1. Introduction
A parametric curve is defined by functions x(t) and y(t) (or x(t), y(t), z(t) in 3D) where t is a parameter varying over an interval. This representation is more general than y = f(x) because it can describe self-intersecting curves, loops and spirals. Classical spirals have elegant parametric representations that reveal their geometric structure.
2. Parametric Representation of Spirals
In polar coordinates r = f(θ), conversion to parametric form is direct: x(θ) = f(θ)cos(θ), y(θ) = f(θ)sin(θ). For the Archimedean spiral (r = aθ): x = aθcos(θ), y = aθsin(θ). For the logarithmic spiral (r = ae^bθ): x = ae^(bθ)cos(θ), y = ae^(bθ)sin(θ).
3. Classical Spirals and Their Properties
- Cornu spiral (clothoid): r = a√θ, curvature proportional to arc length, used in road engineering
- Cotes spiral: r = ae^(bθ) with imaginary b, trajectory in a central force field
- Lituus: r² = a²/θ, asymptote to the polar axis
- Poinsot spiral: r = a/cos(bθ), regular intersections with radii
4. Lissajous Curves
Lissajous curves are defined by x = A sin(at + δ), y = B sin(bt). When a/b is rational, the curve is closed. When a/b is irrational, the curve densely fills a rectangle. These curves appear in physics (coupled oscillations), electronics (signal visualization) and music (harmonic representation).
5. Practical Applications
The clothoid (Cornu spiral) is used in road and railway design to ensure smooth transitions between straight sections and circular curves: its curvature grows linearly with arc length, minimizing lateral forces on vehicles. Parametric spirals are also used in Fresnel lens design, spiral antennas and robot trajectories.
6. Conclusion
Parametric representation unifies the different families of spirals in a common mathematical framework. It allows computing geometric properties (arc length, curvature, area) and designing curves with specific properties for engineering applications.