Question

Alternative derivation of curvature Derive the computational formula for curvature using the following steps. a. Use the tangential and normal components of the acceleration to show that \(\mathbf{v} \times \mathbf{a}=\kappa|\mathbf{v}|^{3} \mathbf{B} .\) (Note that \(\mathbf{T} \times \mathbf{T}=\mathbf{0} .\) ) b. Solve the equation in part (a) for \(\kappa\) and conclude that \(\kappa=\frac{|\mathbf{v} \times \mathbf{a}|}{|\mathbf{v}|^{3}},\) as shown in the text.

Short answer

Answer: The computational formula for curvature is given by \(\kappa = \frac{|\mathbf{v} \times \mathbf{a}|}{|\mathbf{v}|^3}\), where \(\kappa\) is the curvature, \(\mathbf{v}\) is the velocity vector, and \(\mathbf{a}\) is the acceleration vector.

Step-by-step solution

Express the velocity vector and acceleration vector in terms of tangent and normal components

The velocity vector \(\mathbf{v}\) can be expressed in terms of its tangent and normal components as: \(\mathbf{v} = |\mathbf{v}| \mathbf{T}.\) The acceleration vector \(\mathbf{a}\) can also be expressed in terms of its tangent and normal components: \(\mathbf{a} = \frac{d^2 \mathbf{r}}{dt^2} = \frac{d |\mathbf{v}|}{dt} \mathbf{T} + |\mathbf{v}| \frac{d \mathbf{T}}{dt},\) where \(\mathbf{r}\) is the position vector, and \(t\) is the parameter.

Calculate the cross product \(\mathbf{v} \times \mathbf{a}\)

Now, perform the cross product between the velocity and acceleration vectors: \(\mathbf{v} \times \mathbf{a} = (|\mathbf{v}| \mathbf{T}) \times \left(\frac{d |\mathbf{v}|}{dt} \mathbf{T} + |\mathbf{v}| \frac{d \mathbf{T}}{dt}\right).\) Since \(\mathbf{T} \times \mathbf{T} = \mathbf{0}\), the first term in the cross product becomes zero. So, we simplify the cross product to: \(\mathbf{v} \times \mathbf{a} = (|\mathbf{v}| \mathbf{T}) \times \left(|\mathbf{v}| \frac{d \mathbf{T}}{dt}\right).\)

Establish the relation between \(|\mathbf{v}|^2 \mathbf{T} \times (\frac{d \mathbf{T}}{dt})\) and \(\kappa |\mathbf{v}|^3 \mathbf{B}\)

From the Frenet-Serret formulas, we know that \(\frac{d \mathbf{T}}{dt} = \kappa |\mathbf{v}| \mathbf{N}\), where \(\kappa\) is the curvature and \(\mathbf{N}\) is the normal vector. We also know that \(\mathbf{B} = \mathbf{T} \times \mathbf{N}\), which is the binormal vector. Therefore, \((|\mathbf{v}| \mathbf{T}) \times \left(|\mathbf{v}| \frac{d \mathbf{T}}{dt}\right) = |\mathbf{v}|^2 \mathbf{T} \times (\kappa |\mathbf{v}| \mathbf{N}) = \kappa |\mathbf{v}|^3 (\mathbf{T} \times \mathbf{N}) = \kappa |\mathbf{v}|^3 \mathbf{B}.\) Thus, we have shown that \(\mathbf{v} \times \mathbf{a} = \kappa |\mathbf{v}|^3 \mathbf{B}\).

Solve for \(\kappa\) in terms of \(|\mathbf{v}|\) and \(|\mathbf{a}|\)

Divide both sides of the equation \(\mathbf{v} \times \mathbf{a} = \kappa |\mathbf{v}|^3 \mathbf{B}\) by \(|\mathbf{v}|^3 \mathbf{B}\): \(\kappa = \frac{\mathbf{v} \times \mathbf{a}}{|\mathbf{v}|^3 \mathbf{B}}.\) Now, take the magnitude of both sides: \(\kappa = \frac{|\mathbf{v} \times \mathbf{a}|}{|\mathbf{v}|^3 |\mathbf{B}|}.\) Since \(|\mathbf{B}| = 1\), we have \(\kappa = \frac{|\mathbf{v} \times \mathbf{a}|}{|\mathbf{v}|^3}.\) We have derived the computational formula for curvature as given in the problem statement.

Key concepts

Frenet-Serret formulas

The Frenet-Serret formulas are fundamental in the study of curves in three-dimensional space. They describe how the tangent, normal, and binormal vectors of a space curve change along the curve. These formulas are pivotal in understanding curvature and torsion, which are measures of how a curve bends and twists.
Key vectors in these formulas include:

• **Tangent vector (\(\mathbf{T}\))**: A unit vector pointing in the direction of the curve's motion.
• **Normal vector (\(\mathbf{N}\))**: A unit vector pointing towards the center of curvature of the curve.
• **Binormal vector (\(\mathbf{B}\))**: A unit vector perpendicular to both the tangent and normal vectors.

The equations are:

• \( \frac{d \mathbf{T}}{dt} = \kappa \mathbf{N} \), where \( \kappa \) is the curvature.
• \( \frac{d \mathbf{N}}{dt} = -\kappa \mathbf{T} + \tau \mathbf{B} \), where \( \tau \) is the torsion.
• \( \frac{d \mathbf{B}}{dt} = -\tau \mathbf{N} \).

These vectors form a coordinate system along the curve, helping to analyze the motion and conformation of curves.

velocity vector

The velocity vector, denoted as \( \mathbf{v} \), represents the rate of change of the curve's position. If the position vector is \( \mathbf{r}(t) \), then the velocity vector is given by \( \mathbf{v} = \frac{d \mathbf{r}}{dt} \). In the context of space curves, it points in the direction of the tangent vector \( \mathbf{T} \), and can be expressed as:
\[ \mathbf{v} = |\mathbf{v}| \mathbf{T} \]Here:

• \( |\mathbf{v}| \) is the speed or magnitude of the velocity vector.
• \( \mathbf{T} \) is the unit tangent vector, indicating the direction of movement.

This expression highlights how velocity combines both direction and speed.
In analyzing curves, understanding the velocity vector is essential,
as it dictates the initial behavior and path orientation of the curve.

acceleration vector

The acceleration vector, symbolized by \( \mathbf{a} \), is the derivative of the velocity vector with respect to time. It describes how the velocity changes as the curve evolves. The acceleration vector can be split into tangential and normal components represented as:\[ \mathbf{a} = \frac{d |\mathbf{v}|}{dt} \mathbf{T} + |\mathbf{v}| \frac{d \mathbf{T}}{dt} \]In this equation:

• \( \frac{d |\mathbf{v}|}{dt} \mathbf{T} \) is the tangential component, indicating how the magnitude of velocity changes.
• \( |\mathbf{v}| \frac{d \mathbf{T}}{dt} \) is the normal component, associated with the change in direction, contributing to the curve's curvature.

The normal component links directly to the Frenet-Serret formula, where \( \frac{d \mathbf{T}}{dt} = \kappa \mathbf{N} \).
This connection shows that curvature affects acceleration's directional change. Understanding this helps in unraveling how curves accelerate in complex paths.

binormal vector

The binormal vector, denoted as \( \mathbf{B} \), is another crucial element in curve analysis. It is defined in the context of the Frenet-Serret formulas and is expressed as:\[ \mathbf{B} = \mathbf{T} \times \mathbf{N} \]Where:

• \( \mathbf{T} \) is the tangent vector.
• \( \mathbf{N} \) is the normal vector.

The binormal vector is orthogonal to both \( \mathbf{T} \) and \( \mathbf{N} \), forming a right-handed coordinate system together. Its primary role is to provide the third dimension in the curve's orientation, perpendicular to the plane formed by \( \mathbf{T} \) and \( \mathbf{N} \).
This makes it essential for analyzing 3D curves' geometric properties, such as twists and turns.
In the derivation of the curvature formula, the binormal vector plays a role in expressing the product of velocity and acceleration vectors, indicating its importance in understanding motion dynamics.