Cycloid
A cycloid is the curve traced by a point on a circle rolling along a line. Arc length per arch = 8r (no ฯ!); area per arch = 3ฯrยฒ. Solves the brachistochrone.
Why This Mathematical Concept Matters
Why: Cycloids appear in gear teeth, Ferris wheels, and roller coasters. The brachistochrone (fastest descent) is a cycloid arc.
How: Arc length = 8r per arch (elegantโno ฯ). Area = 3ฯrยฒ per archโexactly 3ร the rolling circle. Parametric: x=r(tโsin t), y=r(1โcos t).
- โArc length 8r has no ฯโunusual for a curve involving circles.
- โThe cycloid solves the brachistochrone: fastest path under gravity.
- โHuygens used cycloidal cheeks for accurate pendulum clocks.
Cycloid โ Path of a Rolling Point
Arc length = 8r per arch. Area = 3ฯrยฒ per arch. The curve traced by a point on a rolling circle.
๐ Examples โ Click to Load
Calculation Settings
โ ๏ธFor educational and informational purposes only. Verify with a qualified professional.
๐งฎ Fascinating Math Facts
Cycloid arc length = 8r per archโno ฯ in the formula!
โ Formula
The brachistochrone (fastest descent) is a cycloid arc.
โ Physics
๐ Key Takeaways
- โข A cycloid is the curve traced by a point on a circle rolling along a line
- โข Arc length of one arch = 8r (elegant formula โ no ฯ!)
- โข Area under one arch = 3ฯrยฒ โ exactly 3ร the area of the rolling circle
- โข The cycloid solves the brachistochrone (fastest descent) and tautochrone problems
- โข Huygens used cycloidal cheeks for accurate pendulum clocks
๐ก Did You Know?
๐ Cycloid Formulas
Parametric Equations
t is the angle (radians) the circle has rotated. One arch: t = 0 to 2ฯ.
Arc Length & Area
For n arches: L = 8rn, A = 3ฯrยฒn.
๐ฏ Expert Tips
Quick Arc Length
One arch = 8r. So r=10m โ 80m per arch. No ฯ needed!
Area vs Circle
Area under one arch = 3ฯrยฒ = 3ร the rolling circle area. Memorable!
Multiple Arches
For n complete arches, multiply by n: L=8rn, A=3ฯrยฒn.
Brachistochrone
The cycloid is the curve of fastest descent โ a bead slides fastest along it.
โ๏ธ Comparison
| Property | Cycloid Arch | Circle |
|---|---|---|
| Arc length | 8r | 2ฯr |
| Area | 3ฯrยฒ | ฯrยฒ |
| Ratio | 4/ฯ โ 1.27 | 1 |
๐ Quick Facts
โ FAQ
What is a cycloid?
The curve traced by a point on a circle as it rolls along a straight line without slipping.
Why is arc length 8r?
Derived from the parametric equations. The integral โซโ(dxยฒ+dyยฒ) from 0 to 2ฯ equals 8r.
Why is area 3ฯrยฒ?
The area under one arch equals 3ร the area of the rolling circle. Integral of y dx gives 3ฯrยฒ.
What is the brachistochrone?
The curve of fastest descent under gravity. Johann Bernoulli proved it's a cycloid (1696).
What is the tautochrone?
A curve where descent time is independent of starting point. The cycloid has this property.
How do pendulum clocks use cycloids?
Huygens placed cycloidal "cheeks" so the pendulum bob follows a cycloid โ constant period.
Curtate vs prolate?
Curtate: point inside the circle. Prolate: point outside. This calculator uses standard (point on circumference).
Units?
Use any length unit: m, cm, in, px. Results match your unit.
๐ Sources
โ ๏ธ Disclaimer: This calculator uses the standard cycloid (point on circumference). Curtate and prolate cycloids have different formulas. Results are for ideal mathematical curves.
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