Machinery's Handbook, 31st Edition
48
RELATIONSHIP BETWEEN COORDINATE SYSTEMS Relationship Between Spherical and Rectangular Coordinates
Spherical to Rectangular
Rectangular to Spherical
2
2
2
x = + + y
r
z
z x 2 y 2 + -----------
=
tan –1
φ
for x 2 + y 2 ≠ 0
φ λ = cos cos φ λ = cos sin
x r y r z r
tan –1 y x
-
=
for x > 0, y > 0
λ
φ = sin
λ π tan –1 y x - = +
for x < 0
λ for x > 0, y < 0 Example 3: Find the spherical coordinates of the point P (3, - 4, - 12). r 3 2 –4 ( ) 2 –12 ( ) 2 + + 13 = = 2 π tan –1 y x - = +
–12 3 2 –4 + --------------- tan –1 12 5 ( ) 2 –--- =
=
=
67.38 ° –
tan –1
φ
λ = The spherical coordinates of P are therefore r = 13, f = - 67.38 ° , and l = 306.87 ° . Cylindrical Coordinates: For problems in which points lie on the surface of a cylinder it is convenient to use cylindrical coordinates. The cylindrical coordinates r , q , z of P coincide with the polar coordinates of the point P ′ in the x , y -plane and the rectangular z -coordinate of P . Formulas for q hold only if x 2 + y 2 ≠ 0; q is undetermined if x = y = 0. Cylindrical to Rectangular Rectangular to Cylindrical z 360 ° tan –1 4 3 –-- + 360 ° 53.13 ° – 306.87 ° = =
1 x 2 y 2 = ------+----- x x 2 y 2 + = ----------- y x 2 y 2 + = -----------
r
θ = cos θ = sin
x r y r z z =
θ cos
P
θ sin
O
z z =
r
θ
y
x
P
Example 4: Given the cylindrical coordinates of a point P , r = 3, q = - 30 ° , z = 51, find the rectangular coordinates. Using the above formulas x = 3cos( - 30 ° ) = 3cos(30 ° ) = 2.598; y = 3sin( - 30 ° ) = - 3sin(30 ° ) = - 1.5; and z = 51. Therefore, the rectangular coordinates of point P are x = 2.598, y = - 1.5, and z = 51.
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