Cone from direction vector

问题

I have a normalized direction vector (from a 3d position to a light position) and I would like this vector to be rotated by some angle so I can create a "cone".

Id like to simulate cone tracing by using the direction vector as the center of the cone and create an X number of samples to create more rays to sample from.

What I would like to know is basically the math behind:

https://docs.unrealengine.com/latest/INT/BlueprintAPI/Math/Random/RandomUnitVectorinCone/index.html

Which seems to do exactly what Im looking for.

回答1:

1) Make arbitrary vector P, perpendicular to your direction vector D. You can choose component with max magnitude, exchange it with middle-magnitude component, negate it, and make min magnitude component zero.

For example, if z- component is maximal and y-component is minimal, you may make such P:

 D = (dx, dy, dz)
 p = (-dz, 0, dx)
 P = Normalize(p)   //unit vector

2) Make vector Q perpendicular both D and P through vector product:

  Q = D x P    //unit vector

3) Generate random point in the PQ plane disk

  RMax = Tan(Phi)  //where Phi is cone angle

  Theta = Random(0..2*Pi)

  r = RMax * Sqrt(Random(0..1))   

  V = r * (P * Cos(Theta) + Q * Sin(Theta))

4) Normalize vector V

Note that distribution of vectors is slightly non-uniform on the sphere segment.(it is uniform on the plane disk). There are methods to generate uniform distribution on the sphere but some work needed to apply them to segment (my first attempt before edit was wrong).

Edit: Modification to make sphere-uniform distribution (not checked thoroughly)

 RMax = Tan(Phi)  //where Phi is cone angle
 Theta = Random(0..2*Pi)
 u  = Random(Cos(Phi)..1)
 r = RMax * Sqrt(1 - u^2)   
 V = r * (P * Cos(Theta) + Q * Sin(Theta))

来源：https://stackoverflow.com/questions/39404576/cone-from-direction-vector