Find the parametrization of the well-known curve, a cycloid, i.e. the trajectory that plans a point of a circumference of radio $$1$$ when the circumference turns on the axis $$x$$:
Development:
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Try to apply some simple method that you already know in order to parametrize the curve: if a coordinate is known according to another one, or if it can be written in a simpler form in some type of coordinates.
From what we can observe in the drawing, it turns out to be difficult to express the curve easily in some type of coordinates, and we do not know $$y$$ according to $$x$$ either.
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In the opposite case, try to describe the evolution of the coordinates $$x$$ and $$y$$ in terms of some parameter.
The difficulty of this exercise lies in choosing the variable that will give us the evolution of $$x$$ and $$y$$. To do so, since we have one wheel that turns and we are concentrating on just one point, we can take $$t$$ to be the angle (in radians) between the radius passing through our particular point, starting by below (the origin) and in a clockwise direction.
Thus, it turns out to be simple to calculate the component $$y(t) =1-\cos (t)$$, as we can see in the drawing.
On the other hand, the component $$x(t)$$ will have 2 parts:
The first one will be the horizontal displacement of the center of the circle, that will be equal to the distance covered on the plane, and this one will be equal to $$t$$, because the length covered is equal to the length of the arch, which in radians is equal to the angle.
The second part is the horizontal position of the chosen point with respect to the center of the circumference, (identical in puple colour in the drawing) that is $$\sin(t)$$, but with a negative sign, since for $$t$$ positive, it is a negative distance. Then $$x(t) =t-\sin(t)$$.
Solution:
The parametrization of the curve known as cycloid is $$\gamma(t)=(1-\cos(t),t-\sin(t))$$