Arc Length, Curvature, and Motion in Space

A vector function carries two kinds of information about its curve. Intrinsic to the shape are its length and how sharply it bends, captured by arc length and curvature. Treating the parameter as time instead reads off velocity and acceleration. The two meet in one identity: a moving particle's acceleration splits into a part that changes its speed and a part that turns it, and the turning part is curvature times speed squared.

Arc length

Approximate a curve by a chain of short segments and add their lengths; refining the chain drives the sum to an integral of speed. Since the speed of $\vec{r}(t)$ is $\Vert {\vec{r}}^{\prime }(t)\Vert$, the length is its integral.

For the helix $\vec{r}(t)=\cos t\hat{ı}+\sin t\hat{ȷ}+t\hat{k}$, ${\vec{r}}^{\prime }(t)=⟨-\sin t,\cos t,1⟩$ has constant length $\Vert {\vec{r}}^{\prime }(t)\Vert =\sqrt{{\sin }^{2}t+{\cos }^{2}t+1}=\sqrt{2}$, so one full turn from $t=0$ to $t=2\pi$ has length ${\int }_0^{2\pi }\sqrt{2}\mathbf{d}t=2\sqrt{2}\pi$. The value is independent of how the curve is parametrized: any parametrization of the same curve gives the same length.

The arc length function

Fixing the lower limit and letting the upper limit vary turns length into a function of position along the curve.

Because $s$ measures distance traveled along the curve, it is a coordinate-free parameter fixed by the shape alone. Reparametrizing by arc length rewrites the curve so that moving one unit of parameter means moving one unit of distance. For the helix, $s(t)={\int }_0^t\sqrt{2}\mathbf{d}u=\sqrt{2}t$, so $t=s/\sqrt{2}$ and

Now $\vec{r}(t(3))$ is the point exactly $3$ units of length along the curve from the start.

Curvature

The unit tangent$\hat{T}$ points along the curve; how fast its direction turns, measured against distance rather than parameter, is the curvature. Using arc length keeps the definition independent of parametrization, and since $\hat{T}$ has constant length, only its direction changes.

The chain rule $\mathbf{d}\hat{T}/\mathbf{d}s=(\mathbf{d}\hat{T}/\mathbf{d}t)/(\mathbf{d}s/\mathbf{d}t)$ with $\mathbf{d}s/\mathbf{d}t=\Vert {\vec{r}}^{\prime }\Vert$ converts this to a form computable in the original parameter:

A circle of radius $a$, parametrized $\vec{r}(t)=a\cos t\hat{ı}+a\sin t\hat{ȷ}$, has $\hat{T}(t)=⟨-\sin t,\cos t⟩$ and $\Vert {\hat{T}}^{\prime }(t)\Vert =1$ while $\Vert {\vec{r}}^{\prime }(t)\Vert =a$, so $\kappa =1/a$: small circles bend hard, large circles bend gently, and a straight line has $\kappa =0$. A more convenient formula avoids computing $\hat{T}$ altogether.

For the twisted cubic $\vec{r}(t)=⟨t,t^2,t^3⟩$, ${\vec{r}}^{\prime }(t)=⟨1,2t,3t^2⟩$ and ${\vec{r}}^{\prime \prime }(t)=⟨0,2,6t⟩$ cross to $⟨6t^2,-6t,2⟩$, so

which equals $2$ at the origin. For a plane curve $y=f(x)$, taking $x$ as the parameter reduces the theorem to

The parabola $y=x^2$ then has $\kappa (x)=2/(1+4x^2{)}^{3/2}$, largest at the vertex ($\kappa (0)=2$) and tending to $0$ as the arms flatten out.

The normal and binormal vectors

At each point of the curve, many vectors are perpendicular to $\hat{T}$. One is singled out by the fact that $\hat{T}$ has constant length, which forces ${\hat{T}}^{\prime }\perp \hat{T}$. Normalizing ${\hat{T}}^{\prime }$ gives the direction the curve is turning, and crossing it with $\hat{T}$ completes an orthonormal triple.

The plane through the point spanned by $\hat{T}$ and $\hat{N}$ is the osculating plane, the plane the curve is momentarily bending within. Inside it sits the osculating circle: the circle tangent to the curve at the point, on the concave side, with the same curvature. Its radius $\rho =1/\kappa$ is the radius of curvature, and its center lies a distance $\rho$ from the point along $\hat{N}$. For $y=x^2$ at the origin, $\kappa =2$, so $\rho =\frac{1}{2}$ and the osculating circle is $x^2+(y-\frac{1}{2}{)}^2=\frac{1}{4}$ — the circle that best matches the parabola's bend there.

Each of the five quantities of the moving frame is computable from ${\vec{r}}^{\prime }$ and ${\vec{r}}^{\prime \prime }$ alone, and each has a geometric job.

Motion in space

Now read $\vec{r}(t)$ as the position of a particle at time $t$. The derivatives that gave tangent and curvature acquire physical names.

Velocity is the tangent vector, so it points along the path; speed is its length, the rate of distance covered. For $\vec{r}(t)=t^3\hat{ı}+t^2\hat{ȷ}$, $\vec{v}(t)=3t^2\hat{ı}+2t\hat{ȷ}$, $\vec{a}(t)=6t\hat{ı}+2\hat{ȷ}$, and the speed is $\sqrt{9t^4+4t^2}$.

Running the derivatives backward, integration recovers motion from its rates. Given acceleration and initial data,

By Newton's second law, $\vec{F}(t)=m\vec{a}(t)$, a known force determines the acceleration, and integrating twice yields the trajectory.

Tangential and normal components of acceleration

Acceleration need not point along the motion. Writing speed as $v=\Vert \vec{v}\Vert$ so that $\vec{v}=v\hat{T}$, and differentiating with the product rule and ${\hat{T}}^{\prime }=\kappa v\hat{N}$, resolves acceleration into two orthogonal pieces.

The tangential part $a_T$ changes the speed; the normal part $a_N=\kappa v^2$ changes the direction and always points toward the concave side. The binormal $\hat{B}$ never appears — acceleration lies entirely in the osculating plane. The $\kappa v^2$ form matches experience: a sharp turn (large $\kappa$) or a high speed both press a passenger against the car door, and doubling the speed quadruples the sideways push.

The same decomposition, applied to a planet moving under the sun's gravity, reproduces Kepler's three laws of planetary motion.