2
3. Relationship between Cartesian and Polar
coordinate systems
O
x
y
II
III
IV
M (
x
,
y
) =
r
M (
r
,
ϕ
)
ϕ
ϕ
ϕ
sin
cos
r
y
r
x
=
=
2
2
y
x
r
+
=
<
+
>
=
0
arctan
0
arctan
x
for
x
y
x
for
x
y
π
ϕ
4. Parallel Translation of Cartesian coordinate system
O
x
y
M (
x
,
y
)
O’
(
g, h
)
X
Y
1.
The coordinates for the point
O’
in the original
coordinate is (
g, h
).
2.
So, we have the relations between the original
coordinates (
Oxy
) and the new coordinates (O’XY)are
x
=
X
+
g
y
=
Y
+
h
Three-dimensional Coordinate Systems
1. Cartesian coordinate system
O
x
z
M (
x
,
y, z
)
x-a
xis and y-axis = two perpendicular horizontal axes
-infinity <
x
< + infinity;
-infinity <
y
< + infinity
z-a
xis = Vertical axis
- infinity <
z
< + infinity
y
M’ (
x
,
y,
0)
M
z
(0, 0
, z
)
M
x
(
x
, 0
,
0)
M
y
(0,
y,
0)
2. Cylindrical Polar coordinate system
O
x
z
M (
r
,
ϕ
, z
)
z-a
xis = Vertical axis
- infinity <
z
< + infinity
y
M’ (
r
,
ϕ
,
0)
r
ϕ
ϕ
=
angle of longitude
0
≤
ϕ
< 360 (2
π
)
r
= radial length
0
≤
r
< + infinity
3. Relationship between Cartesian and Cylindrical
polar coordinate systems
O
x
z
M (
x
,
y, z
)
y
M’ (
x
,
y,
0)
M
z
(0, 0
, z
)
M
x
(
x
, 0
,
0)
M
y
(0,
y,
0)
= (
r
,
ϕ
, z
)
r
ϕ
= (
r
,
ϕ
,
0)
z
z
r
y
r
x
=
=
=
ϕ
ϕ
sin
cos
x
y
y
x
r
arctan
2
2
=
+
=
ϕ
4. Spherical polar coordinate system
O
x
z
M
ϕ
-a
xis =
angle of longitude; 0
≤
ϕ
< 360
0
(2
π
)
R-a
xis = radial length; 0
≤
R
< + infinity
y
M’ (
x
,
y,
0)
M
z
(0, 0
, z
)
ϕ
γ
γ
-a
xis =
angle of latitude; 0
≤
γ
< 180
0
(
π
)
(
R
,
ϕ
,
γ
)
R
(0,
ϕ
, r
)