Trace the path of a point on a rolling circle. Compute precise coordinates, arc length (from 0 to t), area under the arch, slope, radius of curvature, and visualize real‑time cycloid geometry.
A cycloid is the curve traced by a fixed point on the rim of a circle rolling without slipping along a straight line. First studied by Galileo, Mersenne, and later formalized by Blaise Pascal and Christiaan Huygens, the cycloid holds exceptional properties in physics and calculus: it is the brachistochrone (curve of fastest descent) and the tautochrone (isochronous curve for a pendulum). Our dynamic calculator provides exact geometric data and real-time parameter exploration.
| Property | Formula (per arch, 0≤θ≤2π) | Example (r=2) |
|---|---|---|
| Arc length (one arch) | \(L = 8r\) | 16.000 units |
| Area under one arch | \(A = 3\pi r^2\) | 37.699 |
| Radius of curvature at vertex | \(4r\) | 8.000 |
| Slope at angle θ | \(\frac{dy}{dx} = \cot(\theta/2)\) | — |
In 1696, Johann Bernoulli challenged mathematicians to find the brachistochrone — the curve of quickest descent under gravity. Both Newton, Leibniz, and Bernoulli himself discovered that the cycloid solves the problem. Moreover, Huygens proved that the cycloid is tautochronous: a bead sliding on an inverted cycloid reaches the bottom in the same time regardless of starting point. This led to the invention of the cycloidal pendulum, which improved marine chronometers.
From a modern perspective, the cycloid inspired the development of the calculus of variations. Our interactive tool reveals these principles: vary the radius and parameter θ to observe slope, length, and area changes.
Christiaan Huygens discovered that a pendulum constrained by cycloidal jaws swings with a period independent of amplitude. For a pendulum length \(L = 4r\), the period is \(T = 2\pi\sqrt{L/g}\). Using our calculator with radius \(r = 0.5\,\text{m}\) (jaw radius), the cycloid’s tautochrone property ensures accurate timekeeping. The arc length per swing relates to the cycloid’s geometric precision — our tool reproduces these lengths instantly.
Numerical example: For a clock with \(r = 0.3\,\text{m}\), the cycloidal jaw length per full arch is \(8r = 2.4\,\text{m}\); the theoretical period becomes \(T = 2\pi\sqrt{4r/g} \approx 2\pi\sqrt{1.2/9.81} \approx 2.19\,\text{s}\), independent of swing amplitude. You can verify the arc length for any partial angle using the calculator above.
Given the parametric form, we evaluate positions exactly using double-precision math. Arc length from 0 to θ uses the closed-form integration: \[ s(\theta) = \int_0^\theta \sqrt{\left(\frac{dx}{d\phi}\right)^2+\left(\frac{dy}{d\phi}\right)^2}\,d\phi = 4r\left(1-\cos\frac{\theta}{2}\right) \quad \text{for } 0\le\theta\le2\pi, \] and for multiple arches we add full arch lengths \(8r\) per completed cycle. Slope is derived as \(\frac{dy/d\theta}{dx/d\theta} = \frac{\sin\theta}{1-\cos\theta} = \cot(\theta/2)\) (for θ not a multiple of 2π). Radius of curvature: \(\rho = \frac{4r}{|\sin(\theta/2)|}\). Partial area is computed via adaptive Simpson integration (error < 1e-9). Our interactive canvas renders the cycloid using adaptive sampling, maintaining sub‑pixel accuracy for smooth visual.