Problem 64 Show that for a plane curve \(\m... [FREE SOLUTION]

Chapter 11: Problem 64

Show that for a plane curve $\mathbf{N}$ points to the concave side of the curve. Hint: One method is to show that $$ \mathbf{N}=(-\sin \phi \mathbf{i}+\cos \phi \mathbf{j}) \frac{d \phi / d s}{|d \phi / d s|} $$ Then consider the cases $d \phi / d s>0$ (curve bends to the left) and $d \phi / d s<0$ (curve bends to the right).

Short Answer

Expert verified

$ \mathbf{N} $ points to the concave side, determined by the sign of $ \frac{d \phi}{d s} $.

Step by step solution

Understand the Problem

We have to show that the normal vector $ \mathbf{N} $ of a plane curve points towards the concave side of the curve, which is associated with the direction of curve bending.

Analyze the Given Normal Vector

The normal vector is given as $ \mathbf{N}=(-\sin \phi \mathbf{i}+\cos \phi \mathbf{j}) \frac{d \phi / d s}{|d \phi / d s|} $. This expression includes a unit vector multiplied by a sign function that determines the direction along the curve.

Consider the Direction of $ \frac{d \phi}{d s} $

The value of $ \frac{d \phi}{d s} $ determines the direction of bending: if $ \frac{d \phi}{d s} > 0 $, the curve bends to the left; if $ \frac{d \phi}{d s} < 0 $, it bends to the right.

Evaluate $ \mathbf{N} $ for $ \frac{d \phi}{d s} > 0 $

When $ \frac{d \phi}{d s} > 0 $, the curve is bending to the left, and our expression $ \mathbf{N} $ remains as is, pointing towards the concave side, which effectively is to the left in standard orientation.

Evaluate $ \mathbf{N} $ for $ \frac{d \phi}{d s} < 0 $

When $ \frac{d \phi}{d s} < 0 $, the negative sign in the expression flips the unit vector direction of $ \mathbf{N} $, ensuring that it points towards the concave side, which is now to the right.

Conclude the Argument

In both cases, $ \mathbf{N} $ points towards the concave side of the curve, corroborating how the sign of $ \frac{d \phi}{d s} $ aligns with the concavity.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Normal Vector

In the study of plane curves, the concept of a **normal vector** is pivotal. A normal vector is perpendicular to the tangent vector, which is tangent to the curve at a given point. The formula for the normal vector, \[ \mathbf{N}=(-\sin \phi \mathbf{i}+\cos \phi \mathbf{j}) \frac{d \phi / d s}{|d \phi / d s|}, \]provides insight into its direction and orientation.

The terms $-\sin \phi \mathbf{i}$ and $+\cos \phi \mathbf{j}$ represent the components of this vector. They are based on trigonometric transformations deriving from the angle $\phi$ formed by the tangent line at that point.
The fraction $\frac{d \phi / d s}{|d \phi / d s|}$ indicates whether the vector maintains its direction or reverses it, depending on the curvature of the line.

A key property of the normal vector is that it always points inward, towards the concave side of the curve. Understanding this concept is essential when analyzing the behavior of curves.

Concave Side

The concave side of a curve is the side that appears 'inwardly curved' when viewed along the direction of the tangent vector. When considering a plane curve, identifying the concave side is crucial for various applications in geometry and physics. Understanding the normal vector's orientation plays into this as it always points toward the concave side, giving a clear indication of which side is concave. The significance of the concave side is reflected in many real-world applications:

In physics, knowing the concave side can help in predicting fields or forces acting on the curve.
In geometry, it affects calculations of areas and volumes when the curve forms part of a boundary.

Identifying the concave side is, therefore, not only about theoretical understanding but also about practical applications.

Curve Bending

Curve bending describes how a curve shifts or changes its direction over a particular segment. In a mathematical sense, curve bending is directly related to the rate of change in the angle $\phi$, represented as $\frac{d \phi}{d s}$.- If $\frac{d \phi}{d s} > 0$, the curve bends to the left.- If $\frac{d \phi}{d s} < 0$, it bends to the right.These conditions dictate the direction of the normal vector and thus, the orientation of the concave side:

When bending to the left, the vector remains unchanged, pointing correctly towards the concave side.
When bending to the right, the vector鈥檚 direction flips, yet still correctly points towards the concave side because of the embedded sign adjustment in its formula.

Understanding curve bending lays a foundation for comprehending how curves behave and why the normal vector behaves as it does.

Direction of Curvature

When examining curves, the **direction of curvature** is an essential characteristic. It refers to whether a curve is bending leftwards or rightwards as one progresses along it.The direction of curvature aligns with the sign of $\frac{d \phi}{d s}$:

Positive values indicate a leftward curvature.
Negative values denote a rightward curvature.

Knowing the direction of curvature is integral to determining many properties and applications of curves:- It helps to calculate the normal vector, which informs about the concave side.- It aids in computer graphics to simulate realistic curve behaviors.Ultimately, the direction of curvature is fundamental in understanding how curves interact with their surroundings.

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Short Answer

Step by step solution

Understand the Problem

Analyze the Given Normal Vector

Consider the Direction of \( \frac{d \phi}{d s} \)

Evaluate \( \mathbf{N} \) for \( \frac{d \phi}{d s} > 0 \)

Evaluate \( \mathbf{N} \) for \( \frac{d \phi}{d s} < 0 \)

Conclude the Argument

Key Concepts

Normal Vector

Concave Side

Curve Bending

Direction of Curvature

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