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4.7 Spherical polar coordinates 171
z
z
P
f
R
O
x
u
y
x
y
Q
Figure4.28
Spherical polar coordinates are (R,θ, φ).
Consider a typical point, P. We look ateach of the three coordinates inturn.
The value ofRis the distance of the point from the origin; that is,Ris the length of
OP. Note thatR 0.
Let Q be the projection of P onto the x--y plane. Then θ is the angle between the
positive x axis and OQ. Thus, θ has the same definition as for polar and cylindrical
polar coordinates. Note that θ can have any value from0 ◦ to360 ◦ .
Consider the line OP. Then φ is the angle between the positive z axis and OP. The
angle φ can have values between 0 ◦ and 180 ◦ . When P is above thex--y plane, then φ
lies between 0 ◦ and 90 ◦ ; when P lies below the x--y plane, then φ is between 90 ◦ and
180 ◦ . When φ = 0 ◦ , then P is on the positivezaxis; when φ = 90 ◦ , P lies in thex--y
plane; when φ = 180 ◦ , Plieson the negativezaxis.
WecandetermineequationswhichrelatetheCartesiancoordinates, (x,y,z),andthe
spherical polar coordinates, (R,θ,φ).
Note that some books describe spherical polar coordinates as (R,φ,θ), that is the
definitions of θ and φ areinterchanged. Be aware of thiswhen reading other texts.
Consider △OPQ.Note that ̸ OQP isaright angle and so
OQ=OPsinφ=Rsinφ
OQ liesinthex--y plane and so
x=OQ cosθ =Rsinφcosθ
y=OQ sinθ =Rsinφsinθ
We also notethat
z=OPcosφ=Rcosφ
Insummarywe have
and
x=Rsinφcosθ
y=Rsinφsinθ
z=Rcosφ
R0, 0φ π(180 ◦ ) 0θ <2π(360 ◦ )