91Ó°ÊÓ

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.

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 \(\alpha(0)=(0,1)\).

A parametrized curve \(\alpha(t)\) has the property that its second derivative \(\alpha^{\prime \prime}(t)\) is identically zero. What can be said about \(\alpha\) ?

Compute the curvature of the ellipse $$ x=a \cos t, \quad y=b \sin t, \quad t \in[0,2 \pi], a \neq b, $$ and show that it has exactly four vertices, namely, the points \((a, 0)\), \((-a, 0),(0, b),(0,-b)\).

Determine the angle of intersection of the two planes \(5 x+3 y+\) \(2 z-4=0\) and \(3 x+4 y-7 z=0\).

Given two planes \(a_{i} x+b_{i} y+c_{i} z+d_{i}=0, i=1,2\), prove that a necessary and sufficient condition for them to be parallel is $$ \frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}, $$ where the convention is made that if a denominator is zero, the corresponding numerator is also zero (we say that two planes are parallel if they either coincide or do not intersect).

A regular parametrized curve \(\alpha\) has the property that all its tangent lines pass through a fixed point. a. Prove that the trace of \(\alpha\) is a (segment of a) straight line. b. Does the conclusion in part a still hold if \(\alpha\) is not regular?

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 \(|k| \geq 1 / r .\)

Prove that a necessary and sufficient condition for the plane $$ a x+b y+c z+d=0 $$ and the line \(x-x_{0}=u_{1} t, y-y_{0}=u_{2} t, z-z_{0}=u_{3} t\) to be parallel is $$ a u_{1}+b u_{2}+c u_{3}=0 $$

One often gives a plane curve in polar coordinates by \(\rho=\rho(\theta)\), \(a \leq \theta \leq b .\) a. Show that the arc length is $$ \int_{a}^{b} \sqrt{\rho^{2}+\left(\rho^{\prime}\right)^{2}} d \theta \text {, } $$ where the prime denotes the derivative relative to \(\theta\). b. Show that the curvature is $$ k(\theta)=\frac{2\left(\rho^{\prime}\right)^{2}-\rho \rho^{\prime \prime}+\rho^{2}}{\left\\{\left(\rho^{\prime}\right)^{2}+\rho^{2}\right\\}^{3 / 2}} $$