The distance between point P1(ρ1, θ1, ϕ1) and P2(ρ2, θ2, ϕ2) given in spherical coordinates

oliviayychengwh

oliviayychengwh

Answered

2021-12-21

The distance between point P1(ρ1, θ1, ϕ1) and P2(ρ2, θ2, ϕ2) given in spherical coordinates extract the formula.

Answer & Explanation

Cheryl King

Cheryl King

Expert

2021-12-22Added 36 answers

Step 1
Given point P1(ρ1, θ1, ϕ1), P2(ρ2, θ2, ϕ2)
Let corresponding coutesian cordinates are,
P1(x1, y1, z1) & P2(x2, y2, z2)
Relation between coutesian cordinates and sherical coordinates are
x1=ρ1, sinθ1, cosϕ1
y1=ρ1, sinϕ1, sinθ1
z1=ρ1, cosθ1
x2=ρ2, sinθ2, cosϕ2
y2=ρ2sinθ2 sinϕ2
z2=ρ2 cosθ2
Distance between paints P1 & P2
d=(x1x2)2+(y1y2)2+(z1z2)2
=x12+x22x1x2+y12+y222y1y2+z12+z222z1z2
=(x12+y12+z12)+(x22+y22+z22)2(x1x2+y1y2+z1z

ambarakaq8

ambarakaq8

Expert

2021-12-23Added 31 answers

Assuming that θ is the latitude and ϕ is the longitude we have that the cartesian coordinates of the first point are:
(ρ1cosθ1cosϕ1, ρ1cosθ1sinϕ1, ρ1sinθ1)
so the distance between the two points is given by:
ρ12+ρ222ρ1ρ2(cosθ1cosθ2cos(ϕ1ϕ2)+sinθ1sinθ2}
nick1337

nick1337

Expert

2021-12-28Added 573 answers

Step 1
The expression of the distance between two vectors in spherical coordinates provided in the other response is usually expressed in a more compact form that is not only easier to remember but is also ideal for capitalizing on certain symmetries when solving problems.
||rr||=(xx)2+(yy)2+(zz)2
=r2+r22rr[sin(θ)sin(θ)cos(ϕ)cos(ϕ)+sin(θ)sin(θ)sin(ϕ)sin(ϕ)+cos(θ)cos(θ)]
=r2+r22rr[sin(θ)sin(θ)(cos(ϕ)cos(ϕ)+sin(ϕ)sin(ϕ))+cos(θ)cos(θ)]
=r2+r22rr[sin(θ)sin(θ)cos(ϕϕ)+cos(θ)cos(θ)]
This form makes it fairly transparent how azimuthal symmetry allows you to automatically eliminate some of the angular dependencies in certain integration problems
Another advantage of this form is that you now have at least two variables, namely ϕ and ϕ,  that appear in the equation only once, which can make finding series expansions w.r.t. these variables a little less of a pain than the others.

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