Math: Distance between two lines

เยี่ยมมาก คิดอยู่ตั้งนานว่าจะ proof ทฤษฎีพื้นฐานนี่ได้ยังไง ในที่สุดก็เจอ

Objective: Given two straight lines in 3D:

• line 1:    r = a + t*b

• line 2:    r = p  +s*q
  
  Show that the distance between those two lines are |(a - p).u|
  where u is a unit vector given by b x q / |b x q|.

The distance is the length of line segment that perpendicular to both line 1 and 2. Thus if we can find the length of that line segment we are done. So there are two subproblems now:

1. Finding direction of the line segment
2. Finding the length of the line segment that bridge line 1 and 2
   

To find direction of the line segment cross product can be used because output from cross product is a vector perpendicular to both input vectors. In other word, b x q gives out the direction we want.

To find the length, scalar projection can be applied. If we know a vector that bridges the two lines (in this case it is a - p), we can find the distance between those two lines by calculating scalar projection on the perpendicular vector. Remember that the distance is actually |a - p| cos theta where theta is an angle between (a - p) and the perpendicular vector.

The scalar projection is |a - p|cos theta = | (a - p) . (b x q) | / | b x q| = | (a - p) . u|.

Q.E.D.