Examples

Parametric circles

The standard equation of a circle of radius $R$ centered at the point $(h,k)$:

$${(x-h)}^2+{(y-k)}^2=R^2$$

This equation says that the distance from a point $(x,y)$ on the circle to the center point $(h,k)$ equals $R$. This fact defines the circle.

Parametric coordinates for the circle:

$$x=h+R\cos t,y=k+R\sin t,t\in [\mathrm{0,2}\pi )$$

For example, the unit circle $x^2+y^2=1$ is parametrized by $x=\cos t$ and $y=\sin t$.

Parametric lines

(1) Parametric coordinate functions for a line:

$$x=a+rt,y=b+st,t\in (-\infty ,+\infty )$$

Compare this to the graph of linear function:

$$y=mx+b\text{some }m,b$$

Vertical lines cannot be described as the graph of a function. We must use $x=a$.

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(2) Parametric lines can describe all lines equally well, including horizontal and vertical lines.

A vertical line $x=a$ is achieved by setting $s=0$ and $r\ne 0$.

A horizontal line $y=b$ is achieved by setting $r=0$ and $s\ne 0$.

A non-vertical line $y=mx+b$ may be achieved by setting $s=m$ and $r=1$, and $a=0$.

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(3) Assuming that $r\ne 0$, the parametric coordinate functions describe a line satisfying:

$$\begin{array}{c}y=b+s\left(\frac{x-a}{r}\right)\\ \\ \gg \gg y=\frac{s}{r}\cdot x+(b-\frac{s}{r}\cdot a)\end{array}$$

and therefore the slope is $m=\frac{s}{r}$ and the $y$-intercept is $b-\frac{s}{r}\cdot a$.

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(4) The point-slope construction of a line has a parametric analogue:

$$\begin{array}{c}\text{point-slope line:}\\ \\ y-a=m(x-b)(x,y)=(a+t,b+mt)\end{array}$$

Parametric ellipses

The general equation of an ellipse centered at $(h,k)$ with half-axes $a$ and $b$ is:

$${\left(\frac{x-h}{a}\right)}^2+{\left(\frac{y-k}{b}\right)}^2=1$$

This equation represents a stretched unit circle:

• by $a$ in the $x$-axis
• by $b$ in the $y$-axis

Parametric coordinate functions for the general ellipse:

$$x=h+a\cos t,y=k+b\sin t,t\in [\mathrm{0,2}\pi )$$

Parametric cycloids

The cycloid is the curve traced by a pen attached to the rim of a wheel as it rolls.

It is easy to describe the cycloid parametrically. Consider the geometry of the situation:

The center $C$ of the wheel is moving rightwards at a constant speed of $1$, so its position is $(t,1)$. The angle is revolving at the same constant rate of $1$ (in radians) because the radius is $1$.

The triangle shown has base $\sin t$, so the $x$ coordinate is $t-\sin t$. The $y$ coordinate is $1-\cos t$.

So the coordinates of the point $P=(x,y)$ are given parametrically by:

$$x=t-\sin t,y=1-\cos t,t>0$$

If the circle has another radius, say $R$, then the parametric formulas change to:

$$x=Rt-R\sin t,y=R-R\cos t,t>0$$