Given the two orange lines in the picture, how can I calculate the center for an arithmetic spiral that will join the two lines whilst simultaneously being tangent to the ends of those lines. There is a similar question that has already been solved that deals with finding the center of logarithmic spirals in How to find the center of a log spiral? but the properties of logarithmic spirals being a constant angle means that it is inapplicable in this case though it does seem close. I believed that it could be possible to constrain the angles to find the center but an arithmetic spiral has different angles at every point of the curve so I would need to somehow use the fact that the radius changes linearly with the change in angle. I don't quite know how to do that though.
Update: I now have something that almost works, thanks to the image provided in the comments and with that I have this spiral that is tangent to one original line and needs a separate line shown in green that is parallel to the unused orange original line. I had just renamed aθ to be either r1 or r2 as that is more relevant in my use case than knowing what the θ is but it should not create the error that is shown in the image.
Here is what the tool for creating the spiral is doing. It requires a circle to be drawn first for the starting diameter of the spiral and takes in the variables of start angle, swept angle and pitch in order to fully define the spiral. All those values are pulled directly from the earlier sketch or calculated for the pitch by multiplying dimension a times 2pi.