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**ÒThe Helen of GeometryÓ**

**Explore Cycloids**

Najia Bao

Good morning! Everyone. Today letÕs
investigate and explore the ÒHelen of GeometryÓ, that is, cycloid. It was the
first time that I heard about cycloid when I was 12 years old. My father, a
mechanical engineer who was just designing a kind of system of cycloidal gears,
told me the history of cycloid, from Galileo, one of the first people to
study the cycloid, to two famous problems about cycloid - Christian HuygensÕ
tautochrone problem and John BernoulliÕs brachistochrone problem. Also, he explained the difference between
cycloidal gears and involute gears and their strength and weakness as well.
Frankly, at that time, I was not interested in what he said. However, after I
took the course *Differential Geometry* (during my graduate study), I master knowledge of cycloids.

A cycloid is the locus of a point on a circle that rolls along a line. Write parametric equations for the cycloid and graph it by GSP.

The cycloid is the locus of a point *P* on the rim of a circle of radius *r* rolling along a straight line (see Figure 1).
Suppose that the circle rolls along the *x*-axis
and one position of *P* is the
origin, letÕs find parametric equations for the cycloid.

Now, we
view the angle of rotation t of the circle as parameter. As we know, *t = 0* when *P*
is at the origin. When the circle has rotated through *t* radians, the distance rolled is *OT = arc
PT = rt*. Suppose the coordinates of *P* be *(x, y)*,
we get that *x = OT – PQ = rt – r sint = r (t - sint)*,* *that
is,* x = r (t -sint)*. Similarly,
we get that *y= r (1-cost)* (see
fig. 1). Therefore the cycloid-the path of the
moving point *P* – has
parametric equations *x = a (t - sint), y = a (1-cost).*

Figure 1

Now we construct the cycloid by GSP.

Figure 2

Now letÕs further
explore cycloids. I want to introduce you hypocycloid and epicycloids. A
hypocycloid is a curve traced out by a fixed point *P* on a circle *C* of radius *b* as *C* rolls on the inside of a circle with center *O* and radius *a*. Its parametric equation is If the
circle C rolls on the outside of the fixed circle, the curve traced out by *P* is called an epicycloid (see fig. 3). Then
its parametric equation is

Figure 3

Maybe some of
you would say that cycloid seems so simple and common but why it is called as ÒHelen of GeometryÓ. That is because
is related to one of the most famous pairs of problems in the history of
calculus. The first problem, called *tautochrone problem*, began with Galileo and completed by Christian
Huygens. The *tautochrone problem*
is that the time required to complete a full swing of a given pendulum is
approximately the same whether it makes a large movement at high speeds or a
small movement at lower speeds. The second problem, called *brachistochrone
problem*, is to determine the path down
which a particle will slide from point A to point B in the shortest time. This
problem was proposed by John Bernoulli in 1696 as a challenge to the
mathematicians of Europe. Newton, Leibniz, IÕHopital, and John and James
Baernoulli all found the correct solution. Just because the two problems
attracted so many great mathematicians and physicists, cycloid is called as the ÒHelen of GeometryÓ.