latitude derivations
latitude from polygon
symbol
description
unit
variable name
\(\lambda\)
longitude
\(degE\)
longitude {:}
\(\lambda^{B}(i)\)
longitude
\(degE\)
longitude_bounds {:,N}
\(\phi\)
latitude
\(degN\)
latitude {:}
\(\phi^{B}(i)\)
latitude
\(degN\)
latitude_bounds {:,N}
The centroid is determined from the normal vector of the polygon area, which is the sum of the area weighted moments of consecutive vertices \(\mathbf{p}(i)\), \(\mathbf{p}(i+1)\) for all polygon edges (with \(\mathbf{p}(N+1):=\mathbf{p}(1)\)).
Convert all polygon corner coordinates defined by \(\phi^{B}(i)\) and \(\lambda^{B}(i)\) into unit sphere points \(\mathbf{p}(i) = [x_{i}, y_{i}, z_{i}]\)
\[\begin{split}\begin{eqnarray} w_{i} & = & \frac{1}{2} \begin{cases} \mathbf{p}(i) \cdot \mathbf{p}(i+1) \lt 0, & \pi - 2 asin(\frac{\Vert\mathbf{p}(i) + \mathbf{p}(i+1)\Vert}{2}) \\ \mathbf{p}(i) \cdot \mathbf{p}(i+1) \ge 0, & 2 asin(\frac{\Vert\mathbf{p}(i) - \mathbf{p}(i+1)\Vert}{2}) \end{cases} \\ \mathbf{p}_{center} & = & \sum_{i}{w_{i} \frac{\mathbf{p}(i) \times \mathbf{p}(i+1)}{\Vert\mathbf{p}(i) \times \mathbf{p}(i+1)\Vert}} \\ \end{eqnarray}\end{split}\]The vector \(\mathbf{p}_{center}\) is converted back to \(\phi\) and \(\lambda\)
latitude from range
symbol
description
unit
variable name
\(\phi\)
latitude
\(degN\)
latitude {:}
\(\phi^{B}(l)\)
latitude boundaries (\(l \in \{1,2\}\))
\(degN\)
latitude_bounds {:,2}
The pattern : for the dimensions can represent {latitude}, or {time,latitude}.
\[\phi = \frac{\phi^{B}(2) + \phi^{B}(1)}{2}\]latitude from vertical profile latitudes
symbol
description
unit
variable name
\(\phi\)
single latitude for the full profile
\(degN\)
latitude {:}
\(\phi(i)\)
latitude for each profile point
\(degN\)
latitude {:,vertical}
\(N\)
number of profile points
The pattern : for the dimensions can represent {time}, or no dimensions at all.
\[\begin{split}\begin{eqnarray} N & = & max(i, \phi(i) \neq NaN) \\ \phi & = & \phi(N/2) \end{eqnarray}\end{split}\]latitude from sensor latitude
symbol
description
unit
variable name
\(\phi\)
latitude
\(degN\)
latitude {:}
\(\phi_{instr}\)
latitude of the sensor
\(degN\)
sensor_latitude {:}
The pattern : for the dimensions can represent {time}, or no dimensions at all.
\[\phi = \phi_{instr}\]