Glossary

3.3 Tangent and normal vectors

3 min read•august 7, 2024

Tangent and normal vectors are essential tools for understanding curves in space. They help us describe a curve's direction and shape at any point, giving us valuable insights into its behavior and properties.

These vectors form the basis of the , a local coordinate system that's crucial for analyzing curve geometry. By using tangent and normal vectors, we can calculate , , and other important characteristics of curves in three-dimensional space.

Tangent and Normal Vectors

Tangent Vectors and Their Properties

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  • T⃗(t)\vec{T}(t) represents the of a curve r⃗(t)\vec{r}(t) at a point
    • Obtained by taking the of the position vector with respect to the parameter tt: T⃗(t)=dr⃗dt\vec{T}(t) = \frac{d\vec{r}}{dt}
    • Always tangent to the curve at the point of interest
  • T^(t)\hat{T}(t) is the normalized tangent vector
    • Calculated by dividing the tangent vector by its : T^(t)=T⃗(t)∣T⃗(t)∣\hat{T}(t) = \frac{\vec{T}(t)}{|\vec{T}(t)|}
    • Has a magnitude of 1 and points in the same direction as the tangent vector
  • Properties of tangent vectors
    • Tangent vectors are perpendicular to the radius of curvature at any point on the curve
    • The direction of the tangent vector indicates the orientation of the curve at that point (increasing or decreasing)

Normal Vectors and Principal Normal Vectors

  • N⃗(t)\vec{N}(t) is perpendicular to the tangent vector at a given point on the curve
    • Lies in the plane containing the curve and the point of interest
    • Points towards the center of curvature of the curve at that point
  • N^(t)\hat{N}(t) is the normalized normal vector
    • Obtained by dividing the normal vector by its magnitude: N^(t)=N⃗(t)∣N⃗(t)∣\hat{N}(t) = \frac{\vec{N}(t)}{|\vec{N}(t)|}
    • Has a magnitude of 1 and points in the same direction as the normal vector
  • Calculating the principal normal vector
    • Can be found using the formula: N^(t)=dT^dt∣dT^dt∣\hat{N}(t) = \frac{\frac{d\hat{T}}{dt}}{|\frac{d\hat{T}}{dt}|}
    • Requires differentiating the unit tangent vector and normalizing the result
  • Relationship between tangent and normal vectors
    • Tangent and normal vectors are orthogonal to each other at any point on the curve
    • The plane containing the tangent and normal vectors at a point is called the

Binormal Vector and Frenet Frame

Binormal Vector and Its Properties

  • B⃗(t)\vec{B}(t) is perpendicular to both the tangent and normal vectors at a given point
    • Completes the orthonormal basis formed by the tangent, normal, and binormal vectors
    • Can be calculated using the cross product of the tangent and normal vectors: B⃗(t)=T⃗(t)×N⃗(t)\vec{B}(t) = \vec{T}(t) \times \vec{N}(t)
  • Properties of the binormal vector
    • The binormal vector is a unit vector, meaning it has a magnitude of 1
    • It is perpendicular to the osculating plane containing the tangent and normal vectors
    • The direction of the binormal vector is determined by the right-hand rule

Frenet Frame and Its Applications

  • Frenet frame, also known as the TNB frame, is an orthonormal basis formed by the tangent, normal, and binormal vectors
    • Consists of three mutually perpendicular unit vectors: T^(t)\hat{T}(t), N^(t)\hat{N}(t), and B⃗(t)\vec{B}(t)
    • Provides a local coordinate system for describing the geometry of a curve at a specific point
  • Applications of the Frenet frame
    • Analyzing the curvature and torsion of a curve
    • Describing the local geometry of a curve in terms of its tangent, normal, and binormal vectors
    • Studying the motion of particles along a curve, such as in physics and engineering problems
  • Osculating plane is the plane containing the tangent and normal vectors at a given point on the curve
    • Represents the best approximating plane to the curve at that point
    • The binormal vector is perpendicular to the osculating plane

Key Terms to Review (17)

Binormal Vector: A binormal vector is a vector that is orthogonal to both the tangent vector and the normal vector of a curve at a given point. It forms part of the Frenet-Serret formulas, which describe the geometric properties of curves in three-dimensional space. The binormal vector is crucial in understanding the curvature and torsion of a curve, which help define how a curve twists and turns in space.
Curvature: Curvature refers to the measure of how a curve deviates from being straight, or how a surface deviates from being flat. It is an essential concept in understanding the geometric properties of curves and surfaces, allowing us to analyze their shape and behavior. Curvature connects directly to tangent and normal vectors, which provide directional information about the curve at a given point, as well as curvature and torsion, which describe how a curve twists in space.
Derivative: The derivative is a fundamental concept in calculus that measures how a function changes as its input changes. In simpler terms, it provides the slope of the tangent line to a curve at any given point, reflecting the rate of change of the function. This concept is essential in understanding motion, growth, and various physical phenomena, as it connects to the behavior of functions in terms of their rates of increase or decrease.
Frenet Frame: The Frenet frame is a coordinate system associated with a curve in space that consists of a unit tangent vector, a unit normal vector, and a unit binormal vector at each point along the curve. This frame provides a way to analyze the geometric properties of the curve, such as its curvature and torsion, by describing how the curve bends and twists in three-dimensional space.
Graphical interpretation: Graphical interpretation refers to the ability to analyze and understand the information conveyed through visual representations, such as graphs and charts. It involves translating mathematical relationships and physical concepts into a visual context, allowing for easier comprehension of complex data. This skill is crucial for interpreting the behavior of functions, trajectories, and various physical phenomena in relation to tangent and normal vectors.
Instantaneous direction: Instantaneous direction refers to the specific orientation of a curve or path at a given point in time, indicating the direction in which an object is moving at that exact moment. This concept is vital when understanding how tangent vectors relate to curves, as the tangent vector at any point provides the instantaneous direction of motion along the curve, revealing the path's behavior at that precise location.
Magnitude: Magnitude refers to the size or length of a vector, providing a numerical measure of its strength or intensity without regard to its direction. This concept is crucial in physics and mathematics as it helps quantify physical quantities such as force, velocity, and displacement, allowing for clearer understanding and application of these vectors in various contexts. Magnitude is also essential in distinguishing between scalar and vector quantities, which is important for operations involving vectors like addition and multiplication.
Motion along a curve: Motion along a curve refers to the movement of an object along a defined path that is not straight, which can be described mathematically using curves in space. This concept is crucial for understanding how objects change their position and direction over time when they follow a non-linear trajectory. The analysis of motion along curves often involves key elements like velocity and acceleration, as well as the tangential and normal vectors that describe the object's behavior at any given point on the curve.
Normal Vector: A normal vector is a vector that is perpendicular to a surface or curve at a given point. It provides crucial information about the orientation of the surface and is used in various applications, including physics and engineering, to analyze forces and motion. Understanding normal vectors is essential when discussing properties like curvature and torsion, as they relate to how curves bend in space.
Normalized vector: A normalized vector is a vector that has been scaled to have a length or magnitude of one. This process is crucial in many mathematical and physical applications as it allows for easier manipulation and comparison of direction without the influence of magnitude. Normalization is often used when dealing with directional quantities, as it simplifies calculations involving angles and projections.
Orthogonality: Orthogonality refers to the concept where two vectors are perpendicular to each other, meaning their dot product equals zero. This fundamental idea extends beyond simple vector operations and plays a crucial role in various mathematical and physical contexts, including the behavior of functions, the nature of coordinate systems, and the analysis of differential equations.
Osculating Plane: The osculating plane at a point on a curve is the plane that best approximates the curve near that point. It is defined by the position vector of the point, the tangent vector, and the normal vector, effectively capturing the local geometric properties of the curve. This concept connects to how curves behave in three-dimensional space, providing insights into curvature and motion.
Perpendicularity: Perpendicularity refers to the geometric relationship where two lines or vectors intersect at a right angle, which is defined as an angle of 90 degrees. This concept is crucial in understanding how tangent and normal vectors relate to curves and surfaces, as the normal vector at a point on a curve is always perpendicular to the tangent vector at that same point, representing distinct but connected directions.
Principal Normal Vector: The principal normal vector is a vector that describes the direction of the curve's curvature at a specific point, providing insight into how the curve is bending. It is perpendicular to the tangent vector and points towards the center of curvature, making it essential for understanding the geometric properties of curves in space.
Tangent vector: A tangent vector is a vector that represents the direction and rate of change of a curve at a specific point. It provides information about how the curve moves through space and is essential in understanding motion, geometry, and analysis in various contexts. Tangent vectors connect to normal vectors, curvature, and differentiation concepts, highlighting their significance in the study of curves and surfaces.
Torsion: Torsion is a measure of how a curve twists in three-dimensional space, quantifying the deviation of the curve from being planar. It describes how much a curve deviates from lying in a two-dimensional plane as it moves through space. Torsion, alongside curvature, helps to define the geometric properties of curves and is essential in understanding the behavior of objects in physical systems.
Unit Tangent Vector: A unit tangent vector is a vector that points in the direction of a curve at a given point and has a magnitude of one. It is derived from the tangent vector, which indicates the direction of the curve's motion, and is obtained by normalizing that vector. This concept is crucial for analyzing curves and their geometric properties, especially in relation to motion and changes in position.
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