Parametric curves and polar coordinates

Up to now, curves have been graphs of functions, \(y = f(x)\). Many curves that engineers care about are not graphs of functions at all: the path of a point on a rotating wheel, the profile of a cam, the outline of a gear tooth, the trajectory of a projectile described as a function of time, the radiation pattern of an antenna, the orbit of a satellite. Such curves are more naturally described in other ways. Parametric equations give the coordinates as functions of a third variable, usually time or an angle; polar coordinates describe a point by its distance and direction from an origin. This lesson shows how to describe curves both ways and how to apply calculus to them: slopes and tangent lines, arc lengths, speeds along a path, and areas enclosed by polar curves.

Parametric equations

A parametric curve is described by a pair of functions:

As the parameter \(t\) runs through its interval, the point \((x, y)\) traces out the curve, with a definite direction. The parameter often has a physical meaning: time for a moving object, an angle for a rotating part, distance along a road. Different parametrizations can trace the same curve at different speeds or in different directions: \(x = \cos t, y = \sin t\) and \(x = \cos 2t, y = \sin 2t\) both trace the unit circle, the second twice as fast.

Eliminating the parameter, by solving one equation for \(t\) and substituting into the other, sometimes gives an ordinary equation in \(x\) and \(y\). For example, \(x = 3\cos t\), \(y = 2\sin t\) gives \(\frac{x^2}{9} + \frac{y^2}{4} = \cos^2 t + \sin^2 t = 1\), an ellipse. Often, though, the parametric form is simpler and more informative than any equation in \(x\) and \(y\).

The cycloid

A point on the rim of a wheel of radius \(r\) rolling along a straight line traces a cycloid. If the wheel has turned through angle \(\theta\), its centre has moved \(r\theta\), and the point is:

The cycloid has cusps where the point touches the ground and momentarily stops. It has remarkable properties: it is the curve of fastest descent between two points under gravity (the brachistochrone), and a bead sliding on an inverted cycloid reaches the bottom in the same time from any starting point (the tautochrone), which seventeenth-century clockmakers used to design pendulums whose period does not depend on amplitude. Related curves, epicycloids and hypocycloids, generated by circles rolling on circles, describe gear tooth profiles and the motion of planetary gear trains.

Slopes and tangent lines

If \(x = f(t)\) and \(y = g(t)\) are differentiable and \(dx/dt \ne 0\), the chain rule gives the slope of the curve:

The curve has a horizontal tangent where \(dy/dt = 0\) (and \(dx/dt \ne 0\)), and a vertical tangent where \(dx/dt = 0\) (and \(dy/dt \ne 0\)). Where both are zero, the curve may have a cusp, as the cycloid does. The second derivative, needed for concavity, is the derivative of \(dy/dx\) with respect to \(t\), divided again by \(dx/dt\): \(\dfrac{d^2y}{dx^2} = \dfrac{d}{dt}\left(\dfrac{dy}{dx}\right) \Big/ \dfrac{dx}{dt}\).

Arc length and speed

The length of a parametric curve follows from the Pythagorean theorem applied to small steps: in time \(dt\), the point moves \(dx\) horizontally and \(dy\) vertically, a distance \(\sqrt{dx^2 + dy^2}\). Adding these:

When \(t\) is time, the integrand is the speed of the moving point, so the arc length is the distance travelled, the integral of speed. The arc length formula for a graph \(y = f(x)\) in Lesson 3 is the special case \(x = t\), \(y = f(t)\).

Most arc length integrals cannot be done in closed form (the ellipse in Lesson 2 is an example) and are evaluated numerically. In computer-aided design and machining, curves are usually represented by parametric polynomials (Bézier curves and splines), and their lengths, needed to plan cutting paths and feed rates, are computed numerically.

Polar coordinates

In polar coordinates, a point is located by its distance \(r\) from the origin (the pole) and the angle \(\theta\) from the positive \(x\)-axis. The conversions to and from Cartesian coordinates are:

Unlike Cartesian coordinates, polar coordinates are not unique: \((r, \theta)\), \((r, \theta + 2\pi)\) and \((-r, \theta + \pi)\) describe the same point. When converting from Cartesian to polar, choose the angle in the correct quadrant (the two-argument arctangent of MATH-100 does this).

Polar coordinates suit problems with a centre: rotating machines, radar and sonar, antenna patterns, orbits, stress around a hole in a plate, and flow around a cylinder. A polar curve \(r = f(\theta)\) gives the distance in each direction:

• \(r = a\) is a circle of radius \(a\) centred at the pole.
• \(r = 2a\cos\theta\) is a circle of radius \(a\) through the pole, centred at \((a, 0)\).
• \(r = a(1 + \cos\theta)\) is a cardioid, a heart-shaped curve.
• \(r = a\cos(n\theta)\) is a rose with \(n\) petals if \(n\) is odd and \(2n\) petals if \(n\) is even.
• \(r = a\theta\) is an Archimedean spiral, with equally spaced turns, the shape of a rolled-up tape, a groove on a vinyl record and many scroll compressors.

Slopes and areas in polar coordinates

A polar curve \(r = f(\theta)\) is a parametric curve with parameter \(\theta\): \(x = f(\theta)\cos\theta\), \(y = f(\theta)\sin\theta\). Its slope therefore follows from the parametric formula, \(dy/dx = (dy/d\theta)/(dx/d\theta)\), and its arc length is:

Areas are found with thin wedges instead of rectangles. A wedge of angle \(d\theta\) and radius \(r\) is approximately a sector with area \(\tfrac{1}{2}r^2\,d\theta\), so the area swept as \(\theta\) runs from \(\alpha\) to \(\beta\) is:

Orbits and conic sections

In polar coordinates with the pole at a focus, every conic section (ellipse, parabola or hyperbola) has the form \(r = \dfrac{p}{1 + e\cos\theta}\), where \(e\) is the eccentricity: \(e < 1\) for an ellipse, \(e = 1\) for a parabola, \(e > 1\) for a hyperbola. Newton showed that this is exactly the shape of an orbit under an inverse-square force, which is why satellite and planetary orbits are conic sections with the central body at a focus. Kepler's second law, that the line from the Sun to a planet sweeps out equal areas in equal times, is a statement about the polar area integral, \(dA/dt = \tfrac{1}{2}r^2\,d\theta/dt\) = constant, and follows from conservation of angular momentum.