draw satellite covarage zone on equirectangular projection - c++

Question

I need to draw borders of the observation zone of satellite on equirectangular projection. I found this formulas (1) and figure:
sin(fi) = cos(alpha) * sin(fiSat) – sin(alpha) * sin (Beta) * cos (fiSat);
sin(lambda) = (cos(alpha) * cos(fiSat) * sin(lambdaSat)) / cos(asin(sin(fi))) +
(sin(alpha) * sin(Beta) * sin(fiSat) * sin(lambdaSat)) / cos(asin(sin(fi))) -
(sin(alpha) * cos(Beta) * cos(lambdaSat))/cos(asin(sin(fi)));
cos(lambda) = (cos(alpha) * cos(fiSat) * cos(lambdaSat)) / cos(asin(sin(fi))) +
(sin(alpha) * sin(Beta) * sin(fiSat) * cos(lambdaSat)) / cos(asin(sin(fi))) -
(sin(alpha) * cos(Beta) * sin(lambdaSat)) / cos(asin(sin(fi)));
Cross-sections of the Earth in various planes:
And equations system (2) with figure:
if sin(lambda) > 0, cos(lambda) > 0 then lambda = asin(sin(lambda));
if sin(lambda) > 0, cos(lambda) < 0 then lambda = 180 - asin(sin(lambda));
if sin(lambda) < 0, cos(lambda) < 0 then lambda = 180 - asin(sin(lambda));
if sin(lambda) < 0, cos(lambda) > 0 then lambda = asin(sin(lambda));
Scheme of reference angles for the longitude of the Earth:
Where: alpha – polar angle;
fiSat, lambdaSat – latitude, longitude of satellite;
Beta – angle which change from 0 to 2*Pi and help to draw the observation zone;
fi, lambda – latitude, longitude of point B on the border of observation zone;
I repeat both (1) and (2) formulas in cycle from 0 to 2*Pi to draw border of observation zone. But I am not quite sure in (2) system of equations.
Inside intervals [-180;-90], [-90;90], [90;180] the zone draws correctly.
Center at -35;45:
Center at 120;60:
Center at -120;-25
But on border of -90 and 90 degree it get messy:
Center at -95;-50
Center at 95;30
Can you help me with formulas(1) and (2) or write another ones?
double deltaB = 1.0*M_PI/180;
observerZone.clear();
for (double Beta = 0.0; Beta <= (M_PI * 2) ; Beta += deltaB){
double sinFi = cos(alpha) * sin(fiSat) - sin(alpha) * sin(Beta) * cos(fiSat);
double sinLambda = (cos(alpha) * cos(fiSat) * sin(lambdaSat))/cos(asin(sinFi)) +
(sin(alpha) * sin(Beta) * sin(fiSat) * sin(lambdaSat))/cos(asin(sinFi)) -
(sin(alpha) * cos(Beta) * cos(lambdaSat))/cos(asin(sinFi));
double cosLambda = (cos(alpha) * cos(fiSat) * cos(lambdaSat))/cos(asin(sinFi)) +
(sin(alpha) * sin(Beta) * sin(fiSat) * cos(lambdaSat))/cos(asin(sinFi)) -
(sin(alpha) * cos(Beta) * sin(lambdaSat))/cos(asin(sinFi));
if (sinLambda > 0) {
if (cosLambda > 0 ){
sinLambda = asin(sinLambda);
sinFi = asin(sinFi);
}
else {
sinLambda = M_PI - asin(sinLambda);
sinFi = asin(sinFi);
}
}
else if (cosLambda > 0) {
sinLambda = asin(sinLambda);
sinFi = asin(sinFi);
}
else {
sinLambda = -M_PI - asin(sinLambda);
sinFi = asin(sinFi);
}
Point point;
point.latitude = qRadiansToDegrees(sinFi);
point.longitude = qRadiansToDegrees(sinLambda);
observerZone.push_back(point);
}

Answer 1

I solve my problem. In (1) equation when calculating cosLambda should be + instead of -.
double cosLambda = (cos(alpha) * cos(fiSat) * cos(lambdaSat))/cos(asin(sinFi)) +
(sin(alpha) * sin(Beta) * sin(fiSat) * cos(lambdaSat))/cos(asin(sinFi)) +
(sin(alpha) * cos(Beta) * sin(lambdaSat))/cos(asin(sinFi));
Sorry for disturbing.