Tangents with Parametric Equations

A tangent line is simply the straight line that matches the direction of a curve at one point, and its slope tells you how steep the curve is right there. The name comes from the Latin tangere, "to touch": up close, the line just grazes the curve. It's good for finding where a path peaks, levels off or turns straight up, and for knowing which way a moving thing is heading at any moment. For a moving point, slope is still rise over run, and the fraction $\frac{dy/dt}{dx/dt}$ is just two pieces: $dy/dt$ (how fast the point moves up) on top and $dx/dt$ (how fast it moves right) on the bottom. If in one second the point goes $3$ units right and $6$ up, the path rose $6$ over a run of $3$, so its slope is $2$. The second-derivative formula looks scarier, but it's the same move done twice; bear with it.

Where the formula comes from. In the last lesson we described curves by a moving point, and the natural next question is the one you asked about every graph in Calculus I: what's the slope? You could eliminate $t$ and differentiate the result, but that's often messy or impossible. There's a shortcut. The chain rule says $\dfrac{dy}{dt} = \dfrac{dy}{dx}\cdot\dfrac{dx}{dt}$, and solving for $\dfrac{dy}{dx}$ gives the formula above. It's the rise-over-run picture from the opener, made official.

Thinking of it as vertical speed over horizontal speed tells you right away where the special tangents are:

• Horizontal tangent: $\dfrac{dy}{dt} = 0$ while $\dfrac{dx}{dt} \ne 0$. For an instant the point isn't moving up or down, only sideways.
• Vertical tangent: $\dfrac{dx}{dt} = 0$ while $\dfrac{dy}{dt} \ne 0$. Now it's only moving up or down.
• If both are zero at once, the point has stopped completely, and the formula gives $\frac00$. Look closer: it's often a cusp, like the bottom of each cycloid arch.

The tangent line at $t = t_0$ passes through $(f(t_0), g(t_0))$ with slope $\dfrac{dy}{dx}\Big|_{t_0}$, just like any tangent line you've written before.

The second derivative. Concavity is how fast the slope changes as $x$ changes. The slope $\frac{dy}{dx}$ is itself a function of $t$, so we use the same trick on it: its rate of change in $t$, divided by how fast $x$ changes in $t$.

A common mistake. It's tempting to think $\dfrac{d^2y}{dx^2}$ is $\dfrac{d^2y/dt^2}{d^2x/dt^2}$, since the first derivative was a ratio of $t$-derivatives and this looks like the same pattern one level up. It isn't. Differentiate the slope $\frac{dy}{dx}$ with respect to $t$, then divide by $dx/dt$ once more.

A car's velocity vector is tangent to its path. Game engines use parametric tangents to orient a camera or a character along a track, and the Bézier lab draws the tangent direction as the curve is traced.