Direction in 3d space described with 2 values: Horizontal and Vertical (x and y) ranging the full 360 degrees (from -180 to +180), thus describing every possible view direction.It seems fairly straight forward to me, that the angle between (0,0) and (5,0) would be 5 degrees.But what is the angle between (0,0) and (5,5). If it was 2 points in 2d I would use Pythagoras theorem; having the difference be ( 5 2 + 5 2 ). But I don't think that would apply in 3d? How do I go about calculating the angle between these 2 directions?

Question

Direction in 3d space described with 2 values: Horizontal and Vertical (x and y) ranging the full 360 degrees (from -180 to +180), thus describing every possible view direction.It seems fairly straight forward to me, that the angle between (0,0) and (5,0) would be 5 degrees.But what is the angle between (0,0) and (5,5). If it was 2 points in 2d I would use Pythagoras theorem; having the difference be       (        5          2        +      5    2    ). But I don&#039;t think that would apply in 3d? How do I go about calculating the angle between these 2 directions?

Regan Holloway · Accepted Answer

Step 1If I understood well, by a &quot;direction&quot; (  ϕ  ,  θ  ) (I assume the angles between 0 and   2  π) you mean the unit vector   u  (  ϕ  ,  θ  )  =  (  cos  ⁡  θ  cos  ⁡  ϕ  ,  cos  ⁡  θ  sin  ⁡  ϕ  ,  sin  ⁡  θ  ). The angle   α between   u  (  θ  ,  ϕ  ) and   u  (      θ    ′    ,      ϕ    ′    ) may be calculated by taking the scalar product   u  (  θ  ,  ϕ  )  ⋅  u  (      θ    ′    ,      ϕ    ′    ) as follows:  cos  ⁡  α  =  cos  ⁡  θ  cos  ⁡      θ    ′    (  cos  ⁡  ϕ  cos  ⁡      ϕ    ′    +  sin  ⁡  ϕ  sin  ⁡      ϕ    ′    )  +  sin  ⁡  θ  sin  ⁡      θ    ′    =  cos  ⁡  θ  cos  ⁡      θ    ′    cos  ⁡  (  ϕ  −      ϕ    ′    )  +  sin  ⁡  θ  sin  ⁡      θ    ′    ,Step 2Hence   α  =      cos          −      1        ⁡  (  cos  ⁡  θ  cos  ⁡      θ    ′    cos  ⁡  (  ϕ  −      ϕ    ′    )  +  sin  ⁡  θ  sin  ⁡      θ    ′    )

Angle difference between 2 directions in 3d

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