🛝Brachistochrone Simulator - Cycloid Race, Tautochrone, Snell’s Law

How to use Brachistochrone Race (Cycloid vs Line, Arc, Parabola)

1. 1Drag A and B on the canvas to set the start and finish (B must be lower than A)
2. 2Tick the curves to race; press "✏️ Draw a curve" to sketch one from A to B
3. 3Press "Start race" to release all beads and compare times and ranks in the table
4. 4Add sliding friction or switch to Moon or Mars gravity and see what changes
5. 5Open "Tautochrone" and "Snell’s law analogy" to explore the cycloid’s special properties

FAQ

Why isn’t the straight line the fastest?

The straight line is shortest, but it starts too gently and the bead gains speed slowly. The fastest path drops steeply first to pick up speed early, then uses that speed to cover the remaining distance. Speed depends only on the height fallen, v = √(2gy), so a slightly longer path that reaches high speed sooner wins. The cycloid strikes the exact balance between steep and shallow, as the calculus of variations proves.

What does a cycloid look like?

A cycloid is the path traced by a point on the rim of a circle rolling along a straight line: x = R(θ − sinθ), y = R(1 − cosθ). As a brachistochrone it hangs upside down, starting vertically and flattening out. When B is farther horizontally than π/2 times the drop, the fastest route actually dips below B and climbs back up. It is very counter-intuitive, and you can see it by dragging B in the tool.

What is the tautochrone, and what does it have to do with clocks?

In a cycloid-shaped bowl a ball released from rest reaches the bottom in π√(R/g) whatever its starting height, so the cycloid is also the tautochrone (equal-time curve). An ordinary pendulum’s period grows with amplitude. In 1673 Huygens hung a pendulum between cycloidal cheeks so the bob moved along a cycloid, making a clock that was isochronous in theory.

How did Bernoulli solve it with optics?

He treated the bead as a ray of light. After falling y its speed is v = √(2gy), like light passing through layers in which it travels faster and faster. Light takes the quickest path (Fermat’s principle), so at every boundary Snell’s law holds: sinφ/v = constant. With infinitely many thin layers, that condition gives exactly a cycloid. The "Snell’s law analogy" tab shows a ray refracted through layers converging on the cycloid.

Is the cycloid still fastest with friction?

Not necessarily. With friction the loss depends on the normal force, which includes both a gravity component and the centripetal term mv²/r, so sharp bends at high speed cost more. The fastest curve with friction is shallower than a cycloid, and when B is far away a bead on the cycloid can even stall partway. The tool integrates the bead-on-a-wire model numerically, so you can compare curves directly.