The last post in this series deals with the Sici spiral (Link1). It is similar to the Nielson spiral. I already mentioned that this spiral is my personal favourite of the three spiral functions presented on my page. On the one hand this is because the curve reminds me of JMLC horns and on the other hand because of the simple relationship of the tangent with respect to the basic rotation angle. In addition, we will see that the tangent vector, when the parameters are selected appropriately, results in a nearly constant length over large areas of the horn curve and only expands towards the horn mouth.
The cartesian parametrization look simple on the first view
\tag{1a}x =-a \cdot Ci({\phi})
\tag{1b}y = a \cdot \left( \frac{\pi}{2}-Si({\phi})\right)
but Ci(\phi) and Si(\phi) are the cosine and sine integrals (Link2). These integrals need to be solved but again as for the Cornu spiral these integrals can be developed as a series expansion. With proper offsets defined the Sici spiral becomes usable as horn profile function.
\tag{2}x_0 =-a \cdot Ci({\phi_s})
\tag{3a}y_0 = y(\phi_s) = a \cdot \left( \frac{\pi}{2}-Si({\phi_s})\right)
\tag{3b}y_0 = y(\phi_0) = a \cdot \frac{\pi}{2}