# Frenet Equations in 2-d Which Result in the Cornu Spiral - Comment

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In summary, Charles Link submitted a new PF Insights post about the Frenet Equations in 2-d and their connection to the Cornu Spiral. The Cornu Spiral has been widely used in road and rail track constructions as well as for roller coasters. The engineer Hans Lorenz first used it in 1938 for the autobahn Vienna - Brno - Wroclaw. This theory has direct applications for Earth bound moving objects and was included in a mathematics challenge by @micromass in October 2016.
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Charles Link submitted a new PF Insights post

Frenet Equations in 2-d Which Result in the Cornu Spiral

Continue reading the Original PF Insights Post.

fresh_42 and Greg Bernhardt
Very nice! The German Wiki page says, that the Cornu spiral is widely used in road and rail track constructions and even for roller coasters.

"The Cornu spiral first has been used by Leopold Oerley in 1937 as geometric element of road construction. Since 1938 the engineer Hans Lorenz used it consequently in the planning for the autobahn Vienna - Brno - Wroclaw. 1954 the Cornu spiral has been made available to engineers in general in a comprehensive book with mathematical tables for track planning (Kasper, Schürba, Lorenz: Die Klotoide als Trassierungselement)."

I had to think about this as I read your considerations on speed and acceleration, which obviously has some very direct applications for Earth bound moving objects. All of a sudden the theory became alive.

@fresh_42 You might find it of interest, I messaged this calculation of the Cornu Spiral to @micromass sometime around September 2016, and he included it as part of his October mathematics challenge (problem #1). @mfb solved it very quickly so I don't know how many others took the time to look it over. :) :)

Great Insight Charles!

## 1. What are Frenet equations in 2-d?

Frenet equations in 2-d are a set of differential equations that describe the curvature and torsion of a curve in two-dimensional space. They are named after French mathematician Jean Frenet.

## 2. How do Frenet equations result in the Cornu spiral?

The Cornu spiral, also known as the Euler spiral, is a special curve that can be described using the Frenet equations. The curvature and torsion values calculated from the Frenet equations are used to generate the coordinates of points on the Cornu spiral curve.

## 3. What is the significance of the Cornu spiral?

The Cornu spiral has many applications in mathematics, physics, and engineering. It is commonly used to model the shape of beams under bending, the path of light through a graded-index medium, and the trajectory of a projectile under the influence of air resistance.

## 4. Are there any real-world examples of the Cornu spiral?

Yes, the Cornu spiral can be observed in various natural phenomena, such as the shape of waves breaking on a shoreline, the shape of a lightning bolt, and the shape of a swinging pendulum. It can also be found in man-made structures, such as the Gateway Arch in St. Louis, Missouri.

## 5. Are there any limitations to using Frenet equations in 2-d to describe the Cornu spiral?

While the Frenet equations in 2-d are a useful tool for generating points on the Cornu spiral, they are limited to describing curves in two-dimensional space. The Cornu spiral itself is a three-dimensional curve, and Frenet equations in 3-d must be used to fully describe it.

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