Transcript Cylindrical and Polar Coordinate Systems

Cylindrical and Polar
Coordinate Systems
Chapter 13.7
Motivating question…

What is the shortest route you could fly
from Edmonton to Amsterdam?
• Edmonton (53.5N, 113.5W)
• Amsterdam (52.7 N, 5.7 E)
…link to Maple worksheet on visualizing
plots in cylindrical and spherical polar
coordinate systems
Spherical Polar Coordinates
x   sin  cos
y   sin  sin 
z   cos 
Going to Amsterdam…
This is the
“great circle”
path
This is the “fly
straight east” path
Fly “Straight East”
r  R0 cos
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.
d  2 Ro cos 
longitude
360
119.2
d  2 (6378.1 km) cos(53 )
360
Edmonton (53.5N, 113.5W)
Amsterdam (52.7 N, 5.7 E) d  7985.6 km
0
Is this the
shortest 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
d  R0
All we need is a
mathematical technique to
find the angle between two
vectors … hmm …
Spherical Polars to Cartesians
Edmonton (53.5N, 113.5W)
Amsterdam (52.7 N, 5.7 E)
x   sin  cos 
y   sin  sin 
z   cos 
<x,y,z>Edmonton
<x,y,z>Amsterdam