In this video I go over the cycloid curve and derive the parametric equations for the case in which the angle inside the circle is between 0 and π/2. The cycloid is formed by tracing a point on the circumference of a circle as it rotates along a straight line. The resulting parametric equations are shown below:
x = r(θ - sinθ)
y = r(1 - cosθ)
Although I derived these equations for the case where θ is between 0 and π/2, these equations do in fact work for all values of θ. I will go over the proof for other values of θ in a later video so stay tuned for that!
Also, these parametric equations can be rearranged to eliminate θ and to write x as a function of y, but the resulting Cartesian equation is much more complicated. I will nonetheless derive the Cartesian form of the cycloid in a later video, as well as the very interesting history of the cycloid, so stay tuned!
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Example:
The curve traced out by a point P on the circumference of a circle as the circle rolls along a straight line is called a cycloid (https://en.wikipedia.org/wiki/Cycloid).
If the circle has radius r and rolls along the x-axis and if one position of P is the origin, find parametric equations for the cycloid.
Solution:
We choose as the parameter the angle of rotation θ of the circle (θ = 0 when P is at the origin).
Suppose the circle has rotated through θ radians (but assume 0 < θ < π/2).
Because the circle has been in contact with the line, we see that the horizontal distance it has traveled from the origin must be the arc length from P to T:
Therefore, the center of the circle, C, is (rθ, r).
Let the coordinates of P be (x, y), then we can solve for the values of x and y:
Therefore, the parametric equations of the cycloid are:
One arch of the cycloid comes from one rotation of the circle and so is described by 0 ≤ θ ≤ 2π.
Although we derived these parametric equations for the case where 0 < θ < π/2, it can be seen that these equations are still valid for other values of θ.
- I will cover this in a later video, so stay tuned!
Although it is possible to eliminate the parameter θ to form an equation with only x and y, the resulting Cartesian equation is very complicated and not as convenient to work with as the parametric equations.
The result is the following equation:
I will derive this also in another video, as well as the history of the cycloid so stay tuned!
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