Cycloids

  • Objective
  • Level
  • Prerequisites
  • Platform
  • Instructor's Notes
  • Credits
  • Generating a hypocycloid.
     
    Objective: To illustrate the power and utility of parametric equations and parametric plotting using cycloids.

    Level: Precalculus or Calculus I.

    Prerequisites: A basic introduction to parametric equations and elementary trigonometry.

    Platform: Any computer algebra system or graphics package capable of plotting parametric equations.

    Instructor's Notes:
     
    Background/suggested prerequisite activities.
    The path of a particle moving in a plane need not trace out the graph of a function, hence we cannot describe the path by expressing y directly in terms of x. An alternate way to describe the path of the particle is to express the coordinates of its points as functions of a third variable using a pair of equations
    x = f(t),      y = g(t).
    Equations of this form are called parametric equations for x and y, and the unknown t is called a parameter. The parameter t may represent time in some instances, an angle in other situations, or the distance a particle has traveled along the path from a designated starting point.

    An easy example of a parametric representation of a curve is obtained by using basic trigonometry to obtain  parametric equations of a circle of radius 1 centered at the origin. We have the relationships between a point (x,y) on the circle and an angle t as shown in the following figure.

    By elementary trigonometry we have the parametric equations
    x = cos(t),     y = sin(t).
    As t goes from 0 to 2p the corresponding points trace out the circle in a counter clockwise direction. The following animation illustrates the process.
         Generating a circle parametrically.
    (For a related demonstration for generation of sine and cosine curves see the Circular Functions demo. )
     
    Parametric equations based on circles.


    The Cycloid:

    A famous curve that was named by Galileo in 1599 is called a cycloid. A cycloid is the path traced out by a point on the circumference of a circle as the circle rolls (without slipping) along a straight line. A cycloid can be drawn by a pencil (chalk or marker) attached to a circular lid which is rolled along a ruler. The following animation illustrates the generation of a cycloid.
     


    If the circle that is rolled has radius a, then the parametric equations of the cycloid are

    x = a(t - sin(t)),     y = a(1 - cos(t))


    where parameter t is the angle through which the circle was rolled. As in the case of the circle, these parametric equations can be derived using elementary trigonometry. To see the basics of the derivation click on the following: The Equations of a Cycloid.

    For some history related to cycloids click on the following: St.Andrews-cycloid information.

    The Epicycloid:

    Take a large circle  centered at the origin.  Place a smaller circle tangent to the original circle at the point where it crosses the positive x-axis and outside of the original circle.  Identify the point of tangency.  See the next figure.

    Next we roll the smaller circle around the larger circle and follow the path of the point of tangency. The resulting curve is called an epicycloid. The shape of the curve generated in this manner depends on the relationship between the radius of the large circle and the small circle. The following animations illustrate three particular cases.
     
     
     
    Cardioid
     
    Nephroid
     Ranunculoid

    With a careful analysis we can show that the parametric equations of an epicycloid using a large circle of radius a and a small circle of radius b, where a > b, are

    x=(a+b)cos(t)-bcos((a+b)t/b) ,    y=(a+b)sin(t)-bsin((a+b)t/b) .
    The epicycloid has been studied by such luminaries as Leibniz, Euler, Halley, Newton and the Bernoullis. The epicycloid curve is of special interest to astronomers and the design of cog-wheels with minimum friction. To experiment with epicycloids see the files available at the end of this module. For more information and an on-line animator click on the following link:
    On-line animator for epicycloid.
     

    The hypocycloid:

    Here take a large circle centered at the origin. Place a smaller circle tangent to the original circle at the point where it crosses the positive x-axis and inside the original circle. Identify the point of tangency. See the next figure.

    Roll the smaller circle around the larger circle and follow the path of the point of tangency. The resulting curve is called an hypocycloid. The shape of the curve generated in this manner depends on the relationship between the radius of the large circle and the small circle. The following two animations illustrate the generation of hypocycloids. (Also see the animation at the beginning of this demo.)
    Again with a careful analysis we can show that the parametric equations of an epicycloid using a large circle of radius a and a small circle of radius b, where a > b, are
    x=(a-b)cos(t)+bcos((a-b)t/b) ,     y=(a-b)sin(t)-bsin((a-b)t/b) .
    Comments:

    I have used demos of this type in several ways.

    • In calculus class we define parametric equations and have the students plot a few by hand before we do the demonstration.  The demonstration then wows them as they realize how difficult it would be to plot these objects by hand.  For the epicycloid I usually use two rolls of tape with different radii, identify a point on the outer one with a magic marker, and then roll it around the other and ask the students what they think the path would look like.  Then using DERIVE (see downloads available below) we plot epicycloids for various radii          and ask questions about what they would expect and (as I already pointed out in discussion) the numbers of times we need to rotate (i.e. what the parameters should be) in order to obtain a closed  epicycloid curve.  We try to get them to state a little theorem about this.
              Each semester for calculus we have four computer labs.  Our 
              students must go to the lab, perform the exercises or experiments on 
              the computer using DERIVE, and then write up their results.  One of
              these labs requires students to plot a collection of functions using 
              parametric equations and polar coordinates.  The main purpose of 
              the lab is to familiarize students with the parametric plotting 
              capabilities of DERIVE.  This lab is assigned after students have
              seen some of the demo material shown above.
    •  I have also used this demo or a variation thereof during various admission  (student recruiting) days.  Generally, here, the audience consists of prospective students and their parents.  It is relative easy to explain what the cycloids are and then it is exciting and informative to see them plotted.
    Downloads available:

    For DERIVE the following file was supplied by Anthony Berard and can be downloaded by clicking on Epi-HypoCycloids.mth. (See the imbedded instructions concerning change of scale.)

    For Matlab the following files were written for this demo and can be downloaded by clicking on the file name: epicycloid.mhypocycloid.m .
     

    Credits:  This demo was submitted by 

    Anthony Berard
    Department of Mathematics 
    Kings College

    and is included in Demos with Positive Impact with his permission.



    DRH 2/14/00   Last updated 5/18/2006

    Since 3/1/2002