Closed-form expressions for the Distance Gradients / Hessians

[INeedAUniqueUsername]

Hi,
I'm implementing IPC in my advisor's physics library. However, the paper does not appear to provide closed-form expressions for the {gradients, Hessians} of the {point-plane, point-line, line-line} distance functions WRT all inputs. Instead, the toolkit only has mangled, auto-generated functions. Since my advisor wants to make sure that I fully understand the math, I need to know the un-mangled expressions for these functions. So far, I've managed to reverse-engineer the following functions below. Could you please provide me the un-mangled forms for all the mangled functions? I'm relatively new to this math (e.g. I have never tried deriving something like the gradient of point-plane distance WRT the plane itself).

Thank you for your time.

VECTOR<T, 12>
Point_Plane_Distance_Gradient(const TV& p, const TV& a, const TV& b, const TV& c)
{
    TV pa = p - a;
    TV ba = b - a;
    TV ca = c - a;
    TV cb = c - b;
    TV normal = ba.Cross(ca);
    T one_over_normal_sq = 1 / normal.Magnitude_Squared();
    T one_over_normal_sq_sq = one_over_normal_sq * one_over_normal_sq;
    T p_distance = pa.Dot(normal);
    T p_distance_sq = p_distance * p_distance;

    return 2 * p_distance * one_over_normal_sq * VECTOR<T, 12>(
        normal,                                                                     //g_p
        VECTOR<T, 9>(
            p_distance * one_over_normal_sq * cb.Cross(normal) - (normal + pa.Cross(cb)), //g_a
            VECTOR<T, 6>(
                ca.Cross(normal) + pa.Cross(ca) * p_distance * one_over_normal_sq,  //g_b
                pa.Cross(ba) - ba.Cross(normal) * p_distance * one_over_normal_sq   //g_c
            )
        )
    );
}

VECTOR<T, 9>
Point_Line_Distance_Gradient(const TV& p, const TV& a, const TV& b) const
{
    TV pa = p - a;
    TV pb = p - b;
    TV ab = a - b;
    T over_ab_sq = 1.0 / ab.Magnitude_Squared();
    TV paXpb = (p - a).Cross(p - b);
    T over_ab_sq_sq = over_ab_sq * over_ab_sq;
    T paXpb_sq = paXpb.Magnitude_Squared();
    TV f = 2 * ab * paXpb_sq * over_ab_sq_sq;
    return 2 * over_ab_sq * VECTOR<T, 9>(
        ab.Cross(paXpb),
        VECTOR<T, 6>(
            f + pb.Cross(paXpb),
            f + pa.Cross(paXpb)
        )
    );

[zfergus]

All of the distance derivatives in the toolkit (except for point-point) are derived using symbolic differentiation, so unfortunately, I do not have the matrix calculus versions of these expressions on hand. I can point you to Kim and Eberle's [2022] "Dynamic Deformables" course (see "Appendix I: Computing the Derivatives of a Triangle (and Edge) Normal"). Shi and Kim [2023] also have some useful derivations in their supplemental.

If your goal is to verify your derivation you can compare it with the existing implementation or use finite differences.

The derivatives were originally derived using the Symbolic Math Toolbox in MATLAB, but I prefer to use SymPy in Python. There is a utility file notebooks/generate_cpp_code.py which I use a lot to generate various gradients/Jacobian/Hessians. While I agree with your advisor that understanding matrix calculus is an important skill, I would also argue that learning how to use symbolic code generation is a good skill to have in your toolbox.