Introduction

In order to understand each geometric transformation remember that the points of the topography are stored in the attribute P, which is a \(N\times 3\) matrix:

\[\begin{split}\mathbf{P} = \begin{bmatrix} x_1 & y_1 & z_1 \\ x_2 & y_2 & z_2 \\ \vdots & \vdots & \vdots \\ x_N & y_N & z_N\end{bmatrix}\end{split}\]

where \(\mathbf{p}_i = \left[x_i\quad y_i \quad z_i\right]\) is the *i*th point of the topography. In practice the altitude \(z_i\) is determined upon a latent function of the coordinates, hence \(z_i = f(x_i, y_i)\).

Translation

Let us define a vector \(\mathbf{v} = \left[x_v\quad y_v\quad z_v \right]\). By calling:

t.translate([xv,yv,zv])

we translate \(\mathbf{P}\) by \(\mathbf{v}\) as:

\[\mathbf{p}_i \leftarrow \mathbf{p}_i + \mathbf{v}\quad\forall\ i = 1,2,\dots,N.\]

Rotation

This method performs the rotation about the centre given three rotation angles \(\theta_x\), \(\theta_y\), \(\theta_z\). The method builds the corresponding rotation matrices:

\[\begin{split}\mathbf{R}_x = \begin{bmatrix} 1 & 0 & 0 \\\ 0 & \cos(\theta_x) & \sin(\theta_x) \\\ 0 & -\sin(\theta_x) & \cos(\theta_x) \\\ \end{bmatrix},\end{split}\]

\[\begin{split}\mathbf{R}_y = \begin{bmatrix} \cos(\theta_y) & 0 & \sin(\theta_y) \\\ 0 & 1 & 0 \\\ -\sin(\theta_y) & 0 & \cos(\theta_y) \end{bmatrix},\end{split}\]

\[\begin{split}\mathbf{R}_z = \begin{bmatrix} \cos(\theta_z) & \sin(\theta_z) & 0 \\\ -\sin(\theta_z) & \cos(\theta_z) & 0 \\\ 0 & 0 & 1 \end{bmatrix},\end{split}\]

\[\mathbf{R} = \mathbf{R}_x\mathbf{R}_y\mathbf{R}_z.\]

Then each point is rotated as:

\[\mathbf{p}_i \leftarrow \mathbf{R}\mathbf{p}_i.\]

This is accomplished via:

t.rotate(t_deg=[t_x, t_y, t_z])

The method supports passing a rotation matrix too:

t.rotate(rot_mat=R)

In case on wishes to rotate about a given centre \(c=\left[c_x\quad c_y\quad c_z\right]^\top\), they would call the wrapper method:

t.rotate_about_centre(c=[c_x, c_y, c_z]t_deg=[t_x, t_y, t_z])

or by providing a rotation matrix rot_mat.

Flip

This method allows mirroring data with respect to a given vector \(\mathbf{v}=\left[v_x\quad v_y\quad v_z \right]\) which represent the outward normal of the intended flipping plane. Assuming one wishes to flip the data about the $yz$ plane, they would define \(\mathbf{v}=\left[-1\quad 1\quad 1 \right]\), and call:

t.flip([-1,1,1])

This operation performs \((\mathbf{p}_i = \left[x_i\quad y_i \quad z_i\right])\):

\[\left[x_i\quad y_i \quad z_i\right] \leftarrow \left[x_i\cdot v_x\quad y_i\cdot vy\quad z_i\cdot vz\right] \quad\forall\ i = 1,2,\dots,N.\]

Cut

Suppose you would like to remove some outliers from topography. Although the same procedure applies to the other axes identically, we focus on the z-axis. We also assume two threshold \(l\) and \(u\), whereby we filter each *i*th datum using the criterion:

\[z_i \leftarrow z_i:\quad z_i > l\quad \text{and}\quad z_i < u.\]

To do so, we call:

t.cut(ax="z", lo=l, up=u, out=False)

If out=True the method keep the points complying with:

\[z_i \leftarrow z_i:\quad z_i < l\quad \text{and}\quad z_i > u.\]