Calculations at a cycloid. A cycloid is the path of a point on the rim of a circle, which rolls on a straight line. The shape calculated here is that between the cycloid and the straight line on which the circle rolls, from one point of tangency of the circle with the straight line to the next.
Enter one value and choose the number of decimal places. Then click Calculate.
Generation of a cycloid.
Formulas:
b = 8 a
c = 2 π a
h = 2 a
p = b + c
A = 3 π a²
pi:
π = 3.141592653589793...
Radius, arc length, base length, height and perimeter have the same unit (e.g. meter), the area has this unit squared (e.g. square meter).
The cycloid has two vertices, one at each point where the straight line and the curve touch. It is axially symmetrical to its perpendicular bisector, which lies on the line of the height of this shape. If a vehicle on wheels travels a distance x, then a single point on the outside of a wheel travels the distance b/c times x, which is 4/π * x, or about 1.27324 times x. The size of the wheel is not important here, as its radius cancels out.
The cycloid was probably already known in ancient times as the shape that is created when a wagon wheel rolls over the plane. However, its course cannot be constructed with compass and ruler and it was not possible to calculate the size for a long time. The first mathematical investigations into this curve were carried out in the sixteenth century, including by Galileo Galilei. In the seventeenth century, the cycloid was comprehensively explored. The first calculation of length and area was made by the Italian Bonaventura Cavalieri in 1629. In 1697, Johann Bernoulli discovered that an inverted, slightly tilted cycloid is a brachistochrone. This is a path on which a frictionless mass, starting at zero speed, moves as quickly as possible from the starting point to the end point due to the force of gravity.
