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Chapter 1: Problem 1

Given the parametrized curve (helix) $$ \alpha(s)=\left(a \cos \frac{s}{c}, a \sin \frac{s}{c}, b \frac{s}{c}\right), \quad s \in R, $$ where \(c^{2}=a^{2}+b^{2}\), a. Show that the parameter \(s\) is the arc length. b. Determine the curvature and the torsion of \(\alpha\). c. Determine the osculating plane of \(\alpha\). d. Show that the lines containing \(n(s)\) and passing through \(\alpha(s)\) meet the \(z\) axis under a constant angle equal to \(\pi / 2\). e. Show that the tangent lines to \(\alpha\) make a constant angle with the \(z\) axis.

Short Answer

Expert verified
s is the arc length since \( \alpha'(s) \) has unit length. Curvature is \( \frac{a}{(a^{2} + b^{2})} \), torsion is \( \frac{b}{a^{2} + b^{2}} \). Osculating plane spanned by tangent and normal vectors. Normal lines meet the z-axis at a constant angle \( \pi /2 \). Tangent lines make a constant angle with the z-axis.

Step by step solution

01

Reevaluate Given Information

Given a parametrized curve (helix): \[ \alpha(s) = \left( a \cos \frac{s}{c}, \ a \sin \frac{s}{c}, \ b \frac{s}{c} \right), \quad s \in \mathbb{R}, \] where \( c^{2} = a^{2} + b^{2} \).
02

Part (a): Prove that s is the Arc Length

1. Compute the derivative of \( \alpha(s) \) with respect to \( s \): \[ \alpha'(s) = \left( -\frac{a}{c} \sin \frac{s}{c}, \ \frac{a}{c} \cos \frac{s}{c}, \ \frac{b}{c} \right). \] 2. Find the magnitude of \( \alpha'(s) \): \[ \| \alpha'(s) \| = \sqrt{\left( -\frac{a}{c} \sin \frac{s}{c} \right)^{2} + \left( \frac{a}{c} \cos \frac{s}{c} \right)^{2} + \left( \frac{b}{c} \right)^{2}}. \] 3. Simplify using \( c^{2} = a^{2} + b^{2} \): \[ \| \alpha'(s) \| = \sqrt{\frac{a^{2}}{c^{2}} + \frac{b^{2}}{c^{2}}} = \sqrt{\frac{a^{2} + b^{2}}{c^{2}}} = \sqrt{\frac{c^{2}}{c^{2}}} = 1. \]It follows that \( s \) is the arc length parameter.
03

Part (b): Determine Curvature and Torsion

1. Compute the second derivative \( \alpha''(s) \): \[ \alpha''(s) = \left( -\frac{a}{c^{2}} \cos \frac{s}{c}, \ -\frac{a}{c^{2}} \sin \frac{s}{c}, \ 0 \right). \] 2. Find the normal vector \( \mathbf{n(s)} \): \[ \mathbf{n(s)} = \frac{\alpha''(s)}{\|\alpha''(s)\|} = \left( -\cos \frac{s}{c}, \ -\sin \frac{s}{c}, \ 0 \right). \] 3. Compute the curvature: \[ \kappa = \frac{\| \alpha''(s) \|}{| \alpha'(s) |} = \frac{\sqrt{\left( \frac{-a}{c^{2}} \cos \frac{s}{c} \right)^{2} + \left( \frac{-a}{c^{2}} \sin \frac{s}{c} \right)^{2}}}{1} = \frac{a}{c^{2}} = \frac{a}{(a^{2} + b^{2})}. \] 4. Compute the derivative of the normal vector: \[ n'(s) = \left( \frac{a}{c^{3}} \sin \frac{s}{c}, \ -\frac{a}{c^{3}} \cos \frac{s}{c}, \ 0 \right). \] 5. Calculate the binormal vector: \[ b(s) = \frac{\alpha'(s) \times \alpha''(s)}{\| \alpha'(s) \times \alpha''(s) \|} = \left( \frac{b}{\sqrt{a^{2} + b^{2}}} \sin \frac{s}{c}, \ -\frac{b}{\sqrt{a^{2} + b^{2}}} \cos \frac{s}{c}, \ \frac{a}{\sqrt{a^{2} + b^{2}}} \right). \] 6. Compute the derivative of \( b(s) \): \[ b'(s) = \frac{a}{c^{2}} \left( \cos \frac{s}{c}, \ \sin \frac{s}{c}, \ 0 \right). \] 7. Compute torsion: \[ \tau = -\left( \frac{(s) \cdot b'(s)}{\| \alpha'(s) \|} \right) = \frac{b}{a^{2} + b^{2}}. \]
04

Part (c): Determine the Osculating Plane

1. The osculating plane is defined by the tangent vector and the normal vector. The normal vector found earlier was \[ \mathbf{n(s)} = \left( -\cos \frac{s}{c}, \ -\sin \frac{s}{c}, \ 0 \right). \]
05

Part (d): Show Lines Through \( \alpha(s) \) and Normal Meet \( z \)-axis at \( \pi /2 \)

1. A line passing through \( \alpha(s) \) with direction \( \mathbf{n(s)} \) can be parameterized as: \[ (x, y, z) = \left( a \cos \frac{s}{c} - t \cos \frac{s}{c}, \ a \sin \frac{s}{c} - t \sin \frac{s}{c}, \ \frac{b s}{c} \right). \] 2. The line meets the \( z \)-axis when \( x = 0 \) and \( y = 0 \), solving we find \[ t = a. \] 3. Substitute this into \( z \): \[ z = \frac{bs}{c}, \] hence meeting the \( z \) axis with a constant angle.
06

Part (e): Show Tangent Lines Make a Constant Angle with the \( z \) Axis

1. The tangent vector is: \[ \alpha'(s) = \left( -\frac{a}{c} \sin \frac{s}{c}, \ \frac{a}{c} \cos \frac{s}{c}, \ \frac{b}{c} \right). \] 2. Compute the angle \( \theta \) made by the tangent vector with the \( z \) axis using the dot product: \[ \cos \theta = \frac{\mathbf{\alpha'(s)} \cdot (0, 0, 1)}{\| \alpha'(s) \|} = \frac{\frac{b}{c}}{1} = \frac{b}{c}. \] 3. Since \( c = \sqrt{a^{2} + b^{2}} \), \[ \cos \theta = \frac{b}{\sqrt{a^{2} + b^{2}}}. \] Thus, the angle \( \theta \) is constant.

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Key Concepts

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

Arc Length
The arc length of a curve is the distance along the curve from one point to another. To show that a parameter is the arc length, we need to prove that the magnitude of the derivative with respect to the parameter is always equal to one.
For the parametrized curve given:\[ \alpha(s) = \left( a \cos \frac{s}{c}, a \sin \frac{s}{c}, b \frac{s}{c} \right), \quad s \in \mathbb{R},\]we find the derivative:\[ \alpha'(s) = \left( -\frac{a}{c} \sin \frac{s}{c}, \frac{a}{c} \cos \frac{s}{c}, \frac{b}{c} \right).\]To show that \( s \) is the arc length, we compute the magnitude of \( \alpha'(s) \):\[ \| \alpha'(s) \| =\sqrt{\left( -\frac{a}{c} \sin \frac{s}{c} \right)^{2} + \left( \frac{a}{c} \cos \frac{s}{c} \right)^{2} + \left( \frac{b}{c} \right)^{2}} = 1.\]This proves \( s \) is indeed the arc length.
Curvature
Curvature measures how sharply a curve bends at a given point. For our curve, we start by finding the second derivative:\[ \alpha''(s) = \left( -\frac{a}{c^{2}} \cos \frac{s}{c}, -\frac{a}{c^{2}} \sin \frac{s}{c}, 0 \right).\]The curvature \( \kappa \) is computed using:\[ \kappa = \frac{\| \alpha''(s) \|}{\| \alpha'(s) \|} = \frac{a}{c^{2}} = \frac{a}{a^{2} + b^{2}}.\]This indicates how the curve bends around a point.
Torsion
Torsion measures the rate of change of the curve's osculating plane. Firstly, we need the normal vector \( \mathbf{n(s)} \) and its derivative:\[ \mathbf{n(s)} = \left( -\cos \frac{s}{c}, -\sin \frac{s}{c}, 0 \right), \quad n'(s) = \left( \frac{a}{c^{3}} \sin \frac{s}{c}, -\frac{a}{c^{3}} \cos \frac{s}{c}, 0 \right).\]The binormal vector \( b(s) \) is found as:\[ b(s) = \left( \frac{b}{\sqrt{a^{2} + b^{2}}} \sin \frac{s}{c}, -\frac{b}{\sqrt{a^{2} + b^{2}}} \cos \frac{s}{c}, \frac{a}{\sqrt{a^{2} + b^{2}}} \right).\]Finally, the torsion \( \tau \) is:\[ \tau = \frac{b}{a^{2} + b^{2}}.\]
Osculating Plane
The osculating plane is defined by the tangent and normal vectors at a point on the curve. It gives the plane that best approximates the curve at that point.
For the given curve, the osculating plane can be determined by:\[ \text{Tangent vector:} \quad \alpha'(s), \quad \text{Normal vector:} \quad \mathbf{n(s)} = \left( -\cos \frac{s}{c}, -\sin \frac{s}{c}, 0 \right).\]These vectors will form the basis for the osculating plane.
Helical Curve
A helical curve, or helix, is a three-dimensional curve that spirals upwards or downwards, similar to a spring. This specific helix demonstrates how the curves in space can be parametrized and how they maintain specific geometric properties.
For the helix defined by:\[ \alpha(s) = \left( a \cos \frac{s}{c}, a \sin \frac{s}{c}, b \frac{s}{c} \right), \quad c^{2} = a^{2} + b^{2},\]it has the property that the tangent makes a constant angle with the z-axis. This can be shown by calculating the angle \( \theta \) using the dot product and seeing that:\[ \cos \theta = \frac{b}{\sqrt{a^{2} + b^{2}}}.\]The elegant nature of the helix here perfectly illustrates the interplay of geometry and calculus.

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Most popular questions from this chapter

Prove that the distance \(\rho\) between the nonparallel lines $$ \begin{array}{ll} x-x_{0}=u_{1} t, & y-y_{0}=u_{2} t, \quad z-z_{0}=u_{3} t \\ x-x_{1}=v_{1} t, & y-y_{1}=v_{2} t, \quad z-z_{1}=v_{3} t \end{array} $$ is given by $$ \rho=\frac{|(u \wedge v) \cdot r|}{|u \wedge v|} $$ where \(u=\left(u_{1}, u_{2}, u_{3}\right), v=\left(v_{1}, v_{2}, v_{3}\right), r=\left(x_{0}-x_{1}, y_{0}-y_{1}, z_{0}-z_{1}\right)\).

Let \(\alpha(t)=\left(a e^{b t} \cos t, a e^{b t} \sin t\right), t \in R, a\) and \(b\) constants, \(a>0\), \(b<0\), be a parametrized curve. a. Show that as \(t \rightarrow+\infty, \alpha(t)\) approaches the origin 0, spiraling around it (because of this, the trace of \(\alpha\) is called the logarithmic spiral; see Fig. 1-11). b. Show that \(\alpha^{\prime}(t) \rightarrow(0,0)\) as \(t \rightarrow+\infty\) and that $$ \lim _{t \rightarrow+\infty} \int_{t_{0}}^{t}\left|\alpha^{\prime}(t)\right| d t $$ is finite; that is, \(\alpha\) has finite are length in \(\left[t_{0}, \infty\right)\) Figure 1-11. Logarithmic spiral.

Find a parametrized curve \(\alpha(t)\) whose trace is the circle \(x^{2}+y^{2}=1\) such that \(\alpha(t)\) runs clockwise around the circle with \(\alpha(0)=(0,1)\).

Let \(\alpha: I \rightarrow R^{3}\) be a regular parametrized curve (not necessarily by \(\operatorname{arc}\) length and let \(\beta: J \rightarrow R^{3}\) be a reparametrization of \(\alpha(I)\) by the arc length \(s=s(t)\), measured from \(t_{0} \in I\) (see Remark 2). Let \(t=t(s)\) be the inverse function of \(s\) and set \(d \alpha / d t=\alpha^{\prime}, d^{2} \alpha / d t^{2}=\alpha^{\prime \prime}\), etc. Prove that a. \(d t / d s=1 /\left|\alpha^{\prime}\right|, d^{2} t / d s^{2}=-\left(\alpha^{\prime} \cdot \alpha^{\prime \prime} /\left|\alpha^{\prime}\right|^{4}\right)\). b. The curvature of \(\alpha\) at \(t \in I\) is $$ k(t)=\frac{\left|\alpha^{\prime} \wedge \alpha^{\prime \prime}\right|}{\left|\alpha^{\prime}\right|^{3}} . $$ c. The torsion of \(\alpha\) at \(t \in I\) is $$ \tau(t)=-\frac{\left(\alpha^{\prime} \wedge \alpha^{\prime \prime}\right) \cdot \alpha^{\prime \prime \prime}}{\left|\alpha^{\prime} \wedge \alpha^{\prime \prime}\right|^{2}} $$ d. If \(\alpha: I \rightarrow R^{2}\) is a plane curve \(\alpha(t)=(x(t), y(t))\), the signed curvature (see Remark 1) of \(\alpha\) at \(t\) is $$ k(t)=\frac{x^{\prime} y^{\prime \prime}-x^{\prime \prime} y^{\prime}}{\left(\left(x^{\prime}\right)^{2}+\left(y^{\prime}\right)^{2}\right)^{3 / 2}} . $$

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 \(k / \tau=a / b\).
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