Hypocycloid and Epicycloid

A Study of Hypocycloids and Epicycloids

1599- The cycloid itself was first discovered by Galileo and Mersenne. Galileo actually originated the term cycloid and was the first to seriously study the curve.

1674- Hypocycloids were first conceived by Roemer while he was studying the best form of gear teeth. Roemer focused on specific types of hypocycloids like the asteroid while investigating the use of cycloidal curves in the manufacture of gear teeth.

1691- Johan Bernoulli worked with this curve a little bit

1725- Daniel Bernoulli discovered the double generation theorem of cycloidal curves.

1745- Euler also worked with this curve in his work involving an optical problem.

If the smaller circle has radius ‘r’ and the larger circle has radius ‘kr’, then the parametric equations follow as:

x(Θ)=r(k-1) cos⁡(Θ)+rcos((k-1)Θ)

y(Θ)=r(k-1) sin⁡(Θ)-rsin((k-1)Θ)

If k is an integer, then the curve is closed and has k number of cusps.

If k is a rational number (k=p/q), expressed in simplest form then the curve has p number of cusps.

If k is an irrational number, then the curve never closes, and fills the space between the larger circle and a circle of radius R-2r.

Each hypocycloid for any value of r is a brachistochrone for the gravitational potential inside a homogeneous sphere of radius kr. A brachistochrone literally means ‘shortest time’ and is the path that carries a point-like body from one place to another in the least amount of time. The body is released at rest from the starting point and is constrained to move without friction along the curve to the end point, while under action of constant gravity.

2 cusps

The curve is a straight line and the circles are called Cardano circles. Girolamo Cardano was the first to describe these hypocycloids and their applications to high speed printing.

Also known as the Tusi couple. This was first proposed in 1247 by 13th century Persian astronomer and mathematician, Nasir-al-Din-al-Tusi; as a solution for the latitudinal motion of inferior planets. This was later used as a substitute of the equant introduced thousand years earlier in Ptolemy’s Almagest.

3 cusps

Deltoid

4 cusps

Astroid

An epicycloid is a plane curve formed by tracing the path of a particular point of a circle, as it rolls (without slipping) along a fixed circle.

140BC -Hipparchus recognized and used epicycloids to create a model for the motion of the moon.

130AD -Ptolemy used combinations of epicycles to predict the positions of the sun, moon and the planets.

1640AD Gerard Desargues used the epicycloid to develop a pump for raising water near Paris

1674AD The epicycloid was named by Ole Roemer

1694AD Philippe de La Hire, credited for inventing the epicycloidal profile for gear teeth

1725AD The double generation theorem of the curve was first noted by Daniel Bernoulli

x(t) = (a+b)*cos(t) - (b)*cos(((a+b)*t)/b)

y(t) = (a+b)*sin(t) - (b)*sin(((a+b)*t)/b)

(where 'a' is the radius of the fixed circle, and 'b' is the radius of the rolling circle)

To get an epicycloid with a certain number of cusps, n, let b = a/n, so that n revolutions of the rolling circle bring the traced point back to it's original starting position.

n = 1, cardioid

n = 2, nephroid

n = 5, ranunculoid

Gear Systems

1) Gear teeth

2) Gear ReductionDrives

3) Gear Pumps

Gear tooth profiles were originally cycloidal.

Modern gear teeth are cut with an involute shape except for watches and clocks.

Cycloidal gear teeth are stronger, the pinion can have fewer teeth, more accurate movement

Involute gear teeth have the same pressure angle throughout whole duration, easier to manufacture, less operating noise

Planetary gear reduction drives

Common Uses

Drills

Radio telescope tracking

(Sumitomo claims that its gear reduction drives can achieve reduction as high as 10 billion revolutions to 1)

Automatic transmissions

http://jwilson.coe.uga.edu/EMT668/emt668.student.folders/BrombacherAarnout/EMT669/cycloids/cycloids.html

http://mathoverflow.net/questions/62543/what-is-the-relation-between-hypocycloids-and-ideals-in-polynomial-rings-as-allu

http://johncarlosbaez.wordpress.com/2013/12/03/rolling-hypocycloids/

http://blogs.scientificamerican.com/roots-of-unity/2013/12/04/hypocycloids-make-you-happy/

https://data.epo.org/publication-server/rest/v1.0/publication-dates/19881019/patents/EP0286760NWA1/document.html

http://msemac.redwoods.edu/~darnold/math50c/CalcProj/sp05/astley/EpicycloidReport.htm

http://www.csparks.com/watchmaking/CycloidalGears/

http://gearcutting.blogspot.com/2008/02/comparison-between-involute-and.html

http://blog.mechguru.com/machine-design/involute-gear-vs-cycloidal-gear/

http://teacher.buet.ac.bd/fhaider/gear.pdf

http://garysclocks.sawdustcorner.com/gears.html

http://garysclocks.sawdustcorner.com/animations.html

http://auto.howstuffworks.com/automatic-transmission2.htm

http://mysite.du.edu/~etuttle/tech/cycloid.htm

http://www.mitsuimiike.co.jp/english/product/power/transmission/images/index_ph001.jpg

Hypocycloids

Gear Pumps

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.

Cycloidal Gear Pumps

History

Math

of

Hypocycloids

Basic Equations

Gear Reduction Drives

Properties

Circadian Cycle of Circles:

Special Cases

Involute

vs

Cycloidal

Gear Teeth

Applications

References

http://www-groups.dcs.st-and.ac.uk/~history/Java/Epicycloid.html

Hypocycloid and Epicycloid Applet

History

Epicycloids

Special Cases

Math

of

Epicycloids