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Hypotrochoid. Thomas Wood Math 50C. Definition. - A curve traced by a point P fixed to a circle with radius r rolling along the inside of a larger, stationary circle with radius R at a constant rate without slipping.

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Thomas Wood

Math 50C


- A curve traced by a point P fixed to a circle with radius r rolling along the inside of a larger, stationary circle with radius R at a constant rate without slipping.

-The name hypotrochoid comes from the Greek word hypo, which means under, and the Latin word trochus, which means hoop.

  • Sir Isaac Newton – English Mathematician


  • Philippe de la Hire – French Mathematician


  • Girard Resargues – French Mathematician


  • Gottfried Wilhelm von Liebniz – German

(1646-1716) Mathematician


Wankel Rotary Engine


parametric equations
Parametric Equations
  • First I found equations for the center of the small circle as it makes its motion around the inside of the large circle.
  • I found that the center point C of the small circle traces out a circle as it rolls along the inside of the circumference of the large circle.

As the point C travels through an angle theta, its

x-coordinate is defined as (Rcosϴ - rcosϴ) and its y-coordinate is defined as (Rsinϴ - rsinϴ). The radius of the circle created by the center point is (R-r).


The more difficult part is to find equations for a point P around the center.

  • As the small circle goes in a circular path from zero to 2π, it travels in a counter-clockwise path around the inside of the large circle. However, the point P on the small circle rotates in a clockwise path around the center point.
  • As the center rotates through an angle theta, the point P rotates through an angle phi in the opposite direction.
  • The point P travels in a circular path about the center of the small circle and therefore has the parametric equations of a circle.
  • However, since phi goes clockwise, x=dcosϕand y=-dsinϕ.

Inner circle

Adding these equations to the equations for the center of the inner circle gives the parametric equations

x=Rcosϴ-rcosϴ +dcosϕ


for a hypotrochoid.


Get phi in terms of theta

  • Since the inner circle rolls along the inside of the stationary circle without slipping, the arc length rϕmust be equal to the arc length Rϴ.



However, since the point P rotates about the circle traced by the center of the small circle, which has radius (R-r), ϕ is equal to (R-r)ϴ



Properties and Special Cases

When r=(R-1), the hypotrochoid draws R loops and has to go from 0 to 2π*r radians to complete the curve. As d increases, the size of the loop decreases. If d ≥ r, there are no longer loops, they become points.

  For example,


If d=r, the point P is on the circumference of the inner circle and this is a

special case of the hypotrochoid called the hypocycloid. For a

hypocycloid, if r (which is equal to d) and R are not both even or both

odd and R is not divisible by r, the hypocycloid traces a star with R




  • Butler, Bill. “Hypotrochoid.” Durango Bill’s Epitrochoids and Hypotrochoids. 26 Nov, 2008. <http://www.durangobill.com/Trochoids.html>.
  • “Hypotrochoid.” 1997. 6 Dec, 2008. <http://www-history.mcs.st- andrews.ac.uk/history/Curves/Hypotrochoid.html>.
  • “Spirograph.” Wikipedia. 2008. 7 Dec, 2008. <http://en.wikipedia.org/wiki/Spirograph>.
  • Wassenaar, Jan. “Hypotrochoid.” 2dcurves.com. 2005. 6 Dec, 2008. <http://www.2dcurves.com/roulette/rouletteh.html#hypotrochoid>
  • Weisstein, Eric W. "Hypotrochoid." MathWorld--A Wolfram Web Resource. 2008. Wolfram Research, Inc. 26 Nov, 2008. <http://mathworld.wolfram.com/Hypotrochoid.html>.