Problem 24 - Thomas Calculus

Question

What can be said about the torsion of a smooth plane curve \(\mathbf{r}(t)=f(t) \mathbf{i}+g(t) \mathbf{j} ?\) Give reasons for your answer.

Step-by-Step Solution

Answer

The torsion of the curve is zero because it's a plane curve, so it doesn't twist out of the plane.

Step 1: Understand the Problem

We have a smooth plane curve given by \(\mathbf{r}(t)=f(t) \mathbf{i} + g(t) \mathbf{j}\). Our task is to determine the torsion of this curve.

Step 2: Recalling Torsion Formula

Torsion measures how a curve twists out of its osculating plane. The torsion \(\tau\) of a space curve is given by \(\tau = \frac{(\mathbf{T}' \times \mathbf{B}) \cdot \mathbf{N}}{||\mathbf{r}'||^2}\), where \(\mathbf{T}\) is the unit tangent vector, \(\mathbf{N}\) is the principal normal vector, and \(\mathbf{B}\) is the binormal vector.

Step 3: Consideration for Plane Curve

For a plane curve like \(\mathbf{r}(t)=f(t) \mathbf{i}+g(t) \mathbf{j}\) (lying entirely in the xy-plane), the binormal vector \(\mathbf{B}\), which is normally defined as \(\mathbf{T} \times \mathbf{N}\), will always point in the direction of the z-axis because the other vectors lie in the xy-plane.

Step 4: Calculation of Torsion for a Plane Curve

Since the curve is confined to the plane, any 'twisting' behavior cannot exist outside this plane; therefore, there is no torsion. For a plane curve, the torsion is identically zero: \(\tau = 0\).

Step 5: Conclusion

Thus, the torsion of any smooth plane curve is always zero because it cannot have any out-of-plane twisting.

Key Concepts

Smooth Plane Curve

A smooth plane curve is a curve that lies entirely within a single plane, usually described by a parametric equation such as \(\mathbf{r}(t) = f(t) \mathbf{i} + g(t) \mathbf{j}\). Smoothness implies that the functions \(f(t)\) and \(g(t)\) are continuously differentiable, ensuring that there are no sharp corners or cusps on the curve. These curves are important because they simplify many aspects of the geometry and calculus involved, as modifications are constrained to a two-dimensional plane. This characteristic makes smooth plane curves ideal for studies involving two-dimensional motion or planar shapes.

Torsion Formula

Torsion refers to the measure of twist of a curve in three-dimensional space, indicating how a curve departs from lying in its osculating plane over a segment. The formula for torsion \(\tau\) is given as:\[ \tau = \frac{(\mathbf{T}' \times \mathbf{B}) \cdot \mathbf{N}}{||\mathbf{r}'||^2} \]where:

• \(\mathbf{T} \): unit tangent vector
• \(\mathbf{N} \): principal normal vector
• \(\mathbf{B} \): binormal vector

This formula is derived from the Frenet-Serret frame, which is a formalism that provides a set of orthonormal vectors to describe a particle's motion along its default trajectory.Since torsion captures a curve's propensity to twist out of a plane, a plane curve inherently has zero torsion because it does not twist out of its initial plane.

Binormal Vector

The binormal vector \(\mathbf{B}\) is a crucial component in understanding the spatial orientation of a curve. Typically defined in three-dimensional space, it is the cross product of the unit tangent vector \(\mathbf{T}\) and the principal normal vector \(\mathbf{N}\):\[ \mathbf{B} = \mathbf{T} \times \mathbf{N} \]For a smooth plane curve, both the tangent and normal vectors lie in the plane, which means that the binormal vector will always be perpendicular to the plane. In the context of the xy-plane, \(\mathbf{B}\) points along the z-axis. This geometric alignment is why plane curves do not experience torsion; there is no component of \(\mathbf{B}\) that can lead the curve out of the plane.

Unit Tangent Vector

The unit tangent vector \(\mathbf{T}\) is a vector that is tangent to a curve at a given point, having unit length. It is defined by the derivative of the position vector \(\mathbf{r}(t)\) with respect to the parameter \(t\):\[ \mathbf{T} = \frac{\mathbf{r}'(t)}{||\mathbf{r}'(t)||} \]This vector is significant because it indicates the direction of the curve's path at any point and remains consistent in magnitude. For plane curves that reside in the xy-plane, \(\mathbf{T}\) has components only along the x and y directions, underscoring the curve's two-dimensional nature. The unit tangent vector is an essential factor in the torsion formula, as it influences the tangential directionality without affecting the twisting out of a plane for plane curves.

Principal Normal Vector

The principal normal vector \(\mathbf{N}\) provides the direction in which a curve is turning or bending at any point, and it is always perpendicular to the unit tangent vector \(\mathbf{T}\):\[ \mathbf{N} = \frac{\mathbf{T}'(t)}{||\mathbf{T}'(t)||} \]This vector gives insight into the curvature of the curve, presenting the immediate path change direction, contrary to \(\mathbf{T}\). For smooth plane curves, \(\mathbf{N}\) lies in the plane, typically showing curvature within the bounds of the x and y axes. As a result, plane curves only exhibit curvature within their own plane, which alongside the constant position of \(\mathbf{B}\), explains why plane curves have zero torsion.