202205271015 Parametrization 001

equation $y=mx+c$ describes a straight line with slope $m$ and $y$ -intercept $c$ , as shown below:

With some scalar parameter $t$ parametrize the equation, in vector representation of units $\hat{\imath}$ and $\hat{\jmath}$ , by

$\mathbf{s}(t)=t\,\hat{\mathbf{i}} + (mt+c)\,\hat{\mathbf{j}}\quad\textrm{where } t\in (-\infty ,\infty)$

whereas for a quadratic equation $y=ax^2+bx+c$ which describes a parabolic curve, its parametric representation is

$\mathbf{s}(t)=t\,\hat{\mathbf{i}} + (at^2+bt+c)\,\hat{\mathbf{j}}\quad\textrm{where } t\in (-\infty ,\infty)$

and similarly for a circle of radius $r$ centered at the origin $O(0,0)$ , its locus is parametrized as

$\mathbf{s}(t)=r\cos t\,\hat{\mathbf{i}}+r\sin t\,\hat{\mathbf{j}}\quad\textrm{where } t\in [-2\pi ,2\pi ]$ .

(to be continued)