# Gaps in a continuous parametric plot problem

I attempted to plot a parametric plot using the following code:

q := Sqrt[((2 + \[Delta])*(4 + \[Delta]))/
((-2 - \[Delta]/2)^(4/(2 + \[Delta])^2)*
(4 + \[Delta]*(6 + \[Delta])))]*
((1 - Cos[\[Theta]])^((\[Delta]*(4*M + \[Delta]))/
(2*(2*M + \[Delta])^2))/(1 + Cos[\[Theta]])^(1/2))*
(-3*M - \[Delta] + M*Cos[\[Theta]])^((2*M^2)/
(2*M + \[Delta])^2) //. {M -> 1, \[Delta] -> 1}

R := ((2*M*Sin[\[Theta]])/Sqrt[(M + \[Delta] - M*Cos[\[Theta]])/
(3*M + \[Delta] - M*Cos[\[Theta]])])*
((1 + q^2)/(2*q)) //. {\[Delta] -> 1, M -> 1}

Chop[ParametricPlot[{{R*((2*q)/(1 + q^2)), R*((1 - q^2)/(1 + q^2))}, {(-R)*((2*q)/(1 + q^2)), R*((1 - q^2)/(1 + q^2))}}, {\[Theta], 0, Pi}, PlotStyle -> {{Black}},
Axes -> None, PlotRange -> All, Exclusions -> None, Mesh -> False]]


However, I obtained such a graph:

where gaps can be seen although the plot should be continuous throughout. In my code, I have also tried several remedies proposed in earlier posts albeit none worked. How do I resolve this issue?

• Apparently if I change the delta value from 1 to for e.g. 1.001, the graph will be completely continuous though I have no idea why that solves the issue.
– user74137
Commented Mar 7, 2021 at 15:30
• Use Chop (or Re) inside (not outside) ParametricPlot to ,,clear'' spurious several \$MachineEpsilon imaginary parts. Commented Mar 7, 2021 at 15:33
• Just Simplify the functions: ParametricPlot[Evaluate[{{R*((2*q)/(1 + q^2)), R*((1 - q^2)/(1 + q^2))}, {(-R)*((2*q)/(1 + q^2)), R*((1 - q^2)/(1 + q^2))}} // Simplify], {\[Theta], 0, Pi}, PlotStyle -> Black, Axes -> None, PlotRange -> All, Exclusions -> None, Mesh -> False] Commented Mar 8, 2021 at 1:39

ParametricPlot[{{R*((2*q)/(1 + q^2)),