(N)Solve does not solve equation

Question

I am trying to calculate where a beam hits a mirror. The mirror is described by this curve (two facing spherical mirrors):

Mirror[x_] := Piecewise[{
   {{Sin[-87.5 °] + 0.5, Cos[-87.5 °]}, x < 0},
   {{Sin[-x ° - 87.5 °] + 0.5, 
     Cos[-x ° - 87.5 °]}, x >= 0 && x < 5},
   {{Sin[x ° + 82.5 °] - 0.5, 
     Cos[x ° + 82.5 °]}, x >= 5 && x < 10}
   }, {Sin[92.5 °] - 0.5, Cos[92.5 °]}]

Now, I am just trying to get the crossing point with a line:

p = {0, 0};
v = {1, 0};
Solve[Evaluate[
  Simplify[Mirror[s].{1, 0}] == (p + v t).{1, 0} &&
   Simplify[Mirror[s].{0, 1}] == (p + v t).{0, 1} 
  ], {s, t}]

The manual projection (.{1, 0}) is done because of the piecewise defined function, as they do not work with the usual Thread - we worked this out here. This code has been working well since then, but it does not for the current Mirror function. I have no idea what the problem could be. Any ideas?

Answer 1

For example:

f[x_] := x Pi/180;
Mirror[x_NumericQ] := 
 Piecewise[{{{Sin[-f[87.5 ]] + 0.5, Cos[-f[87.5 ]]}, x < 0}, 
            {{Sin[-x - f[87.5 ]] + 0.5, Cos[-x - f[87.5 ]]}, x >= 0 && x < f[5]}, 
            {{Sin[x + f[82.5 ]] - 0.5, Cos[x + f[82.5 ]]}, x >= f[5] && x < f[10]}},      
             {Sin[f[92.5 ]] - 0.5, Cos[f[92.5 ]]}]
p = {0, 0};
v = {1, 0};
FindRoot[Mirror[s].{1, 0} == (p + v t).{1, 0} && 
         Mirror[s].{0, 1} == (p + v t).{0, 1}, 
         {{s, 0}, {t, 0}}]
(*
{s -> 0.0436332, t -> -0.5}
*)

graphical insight:

ParametricPlot[{Mirror[t], (p + v t)}, {t, -.6, .6},  AxesOrigin -> {Automatic, -.04}]