Theory

Theory 1

Parametric curves are curves traced by the path of a ‘moving’ point. An independent parameter, such as $t$ for ‘time’, controls both $x$ and $y$ values through Cartesian coordinate functions $x(t)$ and $y(t)$. The coordinates of the moving point are $(x(t),y(t))$.

Parametric curve

A parametric curve is a function from parameter space $ℝ$ to the plane ${ℝ}^2$ given in terms of coordinate functions:

$$t⟼(x(t),y(t))$$

Other notations

Be aware that sometimes the coordinate functions are written with $f$ and $g$ (or yet other letters) like this:

$$(x,y)=(f(t),g(t))$$

Or simply equating coordinate letters with functions: $x=f(t),y=g(t)$

Sometimes a different parameter is used, like $s$ or $u$.

For example, suppose:

$$x=t^2-2t,y=t+1$$

The curve traced out is a parabola that opens horizontally:

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Given a parametric curve, we can create an equation satisfied by $x$ and $y$ variables by solving for $t$ in either coordinate function (inverting either $f$ or $g$) and plugging the result into the other function.

In the example:

$$\begin{array}{c}y=t+1\gg \gg t=y-1\\ \\ \gg \gg x=t^2-2t\gg \gg x={(y-1)}^2-2(y-1)\\ \\ \gg \gg x=y^2-4y+3\gg \gg x={(y-2)}^2-1\end{array}$$

This is the equation of a parabola centered at $(-\mathrm{1,2})$ that opens to the right.

Image of a parametric curve

The image of a parametric curve is the set of output points $(x(t),y(t))$ that are traversed by the moving point.

A parametric curve has hidden information that isn’t contained in the image:

• The time values $t$ when the moving point is found in various locations.
• The speed at which the curve is traversed.
• The direction in which the curve is traversed.

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We can reparametrize a parametric curve to use a different parameter or different coordinate functions while leaving the image unchanged.

In the previous example, shift $t$ by $1$:

$$\begin{array}{c}x={(t+1)}^2-2(t+1),y=(t+1)+1\\ \\ \gg \gg x=t^2-1,y=t+2\end{array}$$

Since the parameter $t$ and the parameter $t+1$ both cover the same values for $t\in (-\infty ,\infty )$, the same curve is traversed. But the moving point in the second, shifted version reaches any given location one unit earlier in time. (When $t=-1$ in the second version, the input to $x(t)$ and $y(t)$ is the same as when $t=0$ in the first one.)