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A cycloid is the path traced by a single fixed point on the rim of a circle as that circle rolls, without slipping, along a straight line — described by the parametric equations x = r(t − sin t), y = r(1 − cos t), where r is the rolling circle's radius and t is the angle it has rotated through (in radians). The curve looks like a series of connected arches, and it has a surprisingly rich history in physics and engineering: it's the solution to both the brachistochrone problem (the shape of a wire down which a bead slides from one point to another, under gravity alone, in the least possible time) and the tautochrone problem (a ball released from any point on an inverted cycloid reaches the bottom in exactly the same time, regardless of starting height) — properties that made cycloidal shapes historically important in pendulum clock design and are still relevant in the tooth profile of certain gear systems.
How it works
Enter the generating circle's radius r and the rotation angle t (the angle, in radians, the circle has rolled through since the tracked point last touched the line). The calculator applies the parametric equations directly: x = r(t − sin t) gives the horizontal position, combining the circle's straight-line rolling distance (rt) with a sinusoidal correction for the point's position on the rim; y = r(1 − cos t) gives the height above the line, which oscillates between 0 (when the tracked point touches the ground, at t = 0, 2π, 4π, ...) and 2r (the highest point of each arch, at t = π, 3π, ...).
- Enter generating circle radius (r).
- Enter rotation angle (parameter t).
- Enter angle unit.
- Click Calculate to see your results.
Examples
r = 1, t = 180° (π radians)
x = 1×(π − sin π) = π − 0 = π ≈ 3.1416. y = 1×(1 − cos π) = 1×(1−(−1)) = 2 — the highest point of the arch, exactly 2r above the line.
r = 2, t = 90° (π/2 radians)
x = 2×(π/2 − sin(π/2)) = 2×(1.5708 − 1) = 2×0.5708 ≈ 1.1416. y = 2×(1 − cos(π/2)) = 2×(1 − 0) = 2.
r = 1, t = 360° (2π radians)
x = 1×(2π − sin(2π)) = 2π − 0 ≈ 6.2832. y = 1×(1 − cos(2π)) = 1×(1−1) = 0 — the point has returned to touch the line again after one full rotation, having traveled a horizontal distance of exactly the circle's circumference, 2πr.
Who should use it
- Physics coursework on the brachistochrone and tautochrone problems.
- Understanding cycloidal gear tooth profiles in mechanical engineering.
- Exploring parametric curve behavior in calculus.
Industry applications
- Physics and mechanical engineering education
- Gear and cam design (cycloidal profiles)
Advantages
- Directly applies the exact parametric equations for any radius and angle.
- Useful for exploring the brachistochrone and tautochrone properties conceptually.
Limitations
- Only models the standard cycloid (a point exactly on the rim) — a point inside or outside the rolling circle traces a related but different curve (a curtate or prolate cycloid).
Common mistakes to avoid
- Using degrees directly in the formula without converting to radians first (the calculator handles this automatically based on your selected unit, but manual calculations must convert first).
- Forgetting that y never exceeds 2r — the highest possible point of any cycloid arch is exactly twice the generating circle's radius.
- Confusing the rotation angle t with the physical position of the circle's center — t is a cumulative rotation measure, not a simple positional angle.
Best practices
- Always convert t to radians before applying the parametric equations by hand.
- Use y's bounded range (0 to 2r) as a quick sanity check on computed results.
- Remember that one full arch corresponds to t ranging from 0 to 2π, spanning a horizontal distance of 2πr.
Tips
- Exploring a different rolling or winding curve? See the Spiral Length Calculator.