Vector functions, curves and motion

Lesson 1 described quantities that vary over a region, such as temperature over a plate. Many engineering problems instead follow something along a path: a vehicle on a highway, the tip of a robot arm, a satellite in orbit, a cutting tool on a machining center, the centerline of a pipe or a cable. The natural description is a vector function, which gives a position vector for each value of a parameter, usually time. This lesson develops the calculus of vector functions. Derivatives give velocity and acceleration; integrals recover position from acceleration; arc length and curvature describe the geometry of the path itself; and splitting acceleration into tangential and normal parts, or into radial and transverse parts, explains the forces felt by a driver on a curve or a part moving along a rotating arm. The ideas reappear in Lesson 7, where integrals along curves are used to compute work and circulation.

Vector functions and space curves

A vector function assigns a vector to each value of a parameter t:

As t varies, the tip of \(\mathbf{r}(t)\) traces a curve in space. The same curve can be traced by different parametrizations, quickly or slowly, forward or backward; the parametrization carries the information about motion, while the curve itself is only the path. Two standard examples:

• A straight line through \(\mathbf{r}_0\) in direction \(\mathbf{v}\): \(\mathbf{r}(t) = \mathbf{r}_0 + t\mathbf{v}\) (MATH-125).
• A circular helix of radius a that rises \(2\pi b\) per turn: \(\mathbf{r}(t) = a\cos t\,\mathbf{i} + a\sin t\,\mathbf{j} + bt\,\mathbf{k}\). Helices describe coil springs, spiral ramps, screw threads, DNA and the path of a charged particle in a magnetic field.

Limits, continuity and derivatives of vector functions are taken component by component.

Velocity, speed and acceleration

If \(\mathbf{r}(t)\) is the position of a particle, its velocity and acceleration are

The velocity vector is tangent to the path and points in the direction of motion; its length is the speed. The unit tangent vector is \(\mathbf{T} = \mathbf{v}/|\mathbf{v}|\). Going the other way, integrating the acceleration twice, with the initial velocity and position as constants of integration, recovers the motion. This is how inertial navigation systems in aircraft and phones estimate position from accelerometer readings, and why small sensor errors make the position estimate drift: an error in acceleration is integrated twice.

Arc length

The distance traveled along the curve from \(t = a\) to \(t = b\) is the integral of the speed:

This extends the arc length formula of MATH-101 to curves in space. Measuring position by arc length s instead of time gives the arc length parametrization, in which the curve is traced at unit speed. Road and rail alignments, pipe routes and tool paths are specified this way, by chainage (distance along the centerline).

Curvature

Curvature measures how sharply a curve bends: how fast the unit tangent turns per unit distance traveled. For a circle of radius R it is \(1/R\), and for a straight line it is zero. In general, the radius of curvature \(\rho = 1/\kappa\) is the radius of the circle that best fits the curve at that point. A convenient formula, valid for any parametrization, is

For a plane curve \(y = f(x)\), this reduces to \(\kappa = |f''|/(1 + f'^2)^{3/2}\), which for small slopes is approximately \(|f''|\), the approximation used in beam theory, where the curvature of the deflected beam is \(M/(EI)\).

Transition curves

A road or railway cannot jump straight from a tangent (zero curvature) to a circular arc (constant curvature): the sideways acceleration would change suddenly, jolting passengers and loads. Designers insert a transition curve, usually a clothoid (Euler spiral), whose curvature increases in proportion to arc length, \(\kappa = s/A^2\). The lateral acceleration then builds up at a steady rate as the vehicle enters the curve. The same idea, limiting the rate of change of acceleration (the jerk), shapes elevator motion profiles, cam designs and robot trajectories.

Tangential and normal acceleration

Acceleration can be split into a part along the path, which changes the speed, and a part perpendicular to it, toward the center of curvature, which changes the direction:

Here \(\mathbf{N}\) is the unit normal, pointing toward the center of curvature. The normal (centripetal) acceleration grows with the square of speed and with curvature, which is why speed limits drop sharply on tight curves. Together with the binormal \(\mathbf{B} = \mathbf{T} \times \mathbf{N}\), the vectors T and N form a moving frame (the Frenet frame) that travels with the particle; the rate at which B turns, the torsion, measures how the curve twists out of a plane.

Motion in polar coordinates

For motion in a plane described by distance r and angle θ, as for a part sliding along a rotating arm or a satellite in orbit, it is natural to use the unit vectors \(\mathbf{e}_r\) (outward) and \(\mathbf{e}_\theta\) (perpendicular, in the direction of increasing θ). Since these vectors turn as θ changes, differentiating \(\mathbf{r} = r\,\mathbf{e}_r\) twice gives

The term \(-r\dot{\theta}^2\) is the centripetal acceleration and \(2\dot{r}\dot{\theta}\) is the Coriolis acceleration, which appears whenever something moves radially on a rotating body.

Circular orbits

For a satellite in a circular orbit of radius r, gravity supplies exactly the normal acceleration: \(v^2/r = GM/r^2\), where \(GM = 3.986 \times 10^{14}\) m³/s² for the Earth. So the orbital speed is \(v = \sqrt{GM/r}\) and the period is \(T = 2\pi r/v = 2\pi\sqrt{r^3/GM}\). At 400 km altitude (r = 6771 km), this gives v = 7.67 km/s and a period of 92.4 minutes, the orbit of the International Space Station. Setting T equal to one sidereal day (86 164 s) gives r = 42 164 km, the geostationary orbit used by communication and weather satellites, which appear fixed in the sky. Here the acceleration is entirely normal: the speed is constant and the satellite falls continuously toward the Earth while moving sideways fast enough to keep missing it. The same balance sets the speed of a vehicle on a banked curve with no reliance on friction, and the magnetic field strength needed to bend a particle beam into a circle in an accelerator.