# 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[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?

• What do you mean by the value of a plot? The y (or x) coordinate? Some other measure? Dec 25, 2015 at 16:18
• @MichaelE2 y coordinate! Dec 25, 2015 at 16:19

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}}}
*)

• This does exactly what I want! Thank you! Dec 25, 2015 at 16:44
• @skrat You're welcome! Dec 25, 2015 at 16:45