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}

Continue reading

Spiral Functions for Horns – The Cornu Spiral

The next spiral function in this series is the so called Cornu spiral (Link1). Other known names are Euler or Fresnel spiral or Clotoide. The most fascinating thing is that the curvature of the Cornu spiral is proportional to it’s arc length. Therefore, it is often used as base function for car lanes or railway traces. The mathematical treatment bear some challenges as the so called Fresnel integrals have to be solved. Fortunately, these integrals can be developed as a series expansion of quite simple terms. Several examples can be found in the web an there is nothing really new about this. The biggest failure for a series expansion happens if it diverges but for the range of parameters needed for a horn it has been found that the used algorithm converges properly throughout.

Continue reading

Spiral Functions for Horns – The Hyperbolic Spiral

It’s been a while since I wrote an article for my blog. As far as I was able to do in time, many other horn profiles have been investigated and programmed with help of my DrBA spread sheet calculator. The blog is therefore clearly behind the current stage of development. Now it is time to reduce the gap somewhat.

When I was working on the True Expansion Tractrix horn, during the research I found something that gave me the base idea of a whole range of new horn profiles. It was a reference to the hyperbolic spiral, which was something like this: if you unroll a hyperbolic spiral on a line and place a virtual pen in the center of the spiral, you get a Tractrix curve. What? A Tractrix curve? If the Tractrix curve is known to be suitable as a horn profile, is this also the case for the hyperbolic spiral?

Continue reading