Point-Line Projection

Result

If line L has the normalized direction vector l, and A is a point on L, the point P projected onto L is:

    PL = A + (AP dot l) * l

Derivation

     P
     |
     |
 A___|______ 
     PL  L

L is defined by the normalized direction vector l and A is a point on L.

By PL we denote the point P projected onto the line L.


     .
  v /|
   / |d
  /__|_____ l
   vl

We can also define the vector v = AP and the vector vl, which is the projection
of v onto l.

    vl = (v dot l) * l

where 'dot' signifies the dot product.

Now we can compute PL:

    PL = A + vl = A + (AP dot l) * l

References

• Wikipedia: Vector Projection