choclo.point.kernel_nu
choclo.point.kernel_nu#
- choclo.point.kernel_nu(easting_p, northing_p, upward_p, easting_q, northing_q, upward_q, distance)[source]#
Second derivative of the inverse of the distance along northing-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:
\[k_{yz}(\mathbf{p}, \mathbf{q}) = \frac{\partial}{\partial y \partial z} \left( \frac{1}{\lVert \mathbf{p} - \mathbf{q} \rVert_2} \right) = \frac{ 3 (y_p - y_q) (z_p - z_q) }{ \lVert \mathbf{p} - \mathbf{q} \rVert_2^5 }\]where \(\lVert \cdot \rVert_2\) refer to the \(L_2\) norm (the Euclidean distance between \(\mathbf{p}\) and \(\mathbf{q}\)).