Cycloid
Calculate cycloid curve coordinates, arc length, and area under one arch using parametric equations. Free online cycloid calculator for math and physics students.
About This Calculator
This cycloid calculator computes the x-coordinate of a point on a cycloid curve using the parametric equation x(t) = t - sin(t) for unit radius. A cycloid is the curve traced by a point on a rolling circle, with applications in gear design, pendulum clocks, and physics (brachistochrone and tautochrone problems). Enter the angle parameter t in radians to find the corresponding x(t) value on the curve.
Frequently Asked Questions
What is a cycloid curve?
A cycloid is the curve traced by a point on the rim of a circular wheel as it rolls along a straight line. Its parametric equations are x(t) = r(t - sin t) and y(t) = r(1 - cos t), where r is the radius of the wheel and t is the angle of rotation.
What are the key properties of a cycloid?
A cycloid has several important properties: the arc length of one arch is 8r, the area under one arch is 3pir^2, and it is the brachistochrone curve -- the path of fastest descent under gravity. It is also the tautochrone curve, where a particle takes the same time to slide down regardless of starting point.
How do you calculate cycloid coordinates?
Cycloid coordinates are calculated using parametric equations x(t) = r(t - sin t) and y(t) = r(1 - cos t), where r is the wheel radius and t ranges from 0 to 2pi for one complete arch. Enter your angle parameter t and radius r into the calculator to get the corresponding x and y coordinates.
Where are cycloids used in real life?
Cycloids are used in gear tooth design (cycloidal gears), clock pendulums (tautochrone property), roller coasters for fastest drops, and in physics for studying brachistochrone problems. They also appear in architecture in arch designs inspired by the curve's structural properties.
Is this cycloid calculator free?
Yes, all calculators on Calculy are completely free to use. There are no hidden charges, subscriptions, or usage limits. You can calculate cycloid coordinates and properties as many times as you need.