# Spiral Functions for Horns – The Sici Spiral

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}

# Consistent Modification of the Spherical Wave Horn

This post will be slightly different because it was written in German. German is the language of the original patents used and the main references of the literature used. Alone, to use the original terminology directly and pay homage to the inventors, I have decided to use the German language as a consistent continuation. I think that nowadays it is no longer an obstacle not to publish in English as powerful algorithms for automatic translation are available. For the captions, however, I have used English for the sake of simplicity. Continue reading

# Ellipsoidal Wave Fronts in Horns

As you might already guess by the name of this website, the spherical wave horn inspired my work a lot. If we assume expanding spherical wave fronts in round horns and want to stretch the round profile to an ellipse then we must inevitably think about ellipsoidal surface areas and of course the associated mathematics. My own learning phase was not easy either, until I found myself able to mathematically master the challenge. I will describe the results of my work in this post and try to give as many details as possible and describe as much math as necessary. The whole stuff is quite complicated and therefore we will simply start with the two-dimensional part and then gradually come to the ellipsoidal surfaces.