choclo.point.kernel_en

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

Second derivative of the inverse of the distance along easting-northing

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:

\[k_{xy}(\mathbf{p}, \mathbf{q}) = \frac{\partial}{\partial x \partial y} \left( \frac{1}{\lVert \mathbf{p} - \mathbf{q} \rVert_2} \right) = \frac{ 3 (x_p - x_q) (y_p - y_q) }{ \lVert \mathbf{p} - \mathbf{q} \rVert_2^5 }\]

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