Parametric Equations and Oriented Paths, precalculus lesson

A delivery drone’s flight log stores two numbers for every second, how far east and how far north the drone is. Neither number alone describes the trip. Together they say where the drone was at each moment, where it started, where it ended and which way it was going. Parametric equations describe paths the same way, and they can draw loops and curves that double back, which no single function $y = f(x)$ can do.

Key terms

Parameter

Definition When both coordinates of a point are given as functions of a third variable, $x = f(t)$ and $y = g(t)$ , the variable $t$ is called the parameter. Each allowed value of $t$ produces exactly one point $(f(t), g(t))$ . The parameter is often time, but it can also be an angle or any other real quantity.

In plain words The parameter is a dial. Set it to a value and the two formulas tell you where the point is.

Example A drone starts at the origin of a map, and after $t$ seconds it is at $x = 3t$ , $y = 4t$ , in meters east and north. At $t = 2$ it is at $(6, 8)$ , which is $\sqrt{36 + 64} = 10$ meters from where it started.

Paired coordinate functions

Definition Parametric equations in the plane pair two functions $f$ and $g$ with the same domain $I$ , written

x = f(t), \qquad y = g(t), \qquad t \in I.

The functions $f$ and $g$ are the coordinate functions. When $f$ and $g$ are continuous on an interval $I$ , the set of points

C = \{ (f(t), g(t)) : t \in I \}

is called a plane curve, and the pair of equations is a parametrization of $C$ .

In plain words One function says where the point is from left to right, the other says where it is up and down, and both read the same clock. Every graph $y = h(x)$ fits this pattern with $x = t$ and $y = h(t)$ , but many curves that are not graphs fit it too.

Example Take $x = 1 - t^2$ and $y = t$ .

$t$	$-2$	$-1$	$0$	$1$	$2$
$(x, y)$	$(-3, -2)$	$(0, -1)$	$(1, 0)$	$(0, 1)$	$(-3, 2)$

The points lie on a parabola that opens to the left. The two points $(0, -1)$ and $(0, 1)$ share the same $x$ , so this curve is not the graph of any function of $x$ .

Parameter interval

Definition The parameter interval is the set $I$ of values of $t$ that are used. When $I = [a, b]$ is a closed interval, the point $(f(a), g(a))$ is the initial point and $(f(b), g(b))$ is the terminal point of the path. The same formulas on a different interval can give a different curve.

In plain words The formulas say how the point moves. The interval says when the motion starts and when it stops.

Example The point $x = \cos t$ , $y = \sin t$ stays on the unit circle, but how much of the circle it covers depends on the interval.

Interval	Initial point	Terminal point	Path
$[0, \frac{\pi}{2}]$	$(1, 0)$	$(0, 1)$	quarter circle in the first quadrant
$[\frac{\pi}{2}, \frac{3\pi}{2}]$	$(0, 1)$	$(0, -1)$	left half of the circle
$[0, 2\pi]$	$(1, 0)$	$(1, 0)$	the whole circle once

Orientation

Definition The orientation of a parametrized curve is the direction in which the point $(f(t), g(t))$ moves along the curve as $t$ increases. On a sketch it is shown with arrows along the curve.

In plain words Orientation is the direction of travel. The set of points alone does not show it, so you read it from how the coordinates change as $t$ grows.

Example Compare $x = \cos t$ , $y = \sin t$ with $x = \cos t$ , $y = -\sin t$ , both for $0 \le t \le 2\pi$ . Both trace the unit circle and both start at $(1, 0)$ . For $0 < t < \frac{\pi}{2}$ the first point has $y > 0$ and reaches $(0, 1)$ , while the second has $y < 0$ and reaches $(0, -1)$ . So the first goes around counterclockwise and the second clockwise.

Tracing speed

Definition Tracing speed measures how fast the point moves along its path per unit of the parameter. If the point travels a distance $L$ along the curve while $t$ runs from $t_1$ to $t_2$ with $t_1 < t_2$ , counting a stretch once for each time it is covered, then its average tracing speed on that interval is

\frac{L}{t_2 - t_1}.

The curve is traced at constant speed when this average is the same on every subinterval. Two cases are worth knowing. The line $x = x_0 + at$ , $y = y_0 + bt$ , with $a$ and $b$ not both zero, is traced at the constant speed $\sqrt{a^2 + b^2}$ , and the circle $x = R\cos(\omega t)$ , $y = R\sin(\omega t)$ with $R > 0$ and $\omega \ne 0$ is traced at the constant speed $R\lvert\omega\rvert$ . The speed at a single instant is made exact in calculus.

In plain words Two parametrizations can draw the same picture at different paces. Tracing speed is that pace, the distance along the curve covered per unit of $t$ .

Example The drone with $x = 3t$ , $y = 4t$ flies at a constant $\sqrt{9 + 16} = 5$ meters per second. Compare the point $x = t^2$ , $y = 0$ for $0 \le t \le 3$ , which moves along the $x$ -axis from $0$ to $9$ .

Interval of $t$	$[0, 1]$	$[1, 2]$	$[2, 3]$
Distance covered	$1$	$3$	$5$

Each unit of $t$ covers more ground than the one before, so this point speeds up. Its average tracing speed over $[0, 3]$ is $\frac{9}{3} = 3$ .

Same curve with different parametrizations

Definition Two parametrizations describe the same curve when they produce the same set of points. They can still differ in orientation, in tracing speed, in where they start and in how many times they pass over each part. One way to build a new parametrization is to substitute $t = h(s)$ , where $h$ is a continuous function that maps an interval $J$ onto $I$ . If $h$ is increasing the orientation is kept, and if $h$ is decreasing it is reversed.

In plain words A curve is a trail, and a parametrization is one particular walk along it. You can walk the same trail faster, slower, backwards or more than once.

Example Each of these traces the arc of $y = x^2$ with $-1 \le x \le 1$ , since in each case $y = x^2$ and $x$ takes every value from $-1$ to $1$ .

Parametrization	Interval	Start	End	Direction
$x = t$ , $y = t^2$	$[-1, 1]$	$(-1, 1)$	$(1, 1)$	left to right
$x = -t$ , $y = t^2$	$[-1, 1]$	$(1, 1)$	$(-1, 1)$	right to left
$x = \sin t$ , $y = \sin^2 t$	$[-\frac{\pi}{2}, \frac{\pi}{2}]$	$(-1, 1)$	$(1, 1)$	left to right

The second row is the first with $t$ replaced by $-t$ , a decreasing substitution, so the direction flips. The third row is the first with $t$ replaced by $\sin t$ , which is increasing on $[-\frac{\pi}{2}, \frac{\pi}{2}]$ , so the direction is kept but the pace changes. Running the third pair on $[-\frac{\pi}{2}, \frac{3\pi}{2}]$ instead takes the point to $(1, 1)$ and back again, so the arc is traced twice.

Check yourself

For $x = 2t - 1$ , $y = 3 - t$ with $0 \le t \le 4$ , find the initial and terminal points and describe the orientation and the tracing speed.

Show the answer

At $t = 0$ the point is $(-1, 3)$ and at $t = 4$ it is $(7, -1)$ . As $t$ increases, $x$ grows and $y$ shrinks, so the point moves right and down along the segment from $(-1, 3)$ to $(7, -1)$ . The segment is traced at the constant speed $\sqrt{2^2 + 1^2} = \sqrt{5}$ .

The point $x = 4\cos(3t)$ , $y = 4\sin(3t)$ moves on a circle. Find the radius, the tracing speed and the time for one full turn.

Show the answer

The radius is $4$ and the tracing speed is $4 \cdot 3 = 12$ . One full turn takes $3t$ through an angle of $2\pi$ , so it takes $t = \frac{2\pi}{3}$ . As a check, the circumference $8\pi$ divided by $\frac{2\pi}{3}$ is $12$ .

Do $x = t$ , $y = t^3$ and $x = -t$ , $y = -t^3$ , both for $-1 \le t \le 1$ , trace the same curve? Do they have the same orientation?

Show the answer

Both satisfy $y = x^3$ with $x$ running over all of $[-1, 1]$ , so they trace the same arc between $(-1, -1)$ and $(1, 1)$ . The first starts at $(-1, -1)$ and moves to the right, while the second starts at $(1, 1)$ and moves to the left. The orientations are opposite.

Parametric Equations and Oriented Paths

Key terms

Parameter

Paired coordinate functions

Parameter interval

Orientation

Tracing speed

Same curve with different parametrizations

Check yourself

How this lesson connects

Builds on

Used later in

Where this leads in Calculus