Section 3 Key Concepts
Parameterized Curves and Orientations.
- A curve (or path) \(C\) in \({\mathbb R}^n\) is the trace left by a moving particle \(P\text{.}\) The coordinates \((x_1,x_2,\dots ,x_n)\) of \(P\) are each functions of time \(t\) called parametric equations for \(C\) with parameter \(t\text{.}\)
- Parameterized curves are typically represented as vector functions with the component functions being the parametric equations. For example, if \(C\) is parameterized by \(x(t), y(t), z(t)\) in 3 - space, then \(C\) is represented by \(\vec{r} (t) =x(t)\vec{i} +y(t)\vec{j}+z(t) \vec{k}\text{.}\)
- An orientation on a curve \(C\) is a specific chosen direction.
Some Standard Curves.
- A line \(L\) line passing through point \(P\) in the direction of the vector \(\vec{v}\) is parameterized by \(\vec{r} (t) =\vec{P} +t \vec{v} .\) We typically call \(\vec{v}\) the direction vector of \(L\text{.}\)
- A circle with radius \(R\) centered at the point \((a,b)\) and oriented counter-clockwise is parameterized by \(\vec{r}(t) =(R\cos (t) +a)\vec{i} +(R\sin (t) +b)\vec{j}\text{.}\)
- The graph of \(y=f(x)\) oriented from left to right is parameterized by \(\vec{r}(t) =t\vec{i} +f(t)\vec{j}\text{.}\) Similarly the graph of \(x=f(y)\) oriented from bottom to top is parameterized by \(\vec{r}(t) =f(t)\vec{i} +t\vec{j}\text{.}\)
Simple Modifications of Curves.
- Domain restriction of the parameter \(t\) can restrict to a specific part of a curve.
- Switch orientation by replacing \(t\) with \(-t\text{.}\)
- Algebraic manipulations such as multiplication by different constants in the parameterization of a circle can create curves such as ellipses.
- Curve concatenation can be performed using piecewise parametric functions.
- Curves in 3 - space can be constructed by first finding parametric equations of projections of curves into one of the coordinate axis.
Vector Equations of Planes.
- Suppose \(\Pi\) is a plane passing through \(P(a,b,c)\) and \(\vec{v}\) and \(\vec{w}\) are non-parallel, non-zero vectors in \(\Pi\text{.}\) Then a vector equation for \(\Pi\) is given by \(\vec{p} (s,t)=\vec{P} +s \vec{v} +t\vec{w}\) where the parameters \(s\) and \(t\) run over all real numbers. The components of this equation are called parametric equations for \(\Pi\text{.}\)