Greenwood press

10
CARTESIAN COORDINATES
An image map uses coordinates 
to define a rectangular “hot spot” 
by noting the opposite corner 
coordinates in an image map.

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with dimensions of 28 pixels by 35 pixels that will link to a new page if the cur-
sor is clicked at a location on the image map between 12 and 40 pixels and
between 35 and 70 pixels. If the cursor is not in this region, then it will not link
to that page. Notice that the coordinate system on the image map is defined dif-
ferently from the standard rectangular system. Since only positive values are
used, this coordinate system uses the opposite of the negative y-coordinates that
are represented in the fourth quadrant of a Cartesian coordinate system.
Desirable locations for fire stations are places where trucks would have equal
access to the entire town. Ideally, they should be situated so that the longest drive
to the edge of town is the same in all directions. A coordinate grid could be super-
imposed on a city map, assigning coordinates to each of the intersections. The 
distance formula, 
d =
(x
1
− x
2
)
2
+ (y
1
− y
2
)
2
, could then be used to deter-
mine relative distances, 
d, of each street based on the coordinates of its end-
points,
(x
1
, y
1
) and (x
2
, y
2
), so that the best possible intersection for the fire sta-
tion could be selected.
On a world map, cities and landmarks are assigned a position according to
how far away they are from the equator (0°latitude) and from the prime merid-
ian in Greenwich, England (0° longitude). For example, Chicago is near 41° N 
87° W, which means that it is 
41
90
in the northern hemisphere and 
87
180
in the west-
ern hemisphere. 
A flight from Chicago to Los Angeles would angle the plane 7° south of west
and expect to travel 31° westward on its journey, because Los Angeles is near the
position 34° N 118° W. The distance 
d traveled between any two cities on the
globe can be determined by the equation 
d = 3963 arccos[sin(latitude
1
) sin(latitude
2
) +
cos(latitude
1
) cos(latitude
2
) cos(longitude
2
− longitude
1
)],
where the position in a spherical coordinate system of two cities are 
(lati-
tude
1
, longitude
1
) and (latitude
2
, longitude
2
) in radians. There are 2π radi-
ans in 360°, so each coordinate should be multiplied by 
2π
360
to convert to radi-
ans. In this case, the Chicago coordinate would convert from (41,87) to approx-
imately (0.7156,1.5184), and the Los Angeles coordinate would convert from