\) are the endpoints of the line segment.