# ParametricPlot not plotting domain end points

Consider the parametric plot

(*prettier Manipulate available at the end*)
ParametricPlot[{Cos@\[Theta], Sin@\[Theta]}^0.01, {\[Theta], 0, \[Pi]/2},
PlotRange -> {{0, 1.1}, {0, 1.1}}, AspectRatio -> 1, Mesh -> All, MeshStyle -> Black]


which gives

The graph point $$(0,1)$$ at $$\theta=\pi/2$$ is not seen. Instead, the graph abruptly stops without reaching the y-axis.
Hot to resolve this?

Note that increasing PlotPoints doesn't work as the new points keep getting added between the leading and the ending points of the mesh. Also, the 'disconnection-from-y-axis' effect is apparent only for sufficiently low values of $$n$$ in the graph of $$(\cos^n\theta,\sin^n\theta)$$, in which case the curve gets progressively slower for $$\theta>\pi/4$$.

Prettier:

Manipulate[
ParametricPlot[{Cos@\[Theta], Sin@\[Theta]}^n, {\[Theta],
0, \[Pi]/2},
Epilog -> {Dashed, {Thin, Gray, Circle[]}, Text[n, {0.1, 1.05}]},
GridLines -> {{1}, {1}}, PlotRange -> {{0, 1.1}, {0, 1.1}},
AspectRatio -> 1, PlotTheme -> "Scientific", Mesh -> All,
MeshStyle -> Black,
Method -> "BoundaryOffset" -> False], {{n, 0.01}, 0.001, 10, 0.01,
Appearance -> "Labeled"}]

• The problem is that your function is not very smooth. It makes a very fast change at Pi/2. MMA has trouble with such specials functions Dec 3, 2023 at 15:18
• @DanielHuber The derivative's $\tan^{n-2}\theta$, so for small $n$ at $\pi/2$, it goes to zero, not blowup (as evident in the plot) - but I take your point (as for $\theta=\pi/2+\epsilon$, the abscissa is complex and graph abruptly ends) Dec 3, 2023 at 15:40
• @lineage: What you need it for? What is your final goal? If you want a parametric equation for a rectangle (or part of it) there are better equations. Dec 3, 2023 at 15:55
• @azerbajdzan Just curious - recreational...puzzled me that it figured out $\theta=0$ but not $\pi/2$ Dec 3, 2023 at 15:59

Adding WorkingPrecision -> 10 solves the problem. Also values for n in manipulate should be exact values.

Manipulate[
ParametricPlot[{Cos@Abs@(\[Theta] + \[Pi]/4),
Sin@Abs@(\[Theta] + \[Pi]/4)}^n, {\[Theta], -\[Pi]/4, \[Pi]/4},
Epilog -> {Dashed, {Thin, Gray, Circle[]}, Text[n, {0.1, 1.05}]},
GridLines -> {{1}, {1}}, PlotRange -> {{0, 1.1}, {0, 1.1}},
AspectRatio -> 1, PlotTheme -> "Scientific", Mesh -> All,
MeshStyle -> Black, WorkingPrecision -> 10,
Method -> "BoundaryOffset" -> False,
Exclusions -> None], {{n, 1/100}, 1/1000, 10, 1/100,
Appearance -> "Labeled"}]


In case a parametric plot of rectangle is needed there are better choices for the equations:

ParametricPlot[{2, 3}*{Abs@Sin@x Sin@x + Abs@Cos@x Cos@x,
Abs@Sin@x Sin@x - Abs@Cos@x Cos@x}, {x, 0, 2 \[Pi]},
AspectRatio -> Automatic]


• +1. That's strange though; reducing the precision makes the plot better? For me , WorkingPrecision->17 and upwards is worse off. Dec 3, 2023 at 16:49