Given is an involute of a circle. The basic circle radius $a$ and a point $A$ on the involute defined by the involute angle $\phi$ and the radius of the involute $\rho$ (and thus the arc length $l$) are known. How can I obtain the position of a following point $B$ on the involute if only the length $d$ of a line segment (comparable with the Euclidean distance) between these two points is available?
Please find following a graphic illustrating the conditions. Image “Points on involute”
As far as I figure out so far, I need to determine the change in the angle $\Delta \phi$ between point $A$ and the following point $B$. Then it would be possible to get the length of the arc $\Delta l$ related to the given length of the line segment $d$ between the two points. I really appreciate any approaches or ideas and not even sure if an applicable solution is possible here.