# latitude derivations

1. 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}

Convert all polygon corner coordinates defined by $$\phi^{B}(i)$$ and $$\lambda^{B}(i)$$ into unit sphere points $$\mathbf{p}(i) = [x_{i}, y_{i}, z_{i}]$$

$$x_{min} = min(x_{i}), y_{min} = min(y_{i}), z_{min} = min(z_{i})$$

$$x_{max} = max(x_{i}), y_{max} = max(y_{i}), z_{max} = max(z_{i})$$

$$\mathbf{p}_{center} = [\frac{x_{min} + x_{max}}{2}, \frac{y_{min} + y_{max}}{2}, \frac{z_{min} + z_{max}}{2}]$$

The vector $$\mathbf{p}_{center}$$ is converted back to $$\phi$$ and $$\lambda$$

2. 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}.

$\phi = \frac{\phi^{B}(2) + \phi^{B}(1)}{2}$
3. 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.

$\begin{split}\begin{eqnarray} N & = & max(i, \phi(i) \neq NaN) \\ \phi & = & \phi(N/2) \end{eqnarray}\end{split}$
4. 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.

$\phi = \phi_{instr}$