arc length between point and linestring where the point is not in between the linestring

[kurogane1031]

Hi I would like to ask your advice regarding finding arc length between a point and a linestring, where the point is not in between the linestring.

For example. given a situation as image below, my intention it to find the arclength from P_x to P_1. Currently my idea is to interpolate the point using splines and find the approximated distance, but it is not efficient. Therefore I wonder if there is an available or appropriate strategy in boost geometry that able to find this,

Thank you very much in advance.