I was reading the paper titled "Rational space curves are not unit-speed" (Authors : Rida T.Farouki, Takis Sakkalis , Link for the paper : https://www.sciencedirect.com/science/article/pii/S0167839607000118).
In the very first line, I found that they consider the rational space curve by taking its 3 components as rational fucntions, but they have kept the denominator for all three components the same. I don't understand whether they are only talking about certain special rational curves of the defined type in the paper, or is it that any rational space curve can be written in that form? If it is the latter case, can someone please explain why it is so?
Here is the section of that paper in JPEG format for reference : Section of paper
Perhaps I'm missing something very obvious... but assume $$x(s) = \frac{p_1(s)}{q_1 (s)},\quad y(s) = \frac{p_2(s)}{q_2 (s)},\quad\mbox{and}\quad z(s) = \frac{p_3(s)}{q_3 (s)}. $$Put $W(s)= q_1 (s)q_2(s)q_3(s)$, and write $$x(s) = \frac{p_1(s)q_2 (s)q_3 (s)}{W (s)},\quad y(s) = \frac{q_1 (s)p_2(s)q_3 (s)}{W (s)},\quad\mbox{and}\quad z(s) = \frac{q_1 (s)q_2 (s)p_3(s)}{W (s)}. $$Now rename the numerators to $X(s)$, $Y(s)$ and $Z(s)$.