Parametric Curves and Their Calculus

A curve in the plane need not be the graph of a function. A particle can loop back over itself, cross its own path, or trace a closed circuit, and no equation $y=F(x)$ can describe such a path because it fails the Vertical Line Test. To address this, let both coordinates depend on a common third variable instead of making $y$ depend on $x$.

The parameter carries information that a bare curve discards. It fixes an orientation (the direction of increasing $t$) and a schedule (which point is occupied when). If $t$ is time, $(f(t),g(t))$ is the position of a moving particle, and the same set of points can be traced by different parametric equations at different speeds and in different directions.

Tracing and orientation

Consider $x=t^2-2t$, $y=t+1$. Tabulating a few values of $t$ and plotting the points reveals a curve that opens to the right, traced upward as $t$ increases.

To confirm the shape, eliminate the parameter. From $y=t+1$ we get $t=y-1$, and substituting into the first equation,

a parabola opening rightward. Eliminating $t$ produces a Cartesian equation for the underlying set of points, but it discards the orientation and the timing. Those are retained only in the parametric form.

Different parametrizations, same points. The equations $x=\cos t$, $y=\sin t$ with $0\le t\le 2\pi$ satisfy $x^2+y^2=1$ and trace the unit circle once counterclockwise from $(1,0)$. The equations $x=\sin 2t$, $y=\cos 2t$ satisfy the same Cartesian equation but start at $(0,1)$ and trace the circle twice clockwise. A curve is a set of points; a parametric curve is that set together with a way of moving along it.

Any function graph $y=F(x)$ is trivially parametric with $x=t$, $y=F(t)$, and any curve $x=G(y)$ becomes $x=G(t)$, $y=t$. Parametric form is strictly more general.

The cycloid

Parametric form is essential when a curve's natural description is kinematic. Roll a circle of radius $r$ along the $x$-axis and track a fixed point $P$ on its rim, starting at the origin. The path $P$ traces is a cycloid.

Take the angle of rotation $\theta$ as the parameter, with $\theta =0$ when $P$ is at the origin. After the circle has turned through $\theta$ radians it has rolled a distance equal to the arc it laid down, so its center sits at $(r\theta ,r)$. Measuring $P$'s displacement from that center gives its coordinates.

The horizontal shift is the rolled distance $r\theta$ minus the horizontal offset $r\sin \theta$ of $P$ from the center; the height is $r$ minus the vertical offset $r\cos \theta$:

One arch corresponds to $0\le \theta \le 2\pi$. Eliminating $\theta$ here yields a Cartesian equation too complicated to be useful, so the parametric description is the one used in practice.

Tangents

The slope of a parametric curve follows from the chain rule. Writing $y$ as a function of $x$ along the curve,¹

and solving gives the slope directly in terms of the parameter derivatives.

Reading the curve as a moving particle, $\mathbf{d}x/\mathbf{d}t$ and $\mathbf{d}y/\mathbf{d}t$ are the horizontal and vertical velocities, and the slope is their ratio. The mnemonic is that the $\mathbf{d}t$'s cancel.

Second derivative. Concavity needs ${\mathbf{d}}^{2}y/\mathbf{d}{x}^2$, obtained by applying the same slope rule to $\mathbf{d}y/\mathbf{d}x$ in place of $y$:

A common error is to write ${\mathbf{d}}^{2}y/\mathbf{d}{x}^2=({\mathbf{d}}^{2}y/\mathbf{d}{t}^2)/({\mathbf{d}}^{2}x/\mathbf{d}{t}^2)$. That is false; the outer derivative acts on the whole slope quotient.

A curve that crosses itself

A single parametric curve can pass through one point at two different times, and the two passages carry different tangents.

Areas

The area under a curve traced by $x=f(t)$, $y=g(t)$ as $t$ runs from $\alpha$ to $\beta$ comes from the substitution rule applied to $A={\int }_a^{b}y\mathbf{d}x$ with $\mathbf{d}x=f^{\prime }(t)\mathbf{d}t$:²

where the limits $\alpha ,\beta$ are the parameter values giving $x=a$ and $x=b$. For the area under one arch of the cycloid, $x=r(\theta -\sin \theta )$ gives $\mathbf{d}x=r(1-\cos \theta )\mathbf{d}\theta$, and $\theta$ runs from $0$ to $2\pi$:

Expanding $(1-\cos \theta {)}^2=1-2\cos \theta +\frac{1}{2}(1+\cos 2\theta )$ and integrating over a full period leaves only the constant terms $1+\frac{1}{2}$:

The area under one arch is exactly three times the area of the rolling circle.

Arc length

For a curve $y=F(x)$ the length is $\int \sqrt{1+(\mathbf{d}y/\mathbf{d}x{)}^2}\mathbf{d}x$. Substituting the parametric slope and $\mathbf{d}x=(\mathbf{d}x/\mathbf{d}t)\mathbf{d}t$ converts this to an integral over $t$; the same formula holds for any traced-once curve, proved directly by polygonal approximation.

The traversed-once condition matters. For the circle $x=\sin 2t$, $y=\cos 2t$ on $[0,2\pi ]$ the integral gives $4\pi$, twice the circumference, because the point rounds the circle twice.

Length of one cycloid arch. With $\mathbf{d}x/\mathbf{d}\theta =r(1-\cos \theta )$ and $\mathbf{d}y/\mathbf{d}\theta =r\sin \theta$,

The half-angle identity $1-\cos \theta =2{\sin }^2(\theta /2)$ turns the square root into $2r\sin (\theta /2)$ on $0\le \theta \le 2\pi$, so

One arch has length $8r$, eight times the generating radius.

Surface area

Rotating a parametric curve about the $x$-axis and adapting the surface-of- revolution formula gives, with $g(t)\ge 0$ and the curve traced once,

where $\mathbf{d}s=\sqrt{(\mathbf{d}x/\mathbf{d}t{)}^2+(\mathbf{d}y/\mathbf{d}t{)}^2}\mathbf{d}t$ is the arc-length element. For the semicircle $x=r\cos t$, $y=r\sin t$, $0\le t\le \pi$, the element is $\mathbf{d}s=r\mathbf{d}t$, and rotation about the $x$-axis sweeps out a sphere:

the standard surface area of a sphere of radius $r$.

The parametric calculus, collected

Every formula above replaces $\mathbf{d}x$ by $(\mathbf{d}x/\mathbf{d}t)\mathbf{d}t$ and integrates over the parameter. The single arc-length element $\mathbf{d}s$ recurs in length, surface area, mass, and centroid computations.

To study an unfamiliar parametric curve, a fixed sequence of steps recovers its shape without ever eliminating the parameter.