Why is there always a horizontal tail line on the parametric curve?

Question

I am trying to create a smooth, piecewisely continuous and differentiable curve by using Mathematica's Floor function, which is as in:

p = ParametricPlot[{(1/4 π (1 + 2 t - 2 Floor[t])) - 
     Cos[(1/4 π (1 + 2 t - 2 Floor[t]))] Sin[(1/
          4 π (1 + 2 t - 2 Floor[t]))] + 
     Floor[t/π] (Pi/2 + 1), 
    2 - Cos[(1/4 π (1 + 2 t - 2 Floor[t]))]^2}, {t, 0, 3 Pi}, 
   ImageSize -> Large, PlotStyle -> Blue, 
   PlotRange -> {{0, 3 Pi}, {1.5, 2.2}}, Frame -> {True, True, False, False}];
p1 = ParametricPlot[{1/
      4 (π + 2 π t - 2 Cos[π (-t + Floor[t])] - 
       2 π Floor[t] + 2 (2 + π) Floor[t/π]), 
    1/2 (3 + Sin[π (t - Floor[t])])}, {t, 0, 2 Pi}, 
   ImageSize -> Large, PlotStyle -> Green, 
   PlotRange -> {{0, 3 Pi}, {1.5, 2.2}}, Frame ->{True, True, False, False}];
GraphicsGrid[{{p1}, {p}}]

It is very strange that the obtained curve always has a horizontal tail at the right side, whenever how the x range is set.

Is it because of the Floor function I am using or any other reason? How can I remove the unexpected horizontal tails?

Answer 1

When you are using Floor you are bound to make a discrete jump at some point, as it only returns you the nearest integer. Depending on your other function this jump may be visible or not. What you are calling a tail is can be a manifestation of a jump. Most Probably it is coming because you are using Floor with a trigonometric function (as you know $ \sin (x) = \sin (x+2\pi))$.

Lets clarify it with an example. I Took your first parametrisation, and make a list plot. First I choose a large interval between data points (d=Pi/10.)

d = Pi/10.;
dat = Table[{(1/4 \[Pi] (1 + 2 t - 2 Floor[t])) - Cos[(1/4 \[Pi] (1 + 2 t - 2 Floor[t]))] Sin[(1/ 4 \[Pi] (1 + 2 t - 2 Floor[t]))] + 
 Floor[t/\[Pi]] (Pi/2 + 1), 2 - Cos[(1/4 \[Pi] (1 + 2 t - 2 Floor[t]))]^2}, {t, Pi, 3 Pi,d}];

ListLinePlot[dat[[All, 1]]]
ListLinePlot[dat[[All, 2]]]
ListLinePlot[dat]

As you can see your plot (3rd figure) has a lot of back and forth movement. This appears because your two consecutive $x$ value difference is more than $\pi$. These movements become less visible when you choose smaller gap say, d=Pi/100.

So in conclusion I can say the tails are coming from a jump in your $x$ values when the jump height is more than $\pi$.

Answer 2

Just an observation, neither nice nor neat:

p = ParametricPlot[{(1/4 \[Pi] (1 + 2 t - 2 Floor[t])) - 
    Cos[(1/4 \[Pi] (1 + 2 t - 2 Floor[t]))] Sin[(1/
         4 \[Pi] (1 + 2 t - 2 Floor[t]))] + Floor[t/\[Pi]] (Pi/2 + 1),
    2 - Cos[(1/4 \[Pi] (1 + 2 t - 2 Floor[t]))]^2}, {t, 0, 6 Pi}, 
  PlotStyle -> Blue, PlotRange -> {{0, 4 Pi}, {0, 2.2}}, 
  Frame -> {True, True, False, False}]
p1 = ParametricPlot[{1/
     4 (\[Pi] + 2 \[Pi] t - 2 Cos[\[Pi] (-t + Floor[t])] - 
      2 \[Pi] Floor[t] + 2 (2 + \[Pi]) Floor[t/\[Pi]]), 
   1/2 (3 + Sin[\[Pi] (t - Floor[t])])}, {t, 0, 6 Pi}, 
  ImageSize -> Large, PlotStyle -> Green, 
  PlotRange -> {{0, 4 Pi}, {0, 2.2}}, 
  Frame -> {True, True, False, False}]