Everything about Cycloid totally explained
A
cycloid is the curve defined by the path of a point on the edge of circular wheel as the wheel rolls along a straight line.
It is an example of a
roulette, a curve generated by a curve rolling on another curve.
The cycloid is the solution to the
brachistochrone problem (for example it's the curve of fastest descent under gravity) and the related
tautochrone problem (for example the period of a ball rolling back and forth inside it doesn't depend on the ball's starting position).
History
The cycloid was first studied by
Nicholas of Cusa and later by
Mersenne. It was named by
Galileo in
1599. In
1634 G.P. de Roberval showed that the area under a cycloid is three times the area of its generating circle. In
1658 Christopher Wren showed that the length of a cycloid is four times the diameter of its generating circle. The cycloid has been called "The Helen of Geometers" as it caused frequent quarrels among 17th century
mathematicians.
Equations
»
The cycloid through the origin, created by a circle of radius
r, consists of the points (
x,
y) with
»
»
where
t is a real
parameter;
rt is the
x-coordinate of the center of the rolling circle.
Using degrees, a more modifiable form of the equation can be found:
x = a(cos(270-t)) + 2πrt/360
y = a(sin(270-t)) + r
This curve is
differentiable everywhere except at the
cusps where it hits the
x-axis, with the derivative tending toward
or
as one approaches a cusp. It satisfies the
differential equation
»
Cycloidal pendulum
If its length is equal to that of half the cycloid, the bob of a
pendulum suspended from the cusp of an inverted cycloid, such that the "string" is constrained between the adjacent arcs of the cycloid, also traces a cycloid path. Such a cycloidal pendulum is
isochronous, regardless of amplitude.
An up-side-down cycloid is called a
tautochrone which is a path of a cycloidal pendulum.
Related curves
Several curves are related to the cycloid. When we relax the requirement that the fixed point be on the edge of the circle, we get the
curtate cycloid and the
prolate cycloid. In the former case, the point tracing out the curve is inside the circle, and, in the latter case, it's outside. A
trochoid refers to any of the cycloid, the curtate cycloid and the prolate cycloid. If we further allow the line on which the circle rolls to be an arbitrary circle then we get the
epicycloid (circle rolling on outside of another circle, point on the rim of the rolling circle), the
hypocycloid (circle on the inside, point on the rim), the
epitrochoid (circle on the outside, point anywhere on circle), and the
hypotrochoid (circle on the inside, point anywhere on circle).
All these curves are
roulettes with a circle rolled along a uniform
curvature. The cycloid, epicycloids, and hypocycloids have the property that each is
similar to its
evolute. If
q is the
product of that curvature with the circle's radius, signed positive for epi- and negative for hypo-, then the curve:evolute
similitude ratio is 1+2
q.
The classic
Spirograph toy traces out hypotrochoid and
epitrochoid curves.
Further Information
Get more info on 'Cycloid'.
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