Table of Contents
In this chapter we focus on what the graphs of parametric equations look like, and how to understand and work with these “parametric curves.”
A parametric equation (in the plane) expresses both coordinates of a point as functions of a parameter $t$:
$$
x = f(t), \quad y = g(t).
$$
As $t$ varies, the point $(x,y) = (f(t),g(t))$ traces out a curve in the $xy$–plane. That traced path is called a parametric curve.
The curve versus the parameter
A key idea is to separate:
- the set of points $(x,y)$ in the plane (the geometric curve), and
- the way the point moves along the curve as $t$ changes (the parametrization).
Different parametrizations can give the same geometric curve, possibly:
- in a different direction, or
- at a different “speed,” or
- with repeated passes over the same points.
Example: the line $y = 2x$.
One parametrization is
$$
x = t,\quad y = 2t.
$$
Another is
$$
x = 2t,\quad y = 4t.
$$
Geometrically, both sets of points describe the same line, but as $t$ varies we move along it at different speeds (in terms of how fast we cover distances with respect to $t$).
Tracing the curve: direction and orientation
Because $t$ has an order (e.g., $t$ increasing from $-1$ to $3$), a parametric curve naturally gets a direction or orientation.
Given $x = f(t)$ and $y = g(t)$:
- The orientation is determined by the order in which $t$ runs through its allowed values.
- As $t$ increases, the point $(f(t), g(t))$ “moves” along the curve.
Example: the circle
$$
x = \cos t,\quad y = \sin t,\quad 0 \le t \le 2\pi.
$$
- At $t = 0$, we get $(1,0)$.
- At $t = \frac{\pi}{2}$, we get $(0,1)$.
- At $t = \pi$, we get $(-1,0)$.
- At $t = \frac{3\pi}{2}$, we get $(0,-1)$.
So, as $t$ increases from $0$ to $2\pi$, the circle is traced counterclockwise exactly once.
If we instead use
$$
x = \cos(-t),\quad y = \sin(-t),
$$
then as $t$ increases, the same circle is traced clockwise. The geometric curve is the same, but the orientation is reversed.
The role of the parameter interval
Parametric curves always come with a domain for $t$, such as $t \in I$ where $I$ might be an interval like $[0,2\pi]$, $(0,\infty)$, etc.
The chosen interval can affect:
- How much of the curve is drawn.
- How many times the curve is traced.
Example: again use $x = \cos t,\ y = \sin t$.
- For $0 \le t \le 2\pi$, you get the whole circle traced once.
- For $0 \le t \le 4\pi$, you still see the same circle, but it is traced twice.
- For $0 \le t \le \pi$, you only see the upper half of the circle.
So when you read or draw a parametric curve, always pay attention to the range of $t$.
Eliminating the parameter (when possible)
Sometimes you can rewrite the parametric equations in the familiar “$y$ as a function of $x$” or in some other Cartesian form by eliminating $t$.
Example: $x = 3\cos t,\ y = 3\sin t$.
We know $\cos^2 t + \sin^2 t = 1$, so:
$$
\frac{x}{3} = \cos t, \quad \frac{y}{3} = \sin t,
$$
and therefore
$$
\left(\frac{x}{3}\right)^2 + \left(\frac{y}{3}\right)^2 = \cos^2 t + \sin^2 t = 1,
$$
which simplifies to
$$
\frac{x^2}{9} + \frac{y^2}{9} = 1 \quad \Rightarrow \quad x^2 + y^2 = 9.
$$
The geometric curve is a circle of radius $3$ centered at the origin. The parametric form adds information: how we go around the circle as $t$ increases.
Note that eliminating $t$ can lose information about:
- orientation (which way we travel), and
- how many times the curve is traced.
Cartesian equations describe the set of points, not the motion.
Examples of common parametric curves
1. Lines
A straight line can be parametrized by choosing a point on the line and a direction.
Let $(x_0,y_0)$ be a point on the line, and let $\langle a,b\rangle$ be a direction vector for the line. Then:
$$
x = x_0 + at,\quad y = y_0 + bt.
$$
As $t$ varies over all real numbers, the point moves along the line indefinitely in both directions.
If you restrict $t$ to an interval, such as $0 \le t \le 1$, you get a line segment from $(x_0,y_0)$ to $(x_0 + a, y_0 + b)$.
2. Circles and ellipses
A circle of radius $r$ centered at the origin is given by:
$$
x = r\cos t,\quad y = r\sin t.
$$
An ellipse centered at the origin with horizontal radius $a$ and vertical radius $b$ is:
$$
x = a\cos t,\quad y = b\sin t.
$$
As $t$ runs from $0$ to $2\pi$, you trace each curve once counterclockwise.
3. Cycloids and “rolling” curves (qualitatively)
There are more elaborate parametric curves, such as:
- Cycloid: the path traced by a point on the rim of a rolling circle.
- Hypotrochoids and epitrochoids: paths traced by points attached to circles rolling inside or outside other circles.
Their formulas are more complicated (and usually introduced later), but they demonstrate how parametric equations naturally model moving points and mechanical motion.
Sketching parametric curves by hand
To get a sense of a parametric curve without plotting many points, use these steps:
- Make a small table of values of $t$ and corresponding $(x,y)$ pairs, especially at easy or “special” values of $t$.
- Plot those points in the coordinate plane.
- Use arrowheads to indicate the direction of motion as $t$ increases, connecting the plotted points in order of increasing $t$.
- Notice if $x$ or $y$ is increasing or decreasing as $t$ passes each value; this helps with the shape and orientation.
Example: $x = t^2 - 1,\ y = t^3 - t$ for $-2 \le t \le 2$.
You might compute:
- $t=-2$: $(x,y)=(3,-6)$
- $t=-1$: $(x,y)=(0,0)$
- $t=0$: $(x,y)=(-1,0)$
- $t=1$: $(x,y)=(0,0)$
- $t=2$: $(x,y)=(3,6)$
Plot these points and connect them in order of $t=-2,-1,0,1,2$. Note that $(0,0)$ is passed twice—once when $t=-1$ and again when $t=1$. This shows how a parametric curve can cross itself, and how the point can visit the same location at different parameter values.
Closed and open parametric curves
A parametric curve may either:
- Close up and form a loop, or
- Remain open and not return to its starting point.
Closed curve
The circle $x = \cos t,\ y = \sin t$ with $0 \le t \le 2\pi$ is a closed curve:
- At $t=0$, $(x,y)=(1,0)$.
- At $t=2\pi$, $(x,y)=(1,0)$ again.
We return to the starting point and the path in between forms a closed loop.
Open curve
The parabola parametrized by
$$
x = t,\quad y = t^2
$$
for all real $t$ never “closes up.” As $t\to\infty$ or $t\to -\infty$, the point goes off to infinity and never returns.
Whether a parametric curve is closed depends on both the formulas and the parameter interval.
Self-intersections
Parametric curves can intersect themselves naturally. A self-intersection occurs when two different parameter values give the same point:
$$
(f(t_1),g(t_1)) = (f(t_2),g(t_2))\quad \text{with}\ t_1 \ne t_2.
$$
Example: the curve $x = \sin 2t,\ y = \sin t$ for $0 \le t \le 2\pi$ has a self-intersection at the origin.
You can see this by noting that:
- $t = 0$ gives $(0,0)$,
- $t = \pi$ gives $(0,0)$ again,
so the curve passes through the origin at least twice.
When sketching, be aware that a given point on the graph might be traced more than once, and at different stages in the motion.
Speed along a parametric curve (intuitive)
Each parametrization has a speed along the curve, which measures how fast the point moves as $t$ changes. The speed at a given $t$ is (intuitively) the rate at which the distance from the starting point increases. Formally, it will be expressed using derivatives in calculus; here we only use an intuitive view.
Consider two parametrizations of the same line segment from $(0,0)$ to $(2,0)$:
- $x = 2t,\ y = 0$ for $0 \le t \le 1$.
- $x = t,\ y = 0$ for $0 \le t \le 2$.
Geometrically, both describe the same path: the horizontal segment from $0$ to $2$ on the $x$–axis.
However:
- In the first, the point goes from $x=0$ to $x=2$ as $t$ runs from $0$ to $1$.
- In the second, it takes twice as long in terms of $t$ (from $t=0$ to $t=2$) to cover the same distance.
They have different speeds with respect to $t$. Parametrizations are therefore not unique, even if the geometric curve is.
Why parametric curves are useful
Parametric curves are especially useful when:
- $y$ is not a single-valued function of $x$ (for example, circles and many loops).
- The geometry comes naturally from motion or time (projectiles, planetary orbits, rotating objects).
- You want to describe a curve in terms of another quantity (like angle, time, or distance) rather than forcing $y$ to be written explicitly in terms of $x$.
They are the natural language for describing curves that arise from physical and geometric processes.
Summary of key ideas
- A parametric curve in the plane is the path traced by $(x,y) = (f(t),g(t))$ as the parameter $t$ varies.
- The geometric shape of the curve can often be found by eliminating $t$, though this can lose information about direction and how the curve is traced.
- The parameter interval matters: it affects how much of the curve you see and how many times it is traced.
- Parametric curves have an inherent orientation and can be closed, open, or have self-intersections.
- The same geometric curve can have many different parametrizations, which may differ in direction and speed.