(See Solution) Consider the parametric curve given by r(t)=(a cos f(t), a sin f(t), b f(t)-b) where a, b>0 and f is some differentiable function: [0,1]

Question: (4 points) Consider the parametric curve given by

\[\mathbf{r}(t)=(a \cos f(t), a \sin f(t), b f(t)-b)\]

where $a, b>0$ and $f$ is some differentiable function: $[0,1] \rightarrow[0,1]$ with $f(0)=1, f(1)=0$ and $f^{\prime}(t)<0$

Find $s(t)=$ the arc length of the curve segment in the interval $[0, t]$, and re-parametrize the curve in terms of $s:=\operatorname{arc}$ length from the point corresponding to $t=0$.
Compute the curvature and determine both parametric and nonparametric equations
for the osculating plane at any given point, all in terms of $s$.
Show that at all points of the curve the tangent lines form the same angle with the $z$ -axis. Determine that angle. Do the same with the normal lines, i.e. the lines that pass through the points and whose direction is the direction of the normal vector.

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Solution: The downloadable solution consists of 3 pages
Deliverable: Word Document

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The Ellipse and its General Equation - MathCracker.com

(See Solution) Consider the parametric curve given by r(t)=(a cos f(t), a sin f(t), b f(t)-b) where a, b>0 and f is some differentiable function: [0,1]

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