Given a parametric equation of a cycloid ($t \in R$):

$$ x(t)=r(t-\sin(t)); \\ y(t)=r(1-\cos(t)). $$

A vector $v=(x'(t),y'(t))$ if is not equals to zero then is a tangent vector to the curve at $(x(t),y(t))$. Given that $||v||$ is a vector norm and that a unit tangent vector (a tangent vector with a length $1$) is:

$$ T=\frac{v}{||v||} $$ to a curve at the same point.

Are there points on the curve at which we could not construct vector $T$ (means that $T$ could not exists at those points). If so what does the curve look like at those points?

My solution: My guess that the only way for which T could not exists is if the vector norm $||v||$ is $0$. Hence equating $||v||= 0$. And from there solve for the value of $t$. But, I do not know how how to proceed from here.

I got x'(t)=r(1-cos(t)) y'(t)=r(sin(t))

hence ||v||= sqrt(x'(t)^2+y'(t)^2) =0

Which leads to the expression 2-2cos(t)=0

solving it t=0

Is this correct?


2 Answers 2


Do what the question instructs: given $$\begin{align*} x(t) &= r(t-\sin t) \\ y(t) &= r(1-\cos t), \end{align*}$$ compute the vector $$\boldsymbol v = ( x'(t), y'(t) ).$$ For what values of $t$ is $\boldsymbol v = (0,0)$?

  • $\begingroup$ Oh that was my solution but it leads me to a weird expression which could not be solved. $\endgroup$
    – ys wong
    Aug 20, 2014 at 7:58
  • $\begingroup$ Are you sure that you're computing the derivative properly? You ought to be able to show that $t$ is an integer multiple of $2\pi$. $\endgroup$ Aug 20, 2014 at 8:05
  • $\begingroup$ We cannot help you if we cannot see your work. $\endgroup$
    – heropup
    Aug 20, 2014 at 8:19
  • $\begingroup$ I got t=0. My working is above $\endgroup$
    – ys wong
    Aug 20, 2014 at 8:42
  • $\begingroup$ What about $t = 2\pi$? Does that work too? How about $t = -4\pi$? You are not restricted to $0 \le t < 2\pi$. $\endgroup$
    – heropup
    Aug 20, 2014 at 13:54

This is quite expected at all $t = 2 \pi n $ points which are singular. The tracing point P on a circle radius r rotates around itself at zero velocity at the cusp of cycloid when circle rolls on line y=0 touching it at these instances.


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