Parametrised Curves in Plane and Space: Full Guide

A parametrised curve describes a path as a function of a parameter t: r(t) = (x(t), y(t)) in the plane, or r(t) = (x(t), y(t), z(t)) in space. Instead of y as a function of x, position itself is a function of t — which handles circles, spirals, and any path a particle actually travels, including ones that fail the vertical-line test.

Parametrisation is the language of motion. Physics tracks projectiles, orbits, and charged particles this way; CAD software and fonts draw with parametrised Bézier curves; animators move characters along parametrised paths. Velocity and acceleration fall out immediately as r′(t) and r″(t).

Key Concepts

• Parameter: often time, but any variable works — arc length s is the geometrically natural choice.
• Velocity and speed: v(t) = r′(t) is the velocity vector (direction of motion); its magnitude |v| is the speed.
• Unit tangent: T = r′/|r′| points along the curve, defined wherever r′ ≠ 0 (regular points).
• Arc length: the true distance travelled along the curve, independent of parametrisation.
• Curvature κ: measures how sharply the curve bends — large κ means a tight turn; a straight line has κ = 0 and a circle of radius R has κ = 1/R.
• Reparametrisation: the same geometric curve can be traversed at different speeds; arc length and curvature do not care.

Formulas

• Arc length: L = ∫ab |r′(t)| dt.
• Plane curvature: κ = |x′y″ − y′x″| / (x′2 + y′2)3/2.
• Space curvature: κ = |r′ × r″| / |r′|3.
• Circle check: r(t) = (Rcos t, Rsin t) gives κ = 1/R, as expected.
• Assumption: r′ ≠ 0 at regular points. Limit: cusps (r′ = 0) need separate treatment.

Worked Example

Study the parabola r(t) = (t, t2) for 0 ≤ t ≤ 1. Velocity r′ = (1, 2t), so speed = √(1+4t2). Arc length L = ∫01 √(1+4t2)dt = [t√(1+4t2)/2 + sinh−1(2t)/4]01 = √5/2 + ln(2+√5)/4 ≈ 1.118 + 0.361 ≈ 1.479. Sanity check: the straight-line distance from (0,0) to (1,1) is √2 ≈ 1.414, and the curved path must be longer — 1.479 > 1.414 confirms it. Curvature: κ = |1·2 − 2t·0|/(1+4t2)3/2 = 2/(1+4t2)3/2. At the vertex t = 0, κ = 2 (the tightest bend); as t grows, κ → 0 and the parabola flattens — matching the picture.

Common Misconceptions

• The parameter must be time. Any parameter works; arc length is often the most convenient for geometry.
• Speed is |r″|. No — speed is |r′|; r″ is acceleration, mixing speeding-up with turning.
• A straight line's curvature is undefined. It is perfectly well defined: κ = 0, since nothing bends.

Real-World Applications

• Projectile and orbital trajectories are parametrised curves with t as time.
• Road and roller-coaster design uses clothoid curves whose curvature changes gradually for comfort.
• CNC machines and 3D printers follow parametrised toolpaths; fonts and vector graphics use Bézier curves.
• Computer animation moves objects along parametrised splines with controlled speed profiles.

A parametrised curve r(t) captures motion and shape that y = f(x) cannot, with velocity r′, speed |r′|, arc length ∫|r′|dt, and curvature κ = |x′y″ − y′x″|/(x′2+y′2)3/2 quantifying bending. The parabola example shows arc length exceeding the straight-line distance and curvature peaking at the vertex. These are the core tools of kinematics, CAD, and graphics.