Cylindrical and Polar Cylindrical and Polar Coordinate SystemsCoordinate Systems

Chapter 13.7Chapter 13.7

Motivating question…Motivating question…

What is the shortest route you could fly What is the shortest route you could fly from Edmonton to Amsterdam?from Edmonton to Amsterdam?• Edmonton (53.5N, 113.5W)Edmonton (53.5N, 113.5W)• Amsterdam (52.7 N, 5.7 E)Amsterdam (52.7 N, 5.7 E)

…link to Maple worksheet on visualizing plots in cylindrical and spherical polar coordinate systems

Spherical Polar CoordinatesSpherical Polar Coordinates

sin cos sin sin cosx y z

Going to Amsterdam…Going to Amsterdam…

This is the “fly straight east” path

This is the “great circle” path

0 cosr R

Fly “Straight East”

Trip distance is the fraction of a circle of this radius one flies when going from Edmonton to Amsterdam. This is easy because the two cities are almost the same latitude.

Edmonton (53.5N, 113.5W)Edmonton (53.5N, 113.5W)Amsterdam (52.7 N, 5.7 E)Amsterdam (52.7 N, 5.7 E)

0

2 cos360

119.22 (6378.1 )cos(53 )

3607985.6

olongitude

d R

d km

d km

Is this the shortest path?

The Geodesic Path…The Geodesic Path…

The shortest path is part of a great circle (one centered on the center of the earth). This is also known as a geodesic path

All we need is a mathematical technique to find the angle between two vectors … hmm …

0d R

Spherical Polars to CartesiansSpherical Polars to Cartesians

Edmonton (53.5N, 113.5W)Edmonton (53.5N, 113.5W)Amsterdam (52.7 N, 5.7 E)Amsterdam (52.7 N, 5.7 E)

sin cos

sin sin

cos

x

y

z

<x,y,z>Edmonton <x,y,z>Amsterdam