2.3 Frenet-Serret formulas

The Frenet-Serret formulas are key tools in Metric Differential Geometry. They describe the movement of a particle along a curve in 3D space, using tangent, normal, and binormal vectors to create a moving coordinate system.

These formulas help us understand a curve's shape by measuring its curvature and torsion. They're used in physics, engineering, and computer graphics to analyze and create curves with specific properties in three-dimensional space.

Definition of Frenet-Serret formulas

• Set of equations describing the kinematic properties of a particle moving along a continuous, differentiable curve in three-dimensional Euclidean space
• Formulas relate the tangent, normal, and binormal unit vectors to each other and to the curvature and torsion of the curve
• Fundamental tool in the study of curves and their geometric properties within the context of Metric Differential Geometry

Derivation of Frenet-Serret formulas

• Derived by considering the motion of a particle along a curve and analyzing the changes in its velocity and acceleration vectors
• Involves the use of vector calculus and the concept of arc length parameterization

Tangent vector

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• Unit vector denoted as $\vec{[T](https://www.fiveableKeyTerm:t)}(s)$ pointing in the direction of the curve at each point
• Defined as the derivative of the position vector $\vec{r}(s)$ with respect to the arc length parameter $s$: $\vec{T}(s) = \frac{d\vec{r}}{ds}$
• Always points in the direction of the curve's velocity

Normal vector

• Unit vector denoted as $\vec{[N](https://www.fiveableKeyTerm:n)}(s)$ perpendicular to the tangent vector and pointing towards the center of curvature
• Defined as the derivative of the tangent vector with respect to the arc length parameter, divided by its magnitude: $\vec{N}(s) = \frac{\frac{d\vec{T}}{ds}}{\left\|\frac{d\vec{T}}{ds}\right\|}$
• Measures how the curve deviates from a straight line

Binormal vector

• Unit vector denoted as $\vec{[B](https://www.fiveableKeyTerm:b)}(s)$ perpendicular to both the tangent and normal vectors, forming a right-handed orthonormal basis
• Defined as the cross product of the tangent and normal vectors: $\vec{B}(s) = \vec{T}(s) \times \vec{N}(s)$
• Measures the torsion or twisting of the curve out of the plane defined by the tangent and normal vectors

Geometric interpretation of Frenet-Serret formulas

• Frenet-Serret frame provides a moving coordinate system along the curve, with the tangent, normal, and binormal vectors as the basis vectors
• Formulas describe how these basis vectors change as the particle moves along the curve

Tangent vector direction

• Always points in the direction of the curve's velocity
• Indicates the instantaneous direction of motion at each point on the curve (e.g., the direction a car is moving along a winding road)

Normal vector direction

• Points towards the center of curvature of the curve
• Indicates the direction in which the curve is turning at each point (e.g., the direction a car's steering wheel points when navigating a turn)

Binormal vector direction

• Perpendicular to the plane containing the tangent and normal vectors
• Indicates the direction of the curve's torsion or twisting (e.g., the direction a roller coaster track twists as it spirals upward)

Curvature in Frenet-Serret formulas

• Measures how much a curve deviates from a straight line at each point
• Quantifies the rate at which the tangent vector changes direction along the curve

Definition of curvature

• Denoted as $\kappa(s)$ and defined as the magnitude of the rate of change of the tangent vector with respect to the arc length parameter: $\kappa(s) = \left\|\frac{d\vec{T}}{ds}\right\|$
• Higher curvature values indicate sharper turns or bends in the curve (e.g., a tight hairpin turn on a mountain road)

Relationship between curvature and tangent vector

• Curvature is directly related to the rate of change of the tangent vector
• As the curvature increases, the tangent vector changes direction more rapidly (e.g., the tangent vector rotates quickly when moving along a highly curved section of a spiral)

Torsion in Frenet-Serret formulas

• Measures how much a curve twists or deviates from a plane at each point
• Quantifies the rate at which the binormal vector changes direction along the curve

Definition of torsion

• Denoted as $\tau(s)$ and defined as the rate of change of the binormal vector with respect to the arc length parameter: $\tau(s) = -\vec{N}(s) \cdot \frac{d\vec{B}}{ds}$
• Positive torsion indicates a right-handed twist, while negative torsion indicates a left-handed twist (e.g., the twisting of a DNA double helix)

Relationship between torsion and binormal vector

• Torsion is directly related to the rate of change of the binormal vector
• As the torsion increases, the binormal vector rotates more rapidly around the tangent vector (e.g., the binormal vector rotates quickly when moving along a highly twisted section of a helix)

Applications of Frenet-Serret formulas

• Frenet-Serret formulas have numerous applications in various fields, including physics, engineering, and computer graphics
• Enable the study and analysis of curves and their properties in three-dimensional space

Curve reconstruction from curvature and torsion

• Given the curvature and torsion functions of a curve, the Frenet-Serret formulas can be used to reconstruct the original curve up to a rigid motion
• Allows for the creation of curves with desired properties or the analysis of existing curves (e.g., designing a roller coaster track with specific curvature and torsion profiles)

Characterization of space curves

• Frenet-Serret formulas provide a way to classify and compare space curves based on their curvature and torsion
• Curves with the same curvature and torsion functions are congruent up to a rigid motion (e.g., identifying similar protein structures based on their backbone curves)

Frenet-Serret formulas in higher dimensions

• Frenet-Serret formulas can be generalized to curves in n-dimensional Euclidean space
• Allows for the study of higher-dimensional curves and their geometric properties

Generalization to n-dimensional space

• In n-dimensional space, the Frenet-Serret frame consists of n orthonormal vectors: the tangent vector and (n-1) normal vectors
• Generalized curvatures, denoted as $\kappa_1(s), \kappa_2(s), \ldots, \kappa_{n-1}(s)$, measure the rate of change of each normal vector

Frenet-Serret frame in higher dimensions

• The generalized Frenet-Serret formulas describe the relationships between the orthonormal vectors and the generalized curvatures
• Provide a complete description of the curve's geometry in n-dimensional space (e.g., analyzing the motion of a particle in a high-dimensional configuration space)

Limitations of Frenet-Serret formulas

• Frenet-Serret formulas have certain limitations and may not be well-defined for all types of curves
• Understanding these limitations is essential for the proper application of the formulas

Non-differentiable points on curves

• Frenet-Serret formulas require the curve to be continuously differentiable
• At non-differentiable points, such as cusps or corners, the formulas may not be well-defined (e.g., the tangent vector may not exist at a sharp corner)

Vanishing curvature or torsion

• If the curvature or torsion vanishes at a point, the corresponding normal or binormal vector may not be well-defined
• Requires special treatment or alternative formulations to handle these cases (e.g., using a different parameterization or considering one-sided limits)

Relationship to other concepts in differential geometry

• Frenet-Serret formulas are closely related to other important concepts in differential geometry
• Understanding these connections provides a deeper insight into the geometric properties of curves and surfaces

Connection to parallel transport

• Parallel transport is the process of moving a vector along a curve while maintaining its angle with respect to the curve's tangent vector
• Frenet-Serret formulas describe how the tangent, normal, and binormal vectors change under parallel transport (e.g., the normal vector remains perpendicular to the tangent vector)

Relationship to geodesics

• Geodesics are curves that represent the shortest path between two points on a surface
• In the context of Frenet-Serret formulas, geodesics are curves with zero geodesic curvature (i.e., the normal vector coincides with the surface normal)
• Frenet-Serret formulas can be used to study the properties of geodesics on surfaces (e.g., analyzing the curvature and torsion of a geodesic on a sphere)