Thomas Wood
Math 50C
12/8/08
Hypotrochoid
Introduction
The hypotrochoid is a member of the family of curves called trochoid curves. The name hypotrochoid comes from the Greek word hypo, which means under, and the Latin word trochus, which means hoop. Therefore, the curve is formed by a small circle rotating “under,” or inside, the radius of a bigger circle. It is defined as 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 point P can be either inside the circle of radius r or it can be outside of the circle, as long as it remains at a constant distance from the center of the inner circle. If the point is within the radius of the inner circle, the curve is called a curtate hypotrochoid. If the point is outside of the inner circle’s radius, it is called a prolate hypotrochoid. The counterpart of the hypotrochoid is the epitrochoid, which is a curve drawn by a point on a circle that rolls along the outside of a fixed circle.
History and Uses
Mathematicians began studying hypotrochoids in the 1600’s. Among them were Sir Isaac Newton, German mathematician Gottfried Wilhelm von Liebniz, and French mathematicians Philippe de la Hire and Girard Resargues. The curve has also been used in engines. The Wankel rotary engine uses the principle of the hypotrochoid, although it is in reverse because the larger circle rotates around a stationary, smaller circle. The engine has a triangular rotor and the inside of the rotor has the shape of a circle, which rotates about a smaller, fixed circle. The rotary engine replaces the pistons, crankshafts, cams, and valves of a regular engine. Another application of the hypotrochoid is the spirograph, which was invented by a British engineer named Denys Fisher in 1965. The spirograph uses plastic circles with gears and several different pencil holes for drawing both hypotrochoids and epitrochoids.
Parametric Equations of the Hypotrochoid
To find parametric equations for a hypotrochoid curve, I first found equations for the center of the inner 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 and its y-coordinate is defined as . The radius of the circle created by the center point is
After finding equations for the center of the circle, 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 , 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 Protates 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 acircle. However, since phi goes clockwise, and . Adding these equations to the equations for the center of the inner circle gives the parametric equations and .
Inner Circle
Since the inner circle rolls along the inside of the stationary circle without slipping, the arc length must be equal to the arc length .
r
However, since the point P rotates about the center of the inner circle, which traces out a circle of radius (, is equal to . Therefore, the equations for a hypotrochoid are
Properties of Hypotrochoids
The inner circle usually has a smaller radius than the fixed circle so the point P rotates around the center of the inner circle at a more frequent rate than the center of the circle rotates about the origin. The ratio of the circumferences of the two circles is = and therefore the ratio is the number of revolutions the small circle goes through during the time the center goes through 1 revolution, or 2 radians.
When the hypotrochoid draws R loops and has to go from radians to complete the curve. As d increases, the size of the loop decreases. If , there are no longer loops, they become points.
For example,
When , you get an ellipse. If , the ellipse gets wider and shorter as d increases. If , the ellipse gets larger as d increases. If , you get a line instead of an ellipse.
For example,
If , 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.
Some examples of hypocycloids:
Another special case is the rose, which is produced when r is greater than R, which actually makes a large circle rotate around a smaller, fixed circle. This is the same thing that occurs with the Wankel rotary engine.
Bibliography
Butler, Bill. “Hypotrochoid.” Durango Bill’s Epitrochoids and Hypotrochoids.26 Nov, 2008.
“Hypotrochoid.” 1997. 6 Dec, 2008. <
“Spirograph.” Wikipedia. 2008. 7 Dec, 2008. <
Wassenaar, Jan. “Hypotrochoid.” 2dcurves.com. 2005. 6 Dec, 2008. <
Weisstein, Eric W. "Hypotrochoid." MathWorld--A Wolfram Web Resource. 2008. Wolfram Research, Inc. 26 Nov, 2008. <