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Chapter 1: Problem 17

In general, a curve \(\alpha\) is called a helix if the tangent lines of \(\alpha\) make a constant angle with a fixed direction. Assume that \(\tau(s) \neq 0, s \in I\), and prove that: * a. \(\alpha\) is a helix if and only if \(k / \tau=\) const. *b. \(\alpha\) is a helix if and only if the lines containing \(n(s)\) and passing through \(\alpha(s)\) are parallel to a fixed plane. *c. \(\alpha\) is a helix if and only if the lines containing \(b(s)\) and passing through \(\alpha(s)\) make a constant angle with a fixed direction. d. The curve $$ \alpha(s)=\left(\frac{a}{c} \int \sin \theta(s) d s, \frac{a}{c} \int \cos \theta(s) d s, \frac{b}{c} s\right), $$ where \(c^{2}=a^{2}+b^{2}\), is a helix, and that \(k / \tau=a / b\).

Short Answer

Expert verified
Recognize helix definition. Prove \(\frac{\text{k}}\tau\), normalize consistency plane orientation/b-in a fixed line.

Step by step solution

01

Understand the Definition

A helix is defined as a curve whose tangent lines make a constant angle with a fixed direction.
02

Given Conditions

We are given that the torsion \(\tau(s) eq 0\) and kernel \(k\) relationships for \(s \in I\).
03

Prove Part (a)

Helix condition is \(k / \tau = \text{{const.}}\): We have curvature \(k\) and torsion \(\tau\). If \(k / \tau \) is constant, there is a constant angle between the tangent vector \(T(s)\) and a fixed direction (say the z-axis).
04

Prove Part (b)

For lines containing \(n(s)\) passing through \(\text{a}(s)\) to be parallel to a fixed plane: Show that the normal \(n(s)\) lies in a fixed plane, indicating consistency.
05

Prove Part (c)

Verify lines containing \(b(s)\) passing through \(\text{a}(s)\) make constant angle with fixed direction: Show constant angle between bi-normal vectors and a fixed direction.
06

Verify Curve

The curve \(\textbf{\textbf{\text{a}}\textbf{\text{(}}\textbf{\text{s}}\textbf{\text{)}}} \text{\right)} = \frac{a}{c} \times sd\textbf{\text{蟺} \text{胃}\textbf{\text{(}}\textbf{\text{s}}\textbf{\text{)}}} ds, \text{\frac{{a}/{c}}\text{cos}\text{\theta}\text{(s)} ds, b/cs\text{\right)\), where \(\textbf{a}\textsuperscript{2} + b\textsuperscript{2}= c\textsuperscript{2} \Is helix, also \frac{\text{k}}{\tau}\) is \fraction{\text{a}}{\text{b}}\textbf\text.

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Key Concepts

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

curvature and torsion
In differential geometry, curvature and torsion are essential properties that describe the way a curve bends and twists in space. They help us understand the shape and behavior of a curve.

**Curvature (k):** Curvature measures how fast a curve deviates from being a straight line. It's a scalar value represented by the symbol **k**. High curvature means the curve bends sharply, while low curvature implies a gentle bend. Mathematically, if a curve is parameterized by its arc length **s**, the curvature is defined as:
\( k(s) = \frac{!!{|| T'(s) ||}{|| r'(s) ||} \)
where **T(s)** is the tangent vector and **r(s)** is the position vector.

**Torsion (蟿):** Torsion measures how a curve twists out of the plane of curvature. It's represented by the symbol **蟿**. Torsion is particularly important for 3D curves and is defined as: \( \tau(s) = \frac{{-b'(s) \bullet (r''(s) \times r'(s))}}{\beta || r''(s) ||} \)
where **b(s)** is the binormal vector and **r'(s)** and **r''(s)** are the first and second derivatives of the position vector.

For a curve to be a helix, the ratio \(\frac{k}{\tau}} \) must be constant. This ratio signifies a constant relationship between bending and twisting, ensuring the curve maintains a constant angle with a fixed direction.
tangent vector
The tangent vector is crucial in understanding curves in differential geometry. It describes the direction of the curve at any given point.

**Definition:** The tangent vector, denoted as **T(s)**, represents the direction in which the curve is heading at point **s**. It is a unit vector that points along the curve's direction of travel. If the curve is parameterized by arc length **s**, the tangent vector is given by: \( T(s) = \frac{{r'(s)}}{|| r'(s) ||} \)
where **r'(s)** is the derivative of the position vector with respect to **s**.

**Key Properties:**
  • The tangent vector is always perpendicular to the normal vector of the curve.
  • It helps in defining the curvature of the curve as it shows how fast the direction is changing.
  • For a helix, the tangent vector makes a constant angle with a fixed direction.

In the context of a helix, the fixed direction is usually aligned with an axis, such as the z-axis. This alignment ensures the curve wraps around the axis uniformly, maintaining a consistent angle between the tangent vector and the predefined direction.
fixed direction
In the context of a helix, the concept of a fixed direction plays a pivotal role. It refers to a constant line of direction that the curve maintains a specific relationship with.

**Definition:** A fixed direction is an unchanging vector in space, often aligned with one of the coordinate axes (like the z-axis).

**Helix Condition:** For a curve to be considered a helix, its tangent vector must make a constant angle with this fixed direction. This condition assures that the curve doesn't just twist randomly in space but follows a predictable and uniform spiraling pattern.

**Fixed Plane:** Another perspective is considering a fixed plane to which the normal or binormal vectors should be parallel. For instance:
  • **Normal Vector:** If the lines containing the normal vector **n(s)** are parallel to a fixed plane, the curve is a helix.
  • **Binormal Vector:** If the lines containing the binormal vector **b(s)** make a constant angle with a fixed direction, the curve is a helix.

The idea of a fixed direction or plane helps simplify the study of helical curves by providing a consistent reference. This consistency is key in proving various properties and conditions related to helices, such as the aforementioned ratios of curvature to torsion.

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Most popular questions from this chapter

If a closed plane curve \(C\) is contained inside a disk of radius \(r\), prove that there exists a point \(p \in C\) such that the curvature \(k\) of \(C\) at \(p\) satisfies \(|k| \geq 1 / r .\)

Consider the map $$ \alpha(t)= \begin{cases}\left(t, 0, e^{-1 / t^{2}}\right), & t>0 \\ \left(t, e^{-1 / t^{2}}, 0\right), & t<0 \\ (0,0,0), & t=0\end{cases} $$ a. Prove that \(\alpha\) is a differentiable curve. b. Prove that \(\alpha\) is regular for all \(t\) and that the curvature \(k(t) \neq 0\), for \(t \neq 0, t \neq \pm \sqrt{2 / 3}\), and \(k(0)=0\). c. Show that the limit of the osculating planes as \(t \rightarrow 0, t>0\), is the plane \(y=0\) but that the limit of the osculating planes as \(t \rightarrow 0, t<0\), is the plane \(z=0\) (this implies that the normal vector is discontinuous at \(t=0\) and shows why we excluded points where \(k=0\) ). d. Show that \(\tau\) can be defined so that \(\tau \equiv 0\), even though \(\alpha\) is not a plane curve.

Show that an equation of a plane passing through three noncolinear points \(p_{1}=\left(x_{1}, y_{1}, z_{1}\right), p_{2}=\left(x_{2}, y_{2}, z_{2}\right), p_{3}=\left(x_{3}, y_{3}, z_{3}\right)\) is given by $$ \left(p-p_{1}\right) \wedge\left(p-p_{2}\right) \cdot\left(p-p_{3}\right)=0, $$ where \(p=(x, y, z)\) is an arbitrary point of the plane and \(p-p_{1}\), for instance, means the vector \(\left(x-x_{1}, y-y_{1}, z-z_{1}\right)\).

Let \(C\) be a plane curve and let \(T\) be the tangent line at a point \(p \in C\). Draw a line \(L\) parallel to the normal line at \(p\) and at a distance \(d\) of \(p\) (Fig. 1-36). Let \(h\) be the length of the segment determined on \(L\) by \(C\) and \(T\) (thus, \(h\) is the "height" of \(C\) relative to \(T\) ). Prove that $$ |k(p)|=\lim _{d \rightarrow 0} \frac{2 h}{d^{2}} $$ where \(k(p)\) is the curvature of \(C\) at \(p\).

Let \(\alpha: I \rightarrow R^{3}\) be a parametrized regular curve (not necessarily by arc length) with \(k(t) \neq 0, \tau(t) \neq 0, t \in I\). The curve \(\alpha\) is called a Bertrand curve if there exists a curve \(\bar{\alpha}: I \rightarrow R^{3}\) such that the normal lines of \(\alpha\) and \(\bar{\alpha}\) at \(t \in I\) are equal. In this case, \(\bar{\alpha}\) is called a Bertrand mate of \(\alpha\), and we can write $$ \bar{\alpha}(t)=\alpha(t)+r n(t) $$ Prove that a. \(r\) is constant. b. \(\alpha\) is a Bertrand curve if and only if there exists a linear relation $$ A k(t)+B \tau(t)=1, \quad t \in I, $$ where \(A, B\) are nonzero constants and \(k\) and \(\tau\) are the curvature and torsion of \(\alpha\), respectively. c. If \(\alpha\) has more than one Bertrand mate, it has infinitely many Bertrand mates. This case occurs if and only if \(\alpha\) is a circular helix.
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