compare equant & kepler models for planet going around sun.
Seek the angle from sun to planet as a function of time, particularly difference in angle
equant model needs cases for tan=0 or infinity and sign of tan
equant model: circle with sun displaced e on one side and the
uniform motion point ("equant") displaced e on the other side
Here we follow orbit phi=0,Pi
The difference looks like a nice sinusoidal at twice the orbital frequency
and an amplitude propto e^2



uu=Table[0,{51}]
qq=Table[0,{51}]
pp=Table[0,{51}]
rr=Table[{0,0},{51}]

eEarth=.0167
eMars=.0934

Do[
uu[[i]]= u /. FindRoot[u-e Sin[u] == .02 Pi (i-1),{u,.02 Pi (i-1) }] ;
rr[[i,1]]= (Cos[uu[[i]]]-e) ;
rr[[i,2]]= Sqrt[1-e^2] Sin[uu[[i]]];
pp[[i]]=ArcTan[rr[[i,1]],rr[[i,2]]];
t=Tan[.02 Pi (i-1)];
cs=Sin[.02 Pi (i-1)] Cos[.02 Pi (i-1)];
 
 If[i==26,
 rr[[i,2]]=Sqrt[1-e^2];
 rr[i,1]]=-2 e];
  If[i<26,
rr[[i,2]]=cs( e + Sqrt[1 + t^2(1 - e^2)]);
rr[i,1]]=rr[[i,2]]/t-2 e];

  If[i>26,
rr[[i,2]]=cs( e - Sqrt[1 + t^2(1 - e^2)]);
rr[i,1]]=rr[[i,2]]/t-2 e];

If[i == 1, rr[[1, 1]] = 1 - e];

 qq[[i]]=ArcTan[rr[[i,1]],rr[[i,2]]]
,
{i,51}]

Max[qq - pp]

.01 0.0000244855
.02 0.0000961244
.03 0.000212287
.04 0.00037047
.05 0.000568293
.06 0.000803493

   A =         0.1957       0.45E-02
   B =          1.949       0.65E-02

Solve[(y/t-e)^2+y^2==1,y]

{{y -> (e/t - Sqrt[1 + t^2 - e^2*t^2]/t)/(1 + t^(-2))}, 
 {y -> (e/t + Sqrt[1 + t^2 - e^2*t^2]/t)/(1 + t^(-2))}}

 1+1/t^2= 1/s^2
 t/s^2=1/(c s)
 c s ( e +- Sqrt[1 + t^2(1 - e^2)])
 
 if t->Infinty, c s t Sqrt[1-e^2] -> Sqrt[1-e^2]