The Cycloid and The Catenary Curves:
An Animated Approach
by 
Bruce Lewis
8/04/99

Overview

This module begins by defining  the cycloid curve.  An animation will visually show what was described in the cycloid's definition.  The student will be asked to find out what happens to the graph of the cycloid when changes are made to the constants in its equation. The definition of a catenary will then be given.  The student will be asked to find out what happens to the graph of the catenary when changes are made to its constant.  We will then run an animation that will view a box being rolled across a catenary-shaped floor.  Next, the definition of a roulette will then be given.  Finally, the student will be given some future ideas to explore.

Definitions and Properties of the Cycloid Curve

The curve traced by a fixed point P on the circumference of a circle as the circle rolls along a straight line in a plane is called a cycloid.   The Cartesian equation of the cycloid is x = at - h sin(t) and y = a - h cos(t).  
	
There are many properties of a cycloid.  The length of the base equals the length of the circumference of the rolling circle.  When a=h, the area under the graph is 3  π(a^2).   The arc length (length of the arch)  is 8a.  An object sliding down a cycloid-shaped wire will finish its descent in a set time, no matter where the object started on the cycloid-shaped wire (which is the tautochrone property).  Also, an object falling sliding down a wire, that can be positioned into any shape, will fall the quickest if the wire is shaped as a cycloid (which is the brachistochrone property).

An Animation which displays the Cycloid

Please run the following program and view the output as an animation.

Clear[x, y, t]\(a = 1.0; 
h = 1.0; 
x[t_] := a\ t - h\ Sin[t]\)y[t_] :=  a  - h Cos[t]cycloid=ParametricPlot[{x[t], y[t]}, {t, 0, 6  π}, 
          AxesLabel -> {"x", "y"}, 
          PlotStyle -> RGBColor[0, 0, 1], 
          DisplayFunction -> Identity];
point1 = {};
\(Do[
	AppendTo[point1, {x[t], y[t]}]; 
			
		
        Show[cycloid, 
          Graphics[Circle[{t, a}, a], 
            PlotRange -> {{0, 6  π}, {0, 2}}], 
          Graphics[Line[{{x[t], y[t]}, {t, a}}]], 
			
          ListPlot[point1, DisplayFunction -> Identity, 
            PlotStyle -> {PointSize[ .02], RGBColor[1, 0, 0]}, 
            PlotJoined -> True, 
            PlotRange -> {{0, 6  π}, {0, 2}}], 
	
          DisplayFunction -> $DisplayFunction, 
          AspectRatio -> Automatic], {t, 0, 6  π, π/6}]\)

\(
\)

Try some Cyloid problems on your own

A cycloid may be graphed by using the following parametric equations.  Use the ParametricPlot command.

x[t_]:= at - h Sin[t] 
y[t_]:= a - h Cos[t]
   
   Try to graph a cycloid when a and h are equal (put your own numbers in for a and h)
   What happens when a and h are different?
   Is there any difference in the graphs when a and/or h is negative?
   Can you translate this curve up, down, left, or right?

The Definition of the Catenary Curve

The catenary curve is the shape of a perfectly flexible chain suspended by its ends and acted upon by gravity.  Try to make this shape with a piece of string.  Just have one of your friends hold one end of a string while you hold the other end up at the same height.  Make sure that the string is not held taut.  The shape of the dangling string is the catenary curve.

Try some catenary problems on your own

A catenary may be graphed by using the following  equation. Use the Plot command.

y = a Cosh[x/a]

Try to graph a catenary curve for differing values of a .
Is there a difference between using positive and negative values for a?
Can you translate this curve up, down, left, or right?

View an interesting program where a square is rolled across a catenary-shaped floor

Please run the following program and view the output as an animation.  

The floor is in the shape of  many catenaries sitting side by side.  Would a bicycle with square wheels actually ride on a floor like this?
Can you you see what path the vertices of the square are following?

\(
x[t_] := Log[Tan[\((2  t + π)\)/4]]\)\(y[t_] := \@2\  - 1/Cos[t]
\)xa[t_] := 
      Log[Tan[(2  t + π)/4]]  + 
        (Sqrt[2]) Cos[(7  π/4)  -  t]ya[t_] := (Sqrt[2])   +  (Sqrt[2]) Sin[(7  π/4)  - t]xb[t_] := 
      Log[Tan[(2  t + π)/4]]  + 
        (Sqrt[2]) Cos[(5  π/4)  -  t]yb[t_] := (Sqrt[2])   +  (Sqrt[2]) Sin[(5  π/4)  - t]xc[t_] := 
      Log[Tan[(2  t + π)/4]]  + 
        (Sqrt[2]) Cos[(3  π/4)  -  t]yc[t_] := (Sqrt[2])   +  (Sqrt[2]) Sin[(3  π/4)  - t]xd[t_] := 
      Log[Tan[(2  t + π)/4]]  + 
        (Sqrt[2]) Cos[(π/4)  -  t]\(yd[t_] := \@2\ \  + \ \@2\ Sin[\((π/4)\)\  - t]
\)\(point2 = {}; 

    Do[AppendTo[point2, {x[t]\  + m\ 2\ Log[Tan[3  π/8]], y[t]}], {m, 0, 
        6, 1}, {t, \(-π\)/4, π/4, 1  π/16}]; 

	
    catenary = 
      Show[ListPlot[point2, DisplayFunction -> Identity, 
          PlotStyle -> {PointSize[ .02], RGBColor[1, 0, 0]}, 
          PlotRange -> {{\(-3\), 13}, {\(-1\), 4}}]]; 


    Do[Show[catenary, 
        Graphics[
          Line[{{xa[t] + m\ 2\ Log[Tan[3  π/8]], ya[t]}, {
                xb[t] + m\ \ 2\ \ Log[Tan[3  π/8]], yb[t]}, {
                xc[t] + m\ 2\ Log[Tan[3  π/8]], yc[t]}, {
                xd[t] + m\ 2\ Log[Tan[3  π/8]], yd[t]}, {
                xa[t] + m\ 2\ Log[Tan[3  π/8]], ya[t]}}]], 
        PlotRange -> {{\(-3\), 13}, {\(-1\), 4}}, 
        DisplayFunction -> $DisplayFunction, 
        AspectRatio -> Automatic], {m, 0, 6, 1}, {t, \(-π\)/4, 
        3  π/16, 1  π/16}]; 
\)

More information on the previous animations -- Roulettes
 
A roulette is defined to be the curve traced by a fixed point on a closed convex curve as that curve rolls without slipping along a second curve.  The first animation above that we did that made a cycloid was a roulette.  The second animation would show a roulette if we drew the curve that is made by one of the vertices of the square as it moves across the catenary-shaped floor.

\(
\)

Follow-up Activities

Find out more about the following:

Do these shapes (cycloid and catenary curves) ever show up in architecture?

Find out more about the tautochrone and brachistone properties of a cycloid.

Have any famous mathematicians ever worked with the cycloid or catenary curves?  
What is the actual  history behind these curves?

Find out more about roulettes and their graphs.

Sources

Calculus with Analytic Geometry by Earl W. Swokowski, Published by Prindle, Weber and Schmidt, July 1980

Famous Curves Index from the MacTutor History of Mathematics archive by O'Connor, Robertson, and Soares found at  the following Internet address http://www-groups.dcs.st-andrews.ac.uk/~history/index.html

Roulette pages from Eric's Treasure Troves of Science by Eric Weisstein at http://www.treasure-troves.com/math/

Paul Kunkel, via Deja News and e-mail at kunkel@cnw.com .  He helped me with the mathematics behind the rolling square over the catenary-shaped floor.  I did all of the Mathematica programming on my own.