# choclo.point.kernel_eu#

choclo.point.kernel_eu(easting_p, northing_p, upward_p, easting_q, northing_q, upward_q, distance)[source]#

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

Important

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

Parameters:
• easting_p (float) – Easting coordinate of point $$\mathbf{p}$$.

• northing_p (float) – Northing coordinate of point $$\mathbf{p}$$.

• upward_p (float) – Upward coordinate of point $$\mathbf{p}$$.

• easting_q (float) – Easting coordinate of point $$\mathbf{q}$$.

• northing_q (float) – Northing coordinate of point $$\mathbf{q}$$.

• upward_q (float) – Upward coordinate of point $$\mathbf{q}$$.

• distance (float) – Euclidean distance between points $$\mathbf{p}$$ and $$\mathbf{q}$$.

Returns:

kernel (float) – 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:

$k_{xz}(\mathbf{p}, \mathbf{q}) = \frac{\partial}{\partial x \partial z} \left( \frac{1}{\lVert \mathbf{p} - \mathbf{q} \rVert_2} \right) = \frac{ 3 (x_p - x_q) (z_p - z_q) }{ \lVert \mathbf{p} - \mathbf{q} \rVert_2^5 }$

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