This document defines and describes several types of curves: - Cyclogons are curves traced by vertices of polygons rolling along a straight line. - Cycloids are curves traced by points on rolling circular wheels. - Hypo/hypocycloids involve a small circle rolling within a larger circle. - Epi/epicycloids involve a small circle rolling around the outside of a larger circle. - Epicyclic gearing uses epicycloid curves to transmit torque through a carrier connecting a planet gear rolling around a stationary sun gear.

1. Cyclo Curves
- CYCLOGON
- CYCLOID
- HYPOTROCHOID
- HYPOCYCLOID
- EPITROCHOID
- EPICYCLOID
- EPICYCLIC GEARING

2. Cyclogon
 A cyclogon is the curve traced by a vertex of a polygon that rolls
without slipping along a straight line.

3. Cycloid
 A cycloid is the curve traced by a point on the rim of a circular wheel
as the wheel rolls along a straight line without slippage.
Tautochrone Curve
If r=a then:

4. Hypotrochoid
 A hypotrochoid is a roulette traced by a point attached to a circle of
radius r rolling around the inside of a fixed circle of radius R, where the
point is a distance d from the center of the interior circle.

5. Hypocycloid
 A hypocycloid is a special plane curve generated by the trace of a fixed
point on a small circle that rolls within a larger circle. It is comparable to
the cycloid but instead of the circle rolling along a line, it rolls within a circle.
R/r = k = a/b :

6. Epitrochoid
 An Epitrochoid is a roulette traced by a point attached to a circle of radius r
rolling around the outside of a fixed circle of radius R, where the point is a
distance d from the center of the exterior circle.

7. Epicycloid
 An epicycloid is a plane curve produced by tracing the path of a chosen point
of a circle which rolls without slipping around a fixed circle. It is a particular
kind of roulette.
R/r = k = a/b :

8. Epicyclic gearing
 An epicyclic gear train consists of two gears mounted so that the center of one gear
revolves around the center of the other. A carrier connects the centers of the two
gears and rotates to carry one gear, called the planet gear, around the other, called
the sun gear. The planet and sun gears mesh so that their pitch circles roll without
slip. A point on the pitch circle of the planet gear traces an Epicycloid curve. In this
simplified case, the sun gear is fixed and the planetary gear(s) roll around the sun
gear.

9. Gear ratio of standard epicyclic gearing
 Sun: The central gear
 Carrier: Holds one or more peripheral Planet gears, all of the same size, meshed with the sun
gear
 Annulus: An outer ring with inward-facing teeth that mesh with the planet gear or gears
is the angular velocity of the Annulus, Sun gear, Planet gears and planet
Carrier respectively, and is the Number of teeth of the Annulus, the Sun gear and
each Planet gear respectively
if

10. Torque ratios of standard epicyclic gearing
 In epicyclic gears, two speeds must be known, in order to determine the
third speed. However, in a steady state condition, only one torque must be
known, in order to determine the other two torques.

11. Thank you! 
seied Mahdi sadat hosseini