choclo.point.kernel_e#

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

Easting component of the gradient of the inverse of the distance

Important

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

Parameters:

easting_pfloat

Easting coordinate of point $\mathbf{p}$.

northing_pfloat

Northing coordinate of point $\mathbf{p}$.

upward_pfloat

Upward coordinate of point $\mathbf{p}$.

easting_qfloat

Easting coordinate of point $\mathbf{q}$.

northing_qfloat

Northing coordinate of point $\mathbf{q}$.

upward_qfloat

Upward coordinate 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:

$$k_x(\mathbf{p},\mathbf{q})=\frac{\partial }{\partial x}\left(\frac{1}{\Vert \mathbf{p}-\mathbf{q}{\Vert }_2}\right)=-\frac{x_p-x_q}{\Vert \mathbf{p}-\mathbf{q}{\Vert }_2}$$

where $\Vert \cdot {\Vert }_2$ refer to the $L_2$ norm (the Euclidean distance between $\mathbf{p}$ and $\mathbf{q}$).