How to find a maximum value in a rotated parametric plot?

Question

Let me just give you a bunch of tables to reproduce the problem. Here they are

factor = {0.21771306781305513`, 0.21268138506316966`, 
   0.21072547572302136`, 0.21280240514184656`};
translation = {{0.07311848609129466`, 
    0.4321749118152892`}, {0.07419850615038676`, 
    0.41942966428645845`}, {0.07495664043451092`, 
    0.4055676791824686`}, {0.075403501661516`, 0.390376772269518`}};
fixed = {{0.15109999999997645`, 
    0.4177298035656685`}, {0.15109999999997636`, 
    0.4050398115591144`}, {0.15109999999997642`, 
    0.39132990005335083`}, {0.15109999999997648`, 0.376458513739336`}};
angles = {-0.2503758967687387`, -0.1877555323951659`, \
-0.12492241866341011`, -0.062224801455231286`};
trans = {0.1983128390296685`, 0.19442477684551443`, 
   0.1897180584133508`, 0.184023014923336`};
angle = {-0.366322981687224`, -0.36996279652454467`, \
-0.3697036572293485`, -0.3636764333664053`};

From the above you should define the following rotation functon:

rotateparametric[parfunc_, fixedpoint_, angle_] := 
  RotationMatrix[angle].(parfunc - fixedpoint) + fixedpoint

This is how the data is used:

Table[
  ParametricPlot[
    rotateparametric[
      factor[[j]]*{Sin[ϕ], -(1 - Cos[ϕ])} + translation[[j]], 
      fixed[[j]], 
      angles[[j]]] - {0, trans[[j]]}, 
    {ϕ, angle[[j]], -angle[[j]]}, 
    PlotRange -> All], 
  {j, 1, Length[factor], 1}]

I have to find the maximum value of this rotated parametric plot. My problem is that I have no idea how to do it. I would normally use MaximumValue or FindMaximum, but due to the rotation, none of the above seems to work. Is there anything I can do?

Answer 1

Does this do what you want?

Table[FindMaximum[
    {0, 1}.(rotateparametric[factor[[j]]*{Sin[ϕ], -(1 - Cos[ϕ])} + 
       translation[[j]], fixed[[j]], angles[[j]]] - {0, trans[[j]]}),
    {ϕ, 0, angle[[j]], -angle[[j]]}],
 {j, 1, Length[factor], 1}]

(*
  {{0.259521, {ϕ -> -0.250376}}, {0.242844, {ϕ -> -0.187756}},
   {0.226868, {ϕ -> -0.124922}}, {0.211446, {ϕ -> -0.0622248}}}
*)