choclo.point.kernel_enu

choclo.point.kernel_enu(easting_p, northing_p, upward_p, easting_q, northing_q, upward_q, distance)

Third derivative of the inverse of the distance along east-north-up.

This is equivalent to the derivatives along east-up-north, up-east-north, north-east-up, north-up-east, and up-north-east.

Important

The coordinates of the two points must be in Cartesian coordinates and have the same units.

Parameters:

easting_p, northing_p, upward_pfloat

Easting, northing and upward coordinates of point \(\mathbf{p}\).

easting_q, northing_q, upward_qfloat

Easting, northing and upward coordinates of point \(\mathbf{q}\).

distancefloat

Euclidean distance between points \(\mathbf{p}\) and \(\mathbf{q}\).

Returns:

kernelfloat

Value of the kernel function.

Notes

Given two points \(\mathbf{p} = (x_p, y_p, z_p)\) and \(\mathbf{q} = (x_q, y_q, z_q)\) defined in a Cartesian coordinate system, compute the following kernel function:

\[ \begin{align}\begin{aligned}k_{xyz}(\mathbf{p}, \mathbf{q}) = k_{xzy} = k_{zxy} = k_{yxz} = k_{yzx} = k_{zyx} = \frac{\partial^3}{\partial x_p \partial y_p \partial z_p} \left( \frac{1}{\lVert \mathbf{p} - \mathbf{q} \rVert_2} \right) = - \frac{ 15(x_p - x_q)(y_p - y_q)(z_p - z_q)\\}{ \lVert \mathbf{p} - \mathbf{q} \rVert_2^7 }\end{aligned}\end{align} \]

where \(\lVert \cdot \rVert_2\) refer to the \(L_2\) norm (the Euclidean distance between \(\mathbf{p}\) and \(\mathbf{q}\)).