# A question about the Frénet–Serret Apparatus

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• swampwiz
In summary, the Frénet–Serret Apparatus is a mathematical tool used to describe the local geometry of a curve in three-dimensional space. It consists of three mutually perpendicular unit vectors, known as the tangential, normal, and binormal vectors, which are calculated using derivatives of the curve's position vector. The apparatus has significance in mathematics for understanding the properties of curved objects and has various real-world applications, but it may have limitations when applied to complex or discontinuous curves.
swampwiz

https://en.wikipedia.org/wiki/Frenet–Serret_formulas

The article says that the unit vector for the tangent direction is:

T = ( dr / ds ) / | dr / ds |

OK, that's the standard way to express a unit vector, but it seems that in this case, | dr / ds | is 1, so what is the point of including this term?

You're correct, there is no point. Wiki has it without the norm in the very next section and gets it correct in other places.

## 1. What is the Frénet–Serret Apparatus?

The Frénet–Serret Apparatus, also known as the Frenet frame or TNB frame, is a mathematical tool used to describe the local geometry of a curve in three-dimensional space. It consists of three mutually perpendicular unit vectors, known as the tangential, normal, and binormal vectors, which are used to describe the orientation and shape of the curve at a specific point.

## 2. How is the Frénet–Serret Apparatus calculated?

The Frénet–Serret Apparatus is calculated using differential geometry and vector calculus. The tangential vector is the unit tangent to the curve, the normal vector is the unit vector perpendicular to the tangent vector and lying in the osculating plane of the curve, and the binormal vector is the unit vector perpendicular to both the tangent and normal vectors. These vectors are calculated using derivatives of the curve's position vector.

## 3. What is the significance of the Frénet–Serret Apparatus in mathematics?

The Frénet–Serret Apparatus is significant in mathematics because it provides a way to understand the local properties of a curved object, such as a curve in space. It allows for the calculation of important quantities such as curvature and torsion, which are used in many fields of mathematics and physics, including differential geometry, differential equations, and mechanics.

## 4. How is the Frénet–Serret Apparatus used in real-world applications?

The Frénet–Serret Apparatus has various real-world applications, such as in computer graphics, robotics, and animation, where it is used to model and control the movement of objects along a curved path. It is also used in the study of fluid dynamics and aerodynamics to analyze the flow of fluids along curved surfaces. Additionally, it has applications in medical imaging, where it is used to describe the shape of biological structures such as blood vessels and bones.

## 5. Are there any limitations to the Frénet–Serret Apparatus?

While the Frénet–Serret Apparatus is a powerful tool for understanding the local geometry of curves, it does have limitations. It is most commonly used for continuous and smooth curves, and may not be applicable for more complex or discontinuous curves. Additionally, the apparatus can only describe the shape of the curve at a single point, and cannot provide information about the overall shape of the curve.

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