Generalize Polarities

[germannp]

Cells can have more than one polarity, e.g. epithelial cells have an apical-basal polarity and can also have planar cell polarity (PCP). We currently only support one polarity per cell, given by theta and phi.

One way to generalize our model is templating the functions in polarity.cuh:

template<typename Pt, float Pt::*theta = &Pt::theta, float Pt::*phi = &Pt::phi>
__device__ __host__ float pol_scalar_product(Pt a, Pt b)
{
    return sinf(a.*theta) * sinf(b.*theta) * cosf(a.*phi - b.*phi) +
           cosf(a.*theta) * cosf(b.*theta);
}

template<typename Pt, float Pt::*theta = &Pt::theta, float Pt::*phi = &Pt::phi>
__device__ __host__ Pt pcp_force(Pt Xi, Pt pj)
{
    Pt dF{0};
    auto prod = pol_scalar_product<Pt, theta, phi>(Xi, pj);
    dF.*theta =
        prod * (cosf(Xi.*theta) * sinf(pj.*theta) * cosf(Xi.*phi - pj.*phi) -
                   sinf(Xi.*theta) * cosf(pj.*theta));
    auto sin_Xi_theta = sinf(Xi.*theta);
    if (fabs(sin_Xi_theta) > 1e-10)
        dF.*phi =
            -prod * sinf(pj.*theta) * sinf(Xi.*phi - pj.*phi) / sin_Xi_theta;

    return dF;
}

...

However, PCP in epithelial cells is within the layer, so it would be better to come up with forces for cellular alignment in the plane orthogonal to the apical-basal polarities.

How many polarities do we actually need?

[germannp]

We should rather speak about tissue polarity instead of PCP in the context of mesenchymal cells. If one is interested in PCP, i.e. the polarization of epithelial cells within the plane one should indeed come up with forces for alignment within the plane, see #14.