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.

.P.P

 Sir Isaac Newton – English Mathematician ( )  Philippe de la Hire – French Mathematician ( )  Girard Resargues – French Mathematician ( )  Gottfried Wilhelm von Liebniz – German ( )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 (Rcos - rcos) and its y-coordinate is defined as (Rsin - rsin). 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 2 π, 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=dcos ϕ and y=-dsin ϕ.

Adding these equations to the equations for the center of the inner circle gives the parametric equations x=Rcos -rcos +dcos ϕ y=Rsin -rsin-dsin ϕ 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 r ϕ must be equal to the arc length R. r ϕ=R ϕ=R /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) r

Therefore, the equations for a hypotrochoid are

When r=(R-1), the hypotrochoid draws R loops and has to go from 0 to 2 π*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, Properties and Special Cases

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 points.

R=2r

r>R

 Butler, Bill. “Hypotrochoid.” Durango Bill’s Epitrochoids and Hypotrochoids. 26 Nov,  “Hypotrochoid.” Dec,  “Spirograph.” Wikipedia Dec,  Wassenaar, Jan. “Hypotrochoid.” 2dcurves.com Dec,  Weisstein, Eric W. "Hypotrochoid." MathWorld -- A Wolfram Web Resource Wolfram Research, Inc. 26 Nov,