Math 201 – Section 3
September 27, 2004
Coordinate Systems
Two dimensions:
Rectangular coordinates:
(x, y)
x and y can take on any real values, and each pair (x, y) corresponds to exactly one point.
Polar coordinates:
(r, q)
r ≥ 0; all pairs with r=0 correspond to the same point (the origin).
q can take on any value but q is defined only mod 2p —
that is, pairs (r, q1) and (r, q2) represent the same point if q1 and q2 differ by a multiple of 2p.
x = r cos q r =
y = r sin q q = atan2 ( y, x ).
( About the atan2 function: “atan2 ( y, x )” means the angle
whose sine and cosine are in the same ratio as y and x.
When x and y are positive, atan2 ( y, x ) = arctan ( y/x ),
but when x or y is negative, atan2 provides an angle in
the appropriate quadrant. Note that atan2 ( 0, 0 ) is
undefined. Excel and C++ both have ATAN2 functions. )
Three dimensions:
Rectangular coordinates:
(x, y, z)
x, y, z can take on any real values; each triple (x, y, z) corresponds to exactly one point.
Cylindrical coordinates:
(r, q, z)
r ≥ 0; all triples with r=0 correspond to the same point (the origin).
q can take on any value but q is defined only mod 2p.
z can take on any value.
x = r cos q r =
y = r sin q q = atan2 ( y, x).
z = z z = z
Spherical coordinates:
(r, q, j)
r ≥ 0; all triples with r=0 correspond to the same point (the origin).
q can take on any value but q is defined only mod 2p.
j must be in the range 0 ≤ j ≤ p.
Points with j = 0 are on the positive z axis, and for each value of r, all triples of the form (r, q, 0) represent the same point (regardless of q).
Points with j = p are on the negative z axis, and for each value of r, all triples of the form (r, q, p) represent the same point (regardless of q).
x = r cos q sin j r =
y = r sin q sin j q = atan2 ( y, x); use any value if x=y=0.
z = r cos j z = atan2 ( , z ).
Latitude, longitude, altitude:
(lat, lon, altitude)
Same as spherical system but with coordinates renamed and slightly adjusted.
lon = q, usually expressed in degrees in the range -180deg < q ≤ +180deg. Positive values represent east longitude, negative values represent west longitude.
lat = 90degrees – j. So, lat is in range -90deg ≤ lat ≤ +90deg. Positive means north.
altitude = r – (radius of earth); so altitude is measured from the earth’s surface (or sea level).
Two-point angular coordinates:
( q, j1, j2 )
q is defined mod 2p.
j1 and j2 are both in the range 0 < j < p.
Points on axis between origins aren’t represented. Triples with either j1 or j2 equal to zero or p don’t represent points.
Homework 5
(Due any time before fall break)
Section 13.7, problems 4, 20, 29, 68*. Practice on 3 and 19 if you like. For 68, convert to rectangular coordinates and find the angle between two vectors. It may be easier to pretend for a while that r = 1.
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