Wrong rotation direction and length of day for retrograde spin with SpiceRotation

[Ajaja]

Describe the bug
QT Info Browser shows wrong rotation direction and length of day for retrograde rotation defined with SpiceRotation

To Reproduce

Modify "Venus" "Sol"
{
     SpiceRotation
     {
        Kernel ["naif0012.tls" "pck00011.tpc"]
        Frame  "IAU_VENUS"
        Period 5832.432
     }
}

QT Info Browser shows:

Sidereal rotation period: 243,018 days
Rotation direction: Prograde
Length of day: 2981,33 days

Additional context
The problem occurs because of the shortcut in spicerotation.h:

    // No notion of an equator for SPICE rotation models
    Eigen::Quaterniond computeEquatorOrientation(double /* tdb */) const override
    {
        return Eigen::Quaterniond::Identity();
    }

But qtinfopanel.cpp needs computeEquatorOrientation() for prograde/retrograde determination:

        bool prograde = false;
        if (orbitalPeriod > 0.0)
        {
            Eigen::Vector3d axis = Eigen::AngleAxisd(rotationModel->equatorOrientationAtTime(t)
                                                     * body->getBodyFrame(t)->getOrientation(t)).axis();
            Eigen::Vector3d orbitNormal = body->getOrbitFrame(t)->getOrientation(t)
                                        * orbit->positionAtTime(t).cross(orbit->velocityAtTime(t));
            prograde = axis.dot(orbitNormal) >= 0;
            double siderealDaysPerYear = orbitalPeriod / rotPeriod;
            double solarDaysPerYear = prograde ? siderealDaysPerYear - 1.0 : siderealDaysPerYear + 1.0;
            if (std::abs(solarDaysPerYear) > 0.0001)
            {
                dayLength = std::abs(orbitalPeriod / solarDaysPerYear);
            }
        }

[Ajaja]

Celestia builds with ENABLE_SPICE=ON for testing are here in artifacts: https://github.com/Ajaja/Celestia/actions (Windows only)

[ajtribick]

Some quick notes from playing around with it so I don't forget when I have time to look at implementing this. Useful summary of time derivatives of rotation matrices is in https://arxiv.org/abs/1609.06088

// get full state vector transformation (x,y,z,dx,dy,dz)
// may not be available even if pxform_c works, depending on loaded kernels
SpiceDouble xform[6][6];
sxform_c(fromFrame, toFrame, t, xform);

// implement some error detection here, then

using VecMap = Eigen::Map<Eigen::Matrix<SpiceDouble, 3, 1>>;
bool prograde = VecMap{xform[0]}.dot(VecMap{xform[4]}) > 0.0;

// if we need obliquity, then we need some more components

Eigen::Matrix<SpiceDouble, 3, 1> axis
{
    VecMap{xform[1]}.dot(VecMap{xform[5]}),
    VecMap{xform[2]}.dot(VecMap{xform[3]}),
    VecMap{xform[0]}.dot(VecMap{xform[4]}),
};

SpiceDouble obliquity = std::acos(axis[2] / axis.norm()); // radians

Will probably need some adjustments to the class model to expose this. Note that we can't just change computeEquatorOrientation as this will affect the rendering as well as the info panel.