0
$\begingroup$
h = -5; k = 0;
ParametricPlot[{{(t - Sin[t] + h), (1 - Cos[t])}, {(Cos[t] - 
     1), (t - Sin[t]) + k} }, {t, 0, 2 Pi}, PlotLabel -> Cycloid_OT, 
 PlotStyle -> {Red, Thick}, GridLines -> Automatic]
ParametricPlot[
 Evaluate@Table[{{(t - Sin[t] + h), (1 - Cos[t])}, {(Cos[t] - 
       1), (t - Sin[t]) + k} }, {h, -5, 5, 1}], {t, 0, 2 Pi}, 
 PlotLabel -> Cycloid_OT, PlotStyle -> {Red, Thick}, 
 GridLines -> Automatic]

The two cycloid curves parametrized below have slopes that multiply to $-1$.

$$[( t-\sin t),(1- \cos t)]$$ $$[\cos t , (t-\sin t)]$$

I am trying to generate a mesh of plots by varying $h$ and $k$.

But $h$ itself is in error.

The plot aims to show that not all intersections are orthogonal, by visible verification.

Thanks for errors correction.

EDIT1:

Plot of two families of cycloids suggested by kglr with plot $Join$ using half interval for (h,k) steps in the intervals $ (-2<x<0),\, (0<y<2) :$

Two family cycloids

$\endgroup$
  • $\begingroup$ try Evaluate[Join@@Table[{{(t - Sin[t] + h), (1 - Cos[t])}, {(Cos[t] - 1), (t - Sin[t]) + k}}, {h, -5, 5, 1}]] in the first argument of ParametricPlot? $\endgroup$ – kglr Aug 2 at 5:22
  • $\begingroup$ Thanks that works for $h$. Next please perturb the $k$ set for a full two parameter net. $\endgroup$ – Narasimham Aug 2 at 5:59
  • $\begingroup$ how about ParametricPlot[ Evaluate[Join[Table[{(t - Sin[t] + h), (1 - Cos[t])}, {h, -5, 5}], Table[{(Cos[t] - 1), (t - Sin[t]) + k}, {k, -5, 5}]]], {t, 0, 2 Pi}, GridLines -> Automatic]? $\endgroup$ – kglr Aug 2 at 6:28
  • $\begingroup$ The plot is ok now.. but shown result is not so encouraging. I had expected more orthogonal reticles. $\endgroup$ – Narasimham Aug 2 at 8:17

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