The path of a point fixed relative to a circle that rolls along a straight line is called a trochoid. The easiest way to visualize this phenomenon is to think of the path of a reflector on a bicycle as someone is riding on a level street. The reflector rotates around the hub of the wheel, but yet the hub of the wheel is moving relative to the ground.
Now think about what happens if the circle is instead rotating around another circle. In other words, the cyclist is now pedaling his/her way around the equator, instead of down the street. Mathematicians call this path an epicycloid. The rest of the world calls them SpiroGraphs!! The parametric equations for these curves are given by:
x(t)=(R+r)cos(t) - p*cos((R+r)t/r)
y(t)=(R+r)sin(t) - p*sin((R+r)t/r)
where R,r, and p are defined below. The applet below allows you to create all the SpiroGraphs your heart desires by varying the values of R, r and p, as well as the following parameters:
Radius1 (R)
Radius of circle (equator) centered at the origin.
Radius2 (r)
Radius of circle (bicycle wheel) centered at (R+r,0)
Position (p)
Distance of Point (reflector) from the center of Circle2, the circle of radius r.
Velocity
Speed at which the SpiroGraph is drawn, with 0 being the slowest and 10 being the fastest. Can be adjusted while drawing is taking place. Sometimes half the fun is seeing your SpiroGraph being drawn! Sometimes not!
Resolution
Controls how precise the SpiroGraph is drawn. For instance, with a Resolution of 360, the points (x(t),y(t)) are plotted for t=0,1,2,3,4,5,...,359,360 degrees. With a Resolution of 180, only the values of t=0,2,4,6,8,10,...,358,360 are used. Be careful, the larger the Resolution, the longer it will take to draw the SpiroGraph. Normally, a value between 270 and 360 will be good enough. With a small value of Radius2 (say Radius2=1), a small change in Resolution can result in remarkably different SpiroGraphs.
Hide/Show
Changing the value of this button will determine whether or not to display the circles/axes while drawing is taking place. Note: SpiroGraphs are drawn much faster if you "Hide" the circles/axes.
Clear
This button will clear the screen of all SpiroGraphs.
Reset
This button stops the current SpiroGraph and returns Circle2 and the Point to its original position
Draw/Pause/Resume
These buttons should be self explanatory.
Red/Green/Blue
You can adjust the values of these sliders to draw your SpiroGraph in virtually any color you like!
PLAY ROUND WITH THE SLIDERS AND HAVE FUN!