Correction to: SN Applied Sciences (2023) 5:346 https://doi.org/10.1007/s42452-023-05558-7

In this article the delimiters “<” and “>” to indicate the range of values were omitted from Eqs. (11), (12), and (19) during production of the published article.

The incorrect equations were as follows:

$$\begin{aligned} & d_{n} = d_{pn} \cos \theta_{H} \cos \theta_{V} \\ & d_{pn} = d_{p} /d_{p,\min } , d_{p} = d_{p,\min } ,d_{p,\max } , d_{pn} = 1,d_{p,\max } /d_{p,\min } \\ \end{aligned}$$
(11)
$$\begin{aligned} & d = \left[ {d_{A} \cos \theta_{H} + d_{B} \cos \left( {\theta_{H} - \theta_{AB} } \right)} \right] \cdot \cos \theta_{V} \\ & d_{A} \,\&\, d_{B} = d_{p,\min } ,d_{p,\max } \\ \end{aligned}$$
(12)
$$\begin{aligned} & d = d_{A} \cos \theta_{H} + d_{B} \cos \left( {\theta_{H} - 120^\circ } \right) + d_{C} \cos \left( {\theta_{H} + 120^\circ } \right) = d_{i} \cos \theta_{H} + d_{q} \sin \theta_{H} = d_{s} \cos \left( {\theta_{H} - \theta_{s} } \right) \\ & d_{A} \,\&\, d_{B} \,\&\, d_{C} = d_{p,\min } ,d_{p,\max } , d_{i} = d_{A} - \frac{{d_{B} + d_{C} }}{2}, d_{q} = \frac{\sqrt 3 }{2}\left( {d_{B} - d_{C} } \right), d_{s} = \sqrt {d_{i}^{2} + d_{q}^{2} } , \theta_{s} = \tan^{ - 1} \left( {\frac{{d_{q} }}{{d_{i} }}} \right) \\ \end{aligned}$$
(19)

The correct equations are as follows:

$$ \begin{array}{ll} {d_{n} = d_{pn} \cos \theta_{H} \cos \theta_{V} } \\ {d_{pn} = {{d_{p} } \mathord{\left/ {\vphantom {{d_{p} } {d_{p,min} }}} \right. \kern-0pt} {d_{p,min} }},d_{p} = \left\langle {d_{p,min} ,d_{p,max} } \right\rangle ,d_{pn} = \left\langle {1,{{d_{p,max} } \mathord{\left/ {\vphantom {{d_{p,max} } {d_{p,min} }}} \right. \kern-0pt} {d_{p,min} }}} \right\rangle } \\ \end{array} $$
(11)
$$ \begin{array}{ll} {d = \left[ {d_{A} \cos \theta_{H} + d_{B} \cos \left( {\theta_{H} - \theta_{AB} } \right)} \right] \cdot \cos \theta_{V} } \\ {d_{A} \,\&\, d_{B} = \left\langle {d_{p,min} ,d_{p,max} } \right\rangle } \\ \end{array} $$
(12)
$$ \begin{array}{ll} {d = d_{A} \cos \theta_{H} + d_{B} \cos \left( {\theta_{H} - 120^{ \circ } } \right) + d_{C} \cos \left( {\theta_{H} + 120^{ \circ } } \right) = d_{i} \cos \theta_{H} + d_{q} \sin \theta_{H} = d_{s} \cos \left( {\theta_{H} - \theta_{s} } \right)} \\ {d_{A}\, \&\, d_{B}\, \&\, d_{C} = \left\langle {d_{p,min} ,d_{p,max} } \right\rangle ,d_{i} = d_{A} - \frac{{d_{B} + d_{C} }}{2},d_{q} = \frac{\sqrt 3 }{2}\left( {d_{B} - d_{C} } \right),d_{s} = \sqrt {d_{i}^{2} + d_{q}^{2} } ,\theta_{s} = \tan^{ - 1} \left( {\frac{{d_{q} }}{{d_{i} }}} \right)} \\ \end{array} $$
(19)

The original article has been corrected.