如何生成与给定平面相距一定距离的随机3D点？

Question

I have a plane from 3 points (triangle). How can I generate a random point to this plane with a specified distance? The projection of the generated point should also be within the triangle.

Answer 1

Let's we have non-collinear points A, B, C.
1. Generate two random values t and u in range 0..1. We can use these values to get uniform distribution in parallelogram formed by vectors AB and AC. To get uniform distribution in triangle, we reflect points which hit the second triangle, about parallelogram diagonal (if-branch of pseudocode)

t = random ( 0, 1 ) 
u = random ( 0, 1 )
if (t + u > 1 ) then
   t = 1 - t
   u = 1 - u

2. Make random point P inside triangle ABC:

P = A + t * AB + u * AC

3. Move P away off ABC plane

   n = normalized(AB x AC) (vector product)

   P' = P + d * n

where d - needed distance