Thomas WoodMath 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.

.P

Sir Isaac Newton – English Mathematician (1643-1727)

Philippe de la Hire – French Mathematician(1640-1718)

Girard Resargues – French Mathematician(1591-1661)

Gottfried Wilhelm von Liebniz – German (1646-1716)Mathematician

Wankel Rotary Engine

Spirograph

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=dcosϕ and y=-dsinϕ.

Adding these equations to the equations for the center of the inner circle gives the parametric equations x=Rcosϴ-rcosϴ +dcosϕ y=Rsinϴ-rsinϴ-dsinϕ for a hypotrochoid.

Inner circle

Get phi in terms of theta

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

rϕ=Rϴϕ=Rϴ/r

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

r

Therefore, the equations for a hypotrochoid are

)sin(sinsin

)cos(coscos

r

rRdrRy

r

rRdrRx

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,

-6 -4 -2 0 2 4 6

-5

-4

-3

-2

-1

0

1

2

3

4

5

x-axis

y-ax

is

R=13, r=12, d=5

-6 -4 -2 0 2 4 6-5

-4

-3

-2

-1

0

1

2

3

4

5

x-axis

y-ax

is

R=6, r=5, d=5

Properties and Special Cases

If d=r, the point P is on the circumference of the inner circle and this is aspecial case of the hypotrochoid called the hypocycloid. For ahypocycloid, if r (which is equal to d) and R are not both even or bothodd and R is not divisible by r, the hypocycloid traces a star with Rpoints.

-4 -2 0 2 4 6

-4

-3

-2

-1

0

1

2

3

4

x-axis

y-ax

is

R=5, r=2, d=2

-25 -20 -15 -10 -5 0 5 10 15 20 25

-15

-10

-5

0

5

10

15

x-axis

y-ax

is

R=20, r=7, d=7

-8 -6 -4 -2 0 2 4 6 8

-6

-4

-2

0

2

4

6

x-axis

y-ax

is

R=12, r=6, d=3

R=2r

-4 -3 -2 -1 0 1 2 3 4 5

-3

-2

-1

0

1

2

3

x-axis

y-ax

is

R=5, r=7, d=2

r>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>.