Parametric Equations of Curves

Try this

For some curves, the relation between $x$ and $y$ is hard to write down directly, yet each of them depends clearly on a third quantity. For a thrown object, both the horizontal position and the height are determined by time; for a point on a circle, both coordinates are determined by the angle turned. Call this quantity the parameter $t$ , and you get the parametric equations of the curve

{x = f (t) y = g (t)

The requirement: the point given by every allowed $t$ lies on the curve, and every point of the curve comes from at least one $t$ .

“Projectile”: a small dot is drawn after each equal time step. What about their horizontal gaps? Their vertical gaps?
“Circle”: what is the parameter $t$ in the picture? As $t$ goes from $0$ to $2 π$ , how many times does the point go around?
“Ellipse”: what are the two dashed circles for? Compare the readouts “eccentric angle $φ$ ” and “ $∠ x O P$ ”. When are they equal?
“Line”: pull $t$ below zero. Which side of $P_{0}$ is $P$ on? How is $∣ P_{0} P ∣$ related to $t$ ?
“Cycloid”: when the circle has rolled one full turn, is $P$ back on the ground? How far has the center moved?

A parameter need not have a physical meaning (time, angle, distance), but choosing one with a geometric meaning often makes a problem easier. The next stop is all about how to choose.

Predict first

For the projectile $x = 5 t, y = 53 t - 5 t^{2}$ , what is the maximum height reached?

Choose an answer first; then you can see the result.

Why

The cycloid: a circle of radius $r$ rolls along a line without slipping, and a fixed point $P$ on its rim traces a path. Finding a direct relation between $x$ and $y$ is nearly hopeless, but with the angle turned, $t$ , as parameter it takes two steps.

Deriving the parametric equations of the cycloid (r = 1)

Step 1 / 5

Starting point and parameter
At the start $P$ is at the origin and the center is at $(0, r)$ . Use the angle $t$ (in radians) through which the circle has turned as the parameter.

$t = 0 : P (0, 0)$
No slipping: distance rolled = arc turned
Look at P from the center
Add them up
One arch

Letting $t$ control $x$ and $y$ separately sidesteps a $y = F (x)$ that cannot be written down at all (try solving $x = t - sin t$ for $t$ ).

Practice

Exercise 1Basic

The curve $x = 2 + 3 cos t, y = - 1 + 3 sin t$ is a circle. Find the square of the distance from its center to the origin.

Exercise 2Basic

On the line $x = 1 + t cos 6 0^{\circ}, y = t sin 6 0^{\circ}$ , find the $x$ -coordinate of the point that lies below and to the left of $P_{0} (1, 0)$ at distance $4$ from it.

Exercise 3Advanced

The highest point of the first arch of the cycloid $x = 2 (t - sin t), y = 2 (1 - cos t)$ has $x$ -coordinate $k π$ . Find $k$ .

Exercise 4Challenge

An object is thrown with initial speed $20$ at elevation angle $α$ (take $g = 10$ , ignore air resistance). Its path is $x = 20 t cos α, y = 20 t sin α - 5 t^{2}$ . As $α$ varies, what is the maximum horizontal range?

Try this

Why

Deriving the parametric equations of the cycloid (r = 1)

Practice

Mistake Museum · Exhibits from this stop

Taking the $\varphi$ in the ellipse’s parametric equations to be the inclination of $OP$

Treating $|t|$ as a distance when the direction vector is not a unit vector