I am using two softwares for the generation of mesh for a CFD problem. Each software has its defintion of skewness for a 2D quadrilateral mesh.
The first software calculates the skewness based on the following expression: $$s_1 = \min(\theta_{\text{max}}-90, 90-\theta_{\text{min}})$$
While the second software calculates it based on the following expression: $$s_2 = \max\left( \dfrac{\theta_{\text{max}} - 90}{90}, \dfrac{90-\theta_{\text{min}}}{90}\right)$$
I am trying to find a link between $s_2$ and $s_1$ so if $s_1$ is known I can easily calculate $s_2$.
In the case where the following relation is statisfied: $$2\theta_\text{min} + 2\theta_\text{max} = 360$$ Then we get: $$\theta_\text{max} = 180 - \theta_\text{min}$$ $$s_1 = \min(90-\theta_\text{min},90-\theta_\text{min}) = 90-\theta_\text{min}$$ Therefore: $$s_2 = \dfrac{s_1}{90}$$ But in the case where the internal angles of the quadrilateral are different $\theta_1 \neq \theta_2 \neq \theta_3 \neq \theta_4$:
Is it possible to calculate $s_2$ if only $s_1$ is known?
Using that $\,\max(a,b)+\min(a,b)=a+b\,$ and $\,\max(a,b)-\min(a,b)=|a-b|\,$:
$\;\require{cancel} 90\, s_2 + s_1 = (\bcancel{90}-\theta_{min})+(\theta_{max} - \bcancel{90}) = \theta_{max}-\theta_{min}$
$\;\require{cancel} 90\, s_2 - s_1 = \big|(90-\theta_{min})-(\theta_{max} - 90)\big|=\big|180-(\theta_{min}+\theta_{max})\big|$
This is insufficient to determine $\,s_2\,$ in general, if you only know $\,s_1\,$ but none of the angles.
If you know $\,\theta_{min} \pm \theta_{max}\,$ then $\,s_2\,$ can be determined from one of the relations above. For example, if $\,\theta_{min}+\theta_{max}=180\,$ then the second equality gives $\,s_2 = \dfrac{1}{90}s_1\,$, as posted.