Deriving the Epicycloid Equations



This diagram will help us to derive the equation for the epicycloid. Let and be the radii of the inner and outer circles, respectively. First, suppose the circle at were not moving, but were centered at the origin instead and still moved around in the same way: counterclockwise, starting from the left. Then we could create equations for moving around as:   Furthermore, we follow the path of around a circle with radius , which is given by the equations:   Adding these will move our outer circle around the inner circle   The problem with these equations is that they assume that point and point are both rotating at the same rate, which is not the case. As goes from to , point completes one full rotation, but point completes a different number of rotation. Finding the relationship between their angular speeds can be done with just a bit of geometry. Start by adjusting the slider for to move the outer circle around the larger circle. As the small circle rolls around, it is going to match up with the large circle. This contact is shown by the orange arcs on the large and small circles. Adjust the slider to . Notice, that at this point, the point of contact between the two circles, has rotated around the center of the large circle to . We'll call this angle of rotation . Also, point has rotated around from the original point of contact. Let this angle be . Click the checkbox to show these two angles. Going back to the orange arcs, these must have the same length because they represent the portions of both circles that were in contact since it started rotating. If is the arc length, then we have:  Solving for :  This shows the relationship between the two angles of rotation. At this point in time, how much has point rotated around ? We can look at where is now in relation to its starting point on the left side of . Click the checkbox for "Starting Position." Notice that the new angle formed is congruent to , because the radii of the two circles are parallel to each other. So we can see that has rotated by Substituting in for we get: For the interval , point completes one rotation, so . This means that . Putting these into our equations from above we get: