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<title>
10.5 - Bagels
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<h2>
10.5 - Bagels, bagels, bagels
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<p>There are many ways to create bagels. Some people prefer boiling them, some prefer baking them, and some prefer defrosting then toasting them. Today we shall be creating bagels by the method of parameterizing them.
<p>In single-variable calculus, it was shown that one can compute the volume of a torus, or doughnut shape, by thinking of it as a circle rotated about a horizontal or vertical line.
<ol>
<li> Parameterize a circle of radius $r$ centered at the origin in the $xy$-plane starting at $(0, r)$. That is, find $x$, $y$, and $z$ in terms of $r$ and $\alpha$.  Let $\alpha$ be the angle between the position vector and the $y$-axis.<br>
<img src="0g/10.5.bagel1.jpg"><br>


<li>Parametrize a circle of radius 1 in the $yz$-plane with center (2, 0) starting at (0, 3). Let $\beta$ be the angle between the position vector and the positive $y$-axis.<br>
<img src="0g/10.5.bagel2.jpg"><br>
<li style="page-break-before:always">
Now we want to characterize a typical point on our bagel, 
so we can write a vector function $\myv s (\alpha, \beta)$ whose range is the entire breakfast treat. To find any specific point we

(a) Move $\alpha$ radians along the horizontal curve, then (b) Rotate $\beta$ radians along the vertical curve. <br>
<img src="0g/10.5.bagel3.jpg"><br>
Find $\myv s(\alpha, \beta)$. 
 That is, in terms of GeoGebra, where...
<tt>
Surface( myx($\alpha$,$\beta$), myy($\alpha$,$\beta$),myz($\alpha$,$\beta$),
        $\alpha$, ___,___, $\beta$, ___,___)
</tt>
Find the three coordinate functions, and fill in the min and max limits of $\alpha$ and $\beta$. 

</ol>

<p><b>Extra Credit</b>: Finish up the work of paramerizing this surface, and then hand your <tt>Surface</tt> to GeoGebra to plot it.

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