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

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[Ï•], -(1 – Cos[Ï•])} + translation[[j]],
fixed[[j]],
angles[[j]]] – {0, trans[[j]]},
{Ï•, 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?

=================

What do you mean by the value of a plot? The y (or x) coordinate? Some other measure?
– Michael E2
Dec 25 ’15 at 16:18

@MichaelE2 y coordinate!
– skrat
Dec 25 ’15 at 16:19

=================

1

=================

Does this do what you want?

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

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

This does exactly what I want! Thank you!
– skrat
Dec 25 ’15 at 16:44

@skrat You’re welcome!
– Michael E2
Dec 25 ’15 at 16:45