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\title{Worksheet 6}
\author{Math 202}
\date{Monday, May 6, 2024}

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This week and next week we're investigating \emph{azimuthal\footnote{Pronounced AZ-a-muthl.} map projections}.  When these are centered at the north or south pole, they map lines of latitude to concentric circles, and lines of longitude to lines radiating out from the center. \bigskip

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In the \emph{gnomonic\footnote{Pronounced no-MON-ic.  I'm not making these words up.} projection}, we imagine a light shining from the center of the earth and casting a shadow onto a plane above.  We've seen that the parallel at a latitude $\beta$ has a circumference of $2 \pi R \cos \beta$, where $R$ is the radius of the earth.  Its shadow on the plane is a bigger circle; what is the radius?  So an east-west mile get stretched by what factor?  Plug in some values of $\beta$ to get a feel for the numbers. \medskip

If you start at the north pole and travel south along a meridian, the corresponding point on the map starts at the center moves out along a line.  Work out a formula for the distance from the center as a function of the latitude $\beta$.  Do you agree that the stretching of north-south miles is given by the derivative of this function?  Find the derivative and plug in some values of $\beta$. \medskip

Does the gnomonic projection preserve areas?  Does it preserve angles? \bigskip

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In the \emph{stereographic projection}, we imagine a light shining up from the south pole.  Now how much does an east-west mile get stretched?  A north-south mile?  Don't forget to plug in some numbers.  Does the projection preserve areas?  Does it preserve angles?

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